thermodynamic law

Physical chemistry

CHAPTER 1

PROPERTIES OF GASES

THE PERFECT GASES

A gas has no deffinite volume or shape; a gas will fill whatever volume is available to it. This property of diffusion implies that the molecular units of a gas are in rapid, random motion, and that they are far enough away from each other that this motion is unimpeded by interactions between the molecules.

The other outstanding characteristic of gases, their low densities, is another indication that the average distance between molecules is very large. The most remarkable property of gases, however, is that to a very good approximation, they all behave the same way in response to changes in temperature and pressure, expanding or contracting by predictable amounts. This is very different from the behavior of liquids or solids, in which the properties of each particular substance must be determined individually.

A. The states of gases

1. Pressure exerted by a gas

Pressure is defined as force per unit area. To visualize this, imagine some gas trapped in a cylinder having one end enclosed by a freely moving piston. In order to keep the gas in the container, a certain amount of weight (force, f) must be placed on the piston so as to exactly balance the force exerted by the gas on the bottom of the piston, and tending to push it up. The pressure of the gas is simply the quotient f=A, where A is the cross-section area of the piston.

Figure 1: Measurement of gas pressure: the barometer and manometer

A modifcation of the barometer, the U-tube manometer, provides a simple device for measuring the pressure of any gas in a container. The U-tube is partially filled with mercury, one end is connected to container, while the other end is left open to the atmosphere. The pressure inside the container is found from the difference in height between the mercury in the two sides of the U-tube. Pressure units :The unit of pressure in the SI system is the pascal (Pa), defined as a force of one newton per square metre (Nm¡2 or kg m¡1s¡2). In chemistry, it is more common to express pressures in units of atmospheres or torr :

1 atm = 1:01325E5 Pa = 760 torr

In meteorology, the pressure unit most commonly used is the bar ; 1 bar = 105 Nm¡2 = 0:987 atm: In engineering work the pound per square inch is often used; standard atmospheric pressure is 14.7 psi.

2. The volume occupied by a gas

The volume of a gas is simply the space in which the molecules of the gas are free to move. If we have a mixture of gases, such as air, the various gases will occupy the same volume at the same time, since they can all move about freely. The volume of a gas can be measured by trapping it above mercury in a calibrated tube known as a gas burette.

The SI unit of volume is the cubic metre, but in chemistry we more commonly use the litre and the millilitre (ml). The cubic

centimetre (cc) is also frequently used; it is very close to 1 ml.

3. The temperature of a gas

If two bodies are at different temperatures, heat will flow from the warmer to the cooler one until their temperatures are the same. This is the principle on which thermometry is based; the temperature of an object is measured indirectly by placing a calibrated device known as a thermometer in contact with it. When thermal equilibrium is obtained, the temperature of the thermometer is the same as the temperature of the object.

Temperature scales . A thermometer makes use of some temperature-dependent quantity, such as the density of a liquid, to allow the temperature to be found indirectly through some easily measured quantity such as the length of a mercury column. The resulting scale of temperature is entirely arbitrary; it is defined by locating its zero point, and the size of the degree unit. The Celsius temperature scale locates the zero point at the freezing temperature of water; the Celsius degree is deffined as 1/100 of the difference between the freezing and boiling temperatures of water at 1 atm pressure.

The Fahrenheit scale is a finer one than the Celsius scale; there are 180 Fahrenheit degrees in the same temperature interval that contains 100 Celsius degrees, so 1C= 9/5 F Since the zero points are also different by 32F, conversion between temperatures expressed on the two scales requires the addition or subtraction of this offset, as well as multiplication by the ratio of the degree size.

Absolute temperature : This temperature, known as absolute zero, corresponds to the total

absence of thermal energy. The temperature scale on which the zero point is ¡273:15 ±C was suggested by Lord Kelvin, and is usually known as the Kelvin scale. Since the size of the Kelvin and Celsius degrees are the same, conversion between the two scales is a simple matter of adding or subtracting 273.15; thus room temperature, 20 ±C, is about 293±K. Because the Kelvin scale is based on an absolute, rather than on an arbitrary zero of temperature, it plays a special significance in scientific calculations; most fundamental physical relations involving temperature are expressed mathematically in terms of absolute temperature. In engineering work, an absolute scale based on the Fahrenheit degree is commonly used; this is known as the Rankine scale.

B. Empirical laws of gas behavior

1. Pressure-volume: Boyle's Law

Robert Boyle3 showed that the volume of air trapped by a liquid in the closed short limb of a J-shaped tube decreased in exact proportion to the pressure produced by the liquid in the long part of the tube. The trapped air acted much like a spring, exerting a force opposing its compression. Boyle called this effect "the spring of the air", and published his results in a pamphlet of that title.

Boyle's law can be expressed as

PV = constant (1)

and is true only if the number of molecules n and the temperature are held constant. This is an equation of inverse proportionality; any change in the pressure is exactly compensated by an opposing change in the volume. As the pressure decreases toward zero, the volume will increase without limit. Conversely, as the pressure is increased, the volume decreases, but can never reach zero. A plot of the pressure of an ideal gas as a function of its volume yields a plot

Figure 2: Boyle's law: pressure-volume isotherms of an ideal gas

whose form is that of a hyperbola. There will be a separate P-V plot for each temperature; a single P-V plot is therefore called an isotherm. A related type of plot with which you should be familiar shows the product PV as a function of P. You should understand why this yields a straight line, and how the position of this line varies with the temperature.

2. Volume and temperature: Charles' law

The discovery that all gases expand by the same amount as the temperature israised was made independently by the French scientists Jacques Charles (1746-1823) and Joseph Gay-Lussac (1778-1850). This relation is now usually stated more explicitly:

the volume of a gas confined against a constant pressure is

directly proportional to the absolute temperature.

3. Volume and number of molecules: Avogadro's law

Gay-Lussac noticed that when two gases react, they do so in volume ratios that can always be expressed as small whole numbers. Thus when hydrogen burns in oxygen, the volume of hydrogen consumed is always exactly twice the volume of oxygen. The Italian scientist Amadeo Avogadro drew the crucial conclusion:

these volume ratios must be related to the relative numbers of molecules that

react, and so equal volumes of gases, measured at the same temperature and

pressure, contain equal numbers of molecules.

V = n mol gas

Avogadro's law thus predicts a directly proportional relation between the number of moles of a gas and its volume.

Figure 3: The law of Charles and Gay-Lussac: temperature dependence of the volume

· The ideal gas equation of state

If the variables n, P, V , and T have known values, then a gas is said to be in a definite state, meaning that all other physical properties of the gas are also defined. The relation between these state variables is known as an equation of state.The ideal gas equation of state can be derived by combining the expressions of Boyle's, Charles', and Avogadro's laws (you should be able to do this!). This equation is usually written

PV = Nrt (2)

where the proportionality constant R is known as the gas constant. This is one of the few equations you must commit to memory in this course; you should also know the common value and units of R. An ideal gas is de¯ned as a hypothetical substance that obeys the ideal gas

equation of state. We will see later that all real gases behave more and more like an ideal gas as the pressure approaches zero. A pressure of only 1 atm is su±ciently close to zero to make this relation useful for most gases at this pressure.

Each point on this surface represents a possible combination of (P; V; T) for an ideal gas. The three sets of lines inscribed on the surface correspond to states in

which one of these three variables is held constant. The curved lines, being lines of constant temperature, are isotherms, and are plots of Boyle's law. The long-dashed lines are isobars and represent Charles' law plots. The short-dashed lines, known as isochors, show all values of (P; T) consistent with various ¯xed volumes.

Figure 4: P-V-T behavior of an ideal gas

Molar volume of a gas: standard temperature and pressure The set of conditions T = 273K and P = 1atm is known as standard temperature and pressure, usually denoted STP. Substituting these values into the ideal gas equation of state and solving for V yields a volume of 22.414 litres for 1 mole. The standard molar volume 22.4 L mol¡1 is a value worth memorizing, but remember also that it is valid only at STP. The molar volume at other temperatures and pressures can easily be found by simple proportion.

· Molecular weight and density of a gas

Since all gases have the same molar volume at the same temperature and pressure,we can easily determine the number of moles contained in a sample of any gas. If, in addition, we measure the mass of the gas, we can determine its molar mass. This is the basis of a simple and widely used procedure for determining the molecular weight of a substance. It is known as the Dumas method, after the French chemist Jean Dumas (1800-1840) who developed it. One simply measures the weight of a known volume of gas and converts this volume to its STP equivalent, using Boyle's and Charles' laws. The weight of the gas divided by its STP volume yields the density of the gas, and the density multiplied by 22.4 L mol¡1 gives the molecular weight.

· Mixtures of gases: Dalton's law of partial pressures

The ideal gas equation of state applies to mixtures just as to pure gases. It was in fact with a gas mixture, ordinary air, that Boyle, Gay-Lussac and Charles did their early experiments. The only new concept we need in order to deal with gas mixtures is the partial pressure. The pressure exerted by a gas depends on the force exerted by each molecular collision with the walls of the container, and on the number of such collisions in a unit of area per unit time. If a gas contains two kinds of molecules, each species will engage in such collisions, and thus make a contribution to the total pressure in exact proportion to its abundance in the mixture. The contribution that each species makes to the total pressure of the gas is known as the partial pressure of that species.

The above is essentially a statement of Dalton's law of partmal pressures. Algebraically, we can express this law by

Dalton himself stated this law in a simple and vivid way:

Every gas is a vacuum to every other gas

.

The partial pressure of any one gas is directly proportional to its abundance in the mixture, and is just the total pressure multiplied by the mole fraction of that gas in the mixture.

C. The kinetic molecular theory of gases

The properties such as temperature, pressure, and volume, together with other properties related to them (density, thermal conductivity, etc.) are known as macroscopic properties of matter; these are properties that can be observed in bulk matter, without reference to its underlying structure or molecular nature. By the late 19th century the atomic theory of matter was su±ciently well accepted that scientists began to relate these macroscopic properties to the behavior of the individual molecules, which are described by the microscopic properties of matter. The outcome of this effort was the kinetic molecular theory of gases. This theory applies strictly only to a hypothetical substance known as an ideal gas; we will see, however, that it describes the behavior of

real gases at ordinary temperatures and pressures quite accurately, and serves as an extremely useful model for treating gases under non-ideal conditions as well.

The basic tenets of the kinetic-molecular theory are as follows:

· A gas is composed of molecules that are separated by average distances that are much greater than the sizes of the molecules themselves. The volume occupied by the molecules of the gas is negligible compared to the volume of the gas itself.

· The molecules of an ideal gas exert no attractive forces on each other, or on the walls of the container.

· The molecules are in constant random motion, and as material bodies, they obey Newton's laws of motion. This means that the molecules move in straight lines until they collide with each other or with the walls of the container.

· Collisions are perfectly elastic; when two molecules collide, they change their directions and kinetic energies, but the total kinetic energy is conserved. Collisions are not \sticky".

· The average kinetic energy of the gas molecules is directly proportional to the absolute temperature. (Notice that the term \average" is very important here; the velocities and kinetic energies of individual molecules will span a wide range of values, and some will even have zero velocity at a given instant.) This implies that all molecular motion would cease if the temperature were reduced to absolute zero.

Kinetic interpretation of gas pressure. The kinetic molecular theory makes it easy to see why a gas should exert a pressure on the walls of a container. Any surface in contact with the gas is constantly bombarded by the molecules. At each collision, a molecule moving with momentum mv strikes the surface. Since the collisions are elastic, the molecule bounces back with the same velocity in the opposite direction. This change in velocity ¢v in a time interval ¢t is equivalent to an acceleration; from Newton's second law F = ma, a force F is thus exerted on the surface of area A, exerting a pressure F=A. Kinetic interpretation of temperature According to the kinetic molecular theory, the average kinetic energy of an ideal gas is directly proportional to thetemperature. Kinetic energy is the energy a body has by virtue of its motion:

As the temperature of a gas rises, the average velocity of the molecules will increase; a doubling of the temperature will increase this velocity by a factor of four. Collisions with the walls of the container will transfer more momentum, and thus more kinetic energy, to the walls. If the walls are cooler than the gas, they will get warmer, returning less kinetic energy to the gas, and causing it to cool until thermal equilibrium is reached. Because temperature depends on the average kinetic energy, the concept of temperature only applies to a statistically meaningful sample of molecules. We will have more to say about molecular velocities and kinetic energies farther on.

Kinetic explanation of Boyle's law Boyle's law is easily explained by the kinetic molecular theory. The pressure of a gas depends on the number of times per second that the molecules strike the surface of the container. If we compress the gas to a smaller volume, the same number of molecules are now acting against a smaller surface area, so the number striking per unit of area,

and thus the pressure, is now greater.

Kinetic explanation of Charles' law Kinetic molecular theory states that an increase in temperature raises the average kinetic energy of the molecules. If the molecules are moving more rapidly but the pressure remains the same, then the molecules must stay farther apart, so that the increase in the rate at which molecules collide with the surface of the container is compensated for by a corresponding increase in the area of this surface as the gas expands.

Kinetic explanation of Avogadro's law If we increase the number of gas molecules in a closed container, more of them will collide with the walls per unit time. If the pressure is to remain constant, the volume must increase in proportion, so that the molecules strike the walls less frequently, and over alarger surface area.

REAL GASES

D. REAL GASES: DEVIATIONS FROM IDEAL BEHAVIOR

A plot of PV as a function of the pressure of an ideal gas yields a horizontal straight line. This implies that any increase in the pressure of the gas is exactly counteracted by a decrease in the volume.

Effects of intermolecular repulsions We know, however, that this relation cannot always be valid; a gas cannot be squeezed out of existence. As a gas is compressed, the individual molecules begin to get in each other's way, giving rise to a very strong repulsive force acts to oppose any further volume decrease. We would therefore expect the PV -vs.-P line to curve upward at high pressures, and this is in fact what is observed for all gases.

Effects of intermolecular attractions At very close distances, all molecules repel each other as their electron clouds come into contact. At greater distances, however, brief statistical °uctuations in the distribution these electron clouds give rise to a universal attractive force between all molecules. The more electrons in the molecule (and thus the greater the molecular weight), the greater is this attractive force. As long as the energy of thermal motion dominates this attractive force, the substance remains in the gaseous state, but at su±ciently low temperatures the attractions dominate and the substance condenses to a liquid or solid.

The universal attractive force described above is known as the dispersion, or London force. There may also be additional (and usually stronger) attractive forces related to charge imbalance in the molecule or to hydrogen bonding. These various attractive forces are often referred to collectively as van der Waals forces.

The effect of intermolecular attractions on the PV-vs.-P plot would be to hold the molecules slightly closer together, so that the volume would decrease more rapidly than the pressure increases. The resulting curve would dip downward as the pressure increases, and this dip would be greater at lower temperatures and for heavier molecules. At higher pressures, however, the stronger repulsive forces would begin to dominate, and the curve will eventually bend upward as before.

The effects of intermolecular interactions are most evident at low temperatures and high pressures; that is, at high densities. As the pressure approaches zero, the behavior of any gas will conform more and more closely to the ideal gas equation of state, which should really be depicted as a limiting relation

Figure 7: PV -vs-P plots for real gases

· Equations of state for real gases

How might we modify the ideal gas equation of state to take into account the effects of intermolecular interactions? The first and most well known answer to this question was offered by the Dutch scientist J.D. van der Waals in 1873.

van der Waals recognized that the molecules themselves take up space that subtracts from the volume of the container, so that the \volume of the gas" V in the ideal gas equation should be replaced by the term (V - b) where b is the excluded volume, typically of the order of 20-100 cm3 mol ^-1 .

The intermolecular attractive forces act to slightly diminish the frequency and intensity of encounters between the molecules and the walls of the container; the effect is the same as if the pressure of the gas were slightly higher than it actually is. This imaginary increase is called the internal pressure, and we can write

P_effective = P_ideal - P_intermal

Thus we should replace the P in the ideal gas equation by

P_ideal = P_effective + P_internal

Since the attractions are between pairs of molecules, the total attractive force is proportional to the square of the number of molecules per volume of space, and thus for a ¯xed number of molecules such as one mole, the force is inversely proportional to the square of the volume of the gas; the smaller the volume, the closer are the molecules and the greater the attractions between pairs (hence the square term) of molecules. The pressure that goes into the corrected ideal gas equation is

in which the constant a expresses the magnitude of the attractive forces in a particular gas and has a value of 10⁶-10⁷ atm cm6 mol ^-2 . The complete van der Waals equation of state thus becomes

Although you do not have to memorize this equation, you are expected to understand it and to explain the signi¯cance of the terms it contains. You should also understand that the van der Waals constants a and b must be determined empirically for every gas. This can be done by plotting the P-V behavior of the gas and adjusting the values of a and b until the van der Waals equation results in an identical plot. The constant a is related in a simple way to the molecular radius; thus the determination of a constitutes an indirect measurment of an important microscopic quantity.

The van der Waals equation is only one of many equations of state for real gases. More elaborate equations are required to describe the behavior of gases over wider pressure ranges. These generally take account of higher-order nonlinear attractive forces, and require the use of more empirical constants. Although we will make no use of them in this course, they are widely employed in engineering work in which the behavior of gases at high pressures must be accurately predicted.

The most striking feature of real gases is that they cease to remain gases as the temperature is lowered and the pressure is increased. The graphs in Fig. 8 illustrate this behavior; as the volume is decreased, the lower-temperature isotherms suddenly change into straight lines. Under these conditions, the pressure remains constant as the volume is reduced. This can only mean that the gas is \disappearing" as w squeeze the system down to a smaller volume. In its place, we obtain a new state of matter, the liquid. In the shaded region of Fig. 8 on the right, two phases, liquid, and gas, are simultaneously present. Finally, at very small volume all the gas has disappeared and only the liquid

phase remains. At this point the isotherms bend strongly upward, reflecting our common experience that a liquid is practically incompressible.

The maximum temperature at which the two phases can coexist is called the critical temperature. The set of (P; V; T) corresponding to this condition is known as the critical point. Liquid and gas can coexist only within the regions indicated in Fig. 8 by the wedge-shaped cross section on the left and the shaded area on the right. An important consequence of this is that a liquid phase cannot exist at temperatures above the critical point.

Figure 8: Critical behavior of a real gas

The critical temperature of carbon dioxide is 31 ^oC, so you can tell whether the temperature is higher or lower than this by shaking a CO₂ flre extinguisher; on a warm day, you will not hear any liquid sloshing around inside. The critical temperature of water is 374 ^oC, and that of hydrogen is only 33 ^ok .

If the region of the almost-vertical isotherms represents the liquid, what is the state of the substance near the left side of either plot, but above the critical point? The answer is that it is a highly-nonideal gas, perhaps best described just as a “fluid" but certainly not a liquid. One intriguing consequence of the very limited bounds of the liquid state is that you could start with a gas at large volume and low temperature, raise the temperature, reduce the volume, and then reduce the temperature so as to arrive at the liquid region at the lower left, without ever passing through the two-phase region, and thus without undergoing condensation!

The supercritical state of matter, as the fluid above the critical point is often called, possesses the flow properties of a gas and the solvent properties of a liquid. Supercritical carbon dioxide is now used to dissolve the caffeine out of coffee beans, and supercritical water has recently attracted interest as a medium for chemically decomposing dangerous environmental pollutants such as PCBs.

CHAPTER 2

Zeroth law of thermodynamics

All pure substances display similar behavior in the gas phase . At 0° C and 1 atmosphere of pressure, one mole of every gas occupies about 22.4 liters of volume. Molar volumes of solids and liquids, on the other hand, vary greatly from one substance to another. In a gas at 1 atmosphere, the molecules are approximately 10 diameters apart. Unlike liquids or solids, gases occupy their containers uniformly and completely. Because molecules in a gas are far apart, it is easier to compress a gas than it is to compress a liquid. In general, doubling the pressure of a gas reduces its volume to about half of its previous value. Doubling the mass of gas in a closed container doubles its pressure. Increasing the temperature of a gas enclosed in a container increases its pressure.

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1. Zeroth law of thermodynamics

“If two thermodynamic systems are each in thermal equilibrium with a third system, then they are in thermal equilibrium with each other”

When two systems, each internally in thermodynamic equilibrium at a different temperature, are brought in diathermic contact with each other they exchange heat to establish a thermal equilibrium between each other.

The zeroth law implies that thermal equilibrium, viewed as a binary relation, is a transitive relation. Thermal equilibrium is furthermore an equivalence relation between any number of system. The law is also a statement about measurability. To this effect the law establishes an empirical parameter, the temperature, as a property of a system so that systems in equilibrium with each other have the same temperature. The notion of transitivity permits a system, for example a gas thermometer, to be used as a device to measure the temperature of another system.

Although the concept of thermodynamic equilibrium is fundamental to thermodynamics, the need to state it explicitly as a law was not widely perceived until Fowler and Planck stated it in the 1930s, long after the first, second, and third law were already widely understood and recognized. Hence it was numbered the zeroth law. The importance of the law as a foundation to the earlier laws is that it defines temperature in a non-circular logistics without reference to entropy, its conjugate variable.

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2. Zeroth law as equivalence relation

Systems are in thermal equilibrium if they do not exchange energy in the form of heat.The zeroth law states that if two systems are in thermal equilibrium with a third system, then they are also in thermal equilibrium with each other. Equally, a system is said to be in thermal equilibrium when it experiences no net change in thermal energy. If A, B, and C are distinct thermodynamic systems, the zeroth law of thermodynamics can then be expressed as:

	"If A and C are each in thermal equilibrium with B, A is also in thermal equilibrium with C."

The preceding sentence asserts that thermal equilibrium is a Euclidean relation between thermodynamic systems. If we also grant that all thermodynamic systems are in thermal equilibrium with themselves, then thermal equilibrium is also a reflexive relation. Relations that are both reflexive and Euclidean are equivalence relations. One consequence of this reasoning is that thermal equilibrium has a transitive relationship between the temperature T of A, B, and C:

If T (A) = T(B)

And T (B) = T(C)

Then T (A) = T(C).

“no heat exchange is possible outside of these systems”

3. Thermal equilibrium between many systems

Many systems are said to be in equilibrium if the small, random exchanges (due to Brownian motion, for example) between them do not lead to a net change in the total energy summed over all systems. A simple example illustrates why the zeroth law is necessary to complete the equilibrium description.

Consider N systems in adiabatic isolation from the rest of the universe, i.e. no heat exchange is possible outside of these N systems, all of which have a constant volume and composition, and which can only exchange heat with one another.

The combined First and Second Laws relate the fluctuations in total energy, δU, to the temperature of the ith system, and the entropy fluctuation in the ith system, as follows:

.

The adiabatic isolation of the system from the remaining universe requires that the total sum of the entropy fluctuations vanishes, or:

That is, entropy can only be exchanged between the N systems. This constraint can be used to rearrange the expression for the total energy fluctuation and obtain:

where is the temperature of any system j we may choose to single out among the N systems. Finally, equilibrium requires the total fluctuation in energy to vanish, in which case:

which can be thought of as the vanishing of the product of an antisymmetric matrix and a vector of entropy fluctuations . In order for a non-trivial solution to exist,

That is, the determinant of the matrix formed by must vanish for all choices of N. However, according to Jacobi's theorem, the determinant of a NxN antisymmetric matrix is always zero if N is odd, although for N even we find that all of the entries must vanish, , in order to obtain a vanishing determinant. Hence at equilibrium. This non-intuitive result means that an odd number of systems are always in equilibrium regardless of their temperatures and entropy fluctuations, while equality of temperatures is only required between an even number of systems to achieve equilibrium in the presence of entropy fluctuations.

The zeroth law solves this odd vs. even paradox, because it can readily be used to reduce an odd-numbered system to an even number by considering any three of the N systems and eliminating one by application of its principle, and hence reduce the problem to even N which subsequently leads to the same equilibrium condition that we expect in every case, i.e., . The same result applies to fluctuations in any extensive quantity, such as volume (yielding the equal pressure condition), or fluctuations in mass (leading to equality of chemical potentials). Hence the zeroth law has implications for a great deal more than temperature alone. In general, we see that the zeroth law breaks a certain kind of asymmetry present in the First and Second Laws.

4. Foundation of temperature

Max Planck and others have stated that the zeroth law implies the definition of a temperature function or more informally, that one can construct a thermometer.

In the space of thermodynamic parameters, zones of constant temperature form a surface, that provides a natural order of nearby surfaces. One may therefore construct a global temperature function that provides a continuous ordering of states. The dimensionality of a surface of constant temperature is one less than the number of thermodynamic parameters, thus, for an ideal gas described with three thermodynamic parameters P, V and n, it is a two-dimensional surface.

For example, if two systems of ideal gases are in equilibrium, then P1V1/N1 = P2V2/N2 where Pi is the pressure in the ith system, Vi is the volume, and Ni is the amount (in moles, or simply the number of atoms) of gas.

The surface PV/N = const defines surfaces of equal temperature, and one may label defining T so that PV/N = RT, where R is some constant. These systems can now be used as a thermometer to calibrate other systems

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CHAPTER 3

First Law of Thermodynamics

1. Introduction

The first law deals with macroscopic properties, work, energy, enthalpy, etc.One of the most fundamental laws of nature is the conservation of energy principle. It simply states that:

· During an interaction, energy can change from one form to another but the total amount of energy remains constant. That is, energy cannot be created or destroyed. Or,

· During an interaction between a system and its surroundings, the amount of energy gained by the system must be exactly equal to the amount of energy lost by the surroundings. A rock falling off a cliff, for example, picks up speed as a result of its potential energy being converted to kinetic energy.

The first law of thermodynamics is simply an expression of the conservation of energy principle, and it asserts that energy is a thermodynamic property.

Energy can cross the boundary of a closed system in two distinct forms: heat and work. It is important to distinguish between these two forms of energy. Therefore, they will be discussed first, to form a sound basis for the development of the first law of thermodynamics.

We can use the principle of conservation of energy to define a function U called the internal energy. When a closed system undergoes a process by which it passes from state A to state B, if the only interaction with its surroundings is in the form of transfer of heat Q to the system, or performance of work W on the system, the change in U will be

ΔU = UB – UA = Q + W 2-1

Note:

· In Equation 2-1 we have defined W as the work done on the system and Q is added to the system. If we had defined W as work done by the system, Equation 2-1 would become ΔU = Q - W.

· For an isolated system there is no heat or work transferred with the surroundings, thus, by definition W = Q = 0 and therefore ΔU = 0.

· The first law of thermodynamics states that this energy difference ΔU depends only on the initial and final states, and not on the path followed between them. Both Q and W have many possible values, depending on exactly how the system passes from A to B, but Q + W = ΔU is invariable and independent of the path. If this were not true, it would be possible, by passing from A to B along one path and then returning from B to A along another, to obtain a net change in the energy of the closed system in contradiction to the principle of conservation of energy.

· For a differential change, Equation 2-1 becomes

dU = dQ +dW 2-2

· For a cyclic process, A→B→A, when the system returns to state A, it has the same U, thus

∫dU = 0 2-3

Next we will take a look separately at the heat transferred (dQ) and the work (dW) exchanged between the system and the surroundings.

Heat Transfer

Heat is defined as the form of energy that is transferred between two systems (or a system and its surroundings) by virtue of a temperature difference. That is, an energy interaction is heat only if it takes place because of a temperature difference. Then it follows that there cannot be any heat transfer between two systems that are at the same temperature

Heat is energy in transition. It is recognized only as it crosses the boundary of a system. Consider the hot baked potato. The potato contains energy, but this energy is heat transfer only as it passes through the skin of the potato (the system boundary) to reach the air, as shown below.

Once in the surroundings, the transferred heat becomes part of the internal energy of the surroundings. Thus, in thermodynamics, the term heat simply means heat transfer.

A process during which there is no heat transfer is called an adiabatic process. There are two ways a process can be adiabatic:

Either the system is well insulated so that only a negligible amount of heat can pass through the boundary, or

Both the system and the surroundings are at the same temperature and therefore there is no driving force (temperature difference) for heat transfer.

An adiabatic process should not be confused with an isothermal process. Even though there is no heat transfer during an adiabatic process, the energy content and thus the temperature of a system can still be changed by other means such as work.

The amount of heat transferred during the process between two states (states 1 and 2) is denoted by Q12, or just Q. Heat transfer per unit mass of a system is denoted q and is determined from

q = Q/m 2-4

Sometimes it is desirable to know the rate of heat transfer (the amount of heat transferred per unit time) instead of the total heat transferred over some time interval. The heat transfer rate is denoted Q , where the overdot stands for the time derivative, or "per unit time." The heat transfer rate Q has the unit kJ/s,which is equivalent to kW. When Q varies with time, the amount of heat transfer during a process is determined by integrating Q over the time interval of the process:

When Q remains constant during the process, the relation reduces to

Heat is a directional (or vector) quantity; the universally accepted sign convention for heat is as follows: Heat transfer to a system is positive, and heat transfer from a system is negative. That is, any heat transfer that increases the energy of a system is positive, and any heat transfer that decreases the energy of a system is negative.

2. Modes of Heat transfer

Heat can be transferred in three different ways: conduction, convection, and radiation. A detailed study of these heat transfer modes is given later. Below we will give a brief description of each mode to familiarize yourselves with the basic mechanisms of heat

transfer.

All modes of heat transfer require the existence of a temperature difference, and all modes of heat transfer are from the high-temperature medium to a lower-temperature one.

Conduction is the transfer of energy from the more energetic particles of a substance to the adjacent less energetic ones as a result of interactions between the particles. Conduction can take place in solids, liquids, or gases. In gases and liquids, conduction is due to the collisions of the molecules during their random motion. In solids, it is due to the combination of vibrations of the

molecules in a lattice and the energy transport by free electrons. A cold canned drink in a warm room, for example, eventually warms up to the room temperature as a result of heat transfer from the room to the drink through the aluminum can by conduction.

It is observed that the rate of heat conduction Qcond through a layer of constant thickness Δx is proportional to the temperature difference ΔT across the layer and the area A normal to the direction of heat transfer, and is inversely proportional to the thickness of the layer. Therefore,

x 2-7

where the constant of proportionality k is the thermal conductivity of the material which is a measure of the ability of a material to conduct heat. Materials such as copper and silver that are good electric conductors are also good heat conductors: kcopper = 401 W/(m.K), and therefore have high k values. Materials such as rubber, wood, and styrofoam are poor conductors of heat (kurethane = 0.026), and therefore have low k values. Diamond has a very high thermal conductivity (k = 2300). In the limiting case of Δx→0, the equation above reduces to the differential form

2-8

which is known as Fourier's law of heat conduction. It indicates that the rate of heat conduction in a direction is proportional to the temperature gradient in that direction. Heat is conducted in the direction of decreasing temperature, and the temperature gradient becomes negative when temperature decreaseswith increasing x. Therefore, a negative sign is added in Eq. 2-8 to make heat transfer in the positive x direction a positive quantity.

Note: Temperature is a measure of the kinetic energies of the molecules. In a liquid or gas, the kinetic energy of the molecules is due to the random motion of the molecules as well as the vibrational and rotational motions. When two molecules possessing different kinetic energies collide, part of the kinetic energy of the more energetic (higher-temperature) molecule is transferred to the less energetic (lowertemperature) particle, in much the same way as when two elastic balls of the same mass at different velocitie collide, part of the kinetic energy of the faster ball is transferred to the slower one. In solids, heat conduction is due to two effects: the lattice vibrational waves induced by the vibrational motions of the molecules positioned at relatively fixed positions in a periodic manner called a lattice, and the energy transported via the free flow of electrons in the solid. The thermal conductivity of a solid is obtained by adding the lattice and the electronic components. The thermal conductivity of pure metals is primarily due to the electronic component whereas the thermal conductivity of nonmetals is primarily due to the lattice component. The lattice component of thermal conductivity strongly depends on the way the molecules are arranged. For example, the thermal conductivity of diamond, which is a highly ordered crystalline solid, is much higher than the thermal conductivities of pure metals.

Convection is the mode of energy transfer between a solid surface and the adjacent liquid or gas which is in motion, and it involves the combined effects of conduction and fluid motion. The faster the fluid motion, the greater the convection heat transfer. In the absence of any bulk fluid motion, heat transfer between a solid surface and the adjacent fluid is by pure conduction The presence of bulk motion of the fluid enhances the heat transfer between the solid surface and the fluid, but it also complicates the determination of heat transfer rates.

· Consider the cooling of a hot block by blowing of cool air over its top surface5-9 shown in figure below. The arrows in the figure indicate the velocity variation of air. Energy is first transferred to the air layer adjacent to the surface of the block by conduction. This energy is then carried away from the surface by convection; that is, by the combined effects of conduction within the air, which is due to random motion of air molecules, and the bulk or macroscopic motion of the air, which removes the heated air near the surface and replaces it by the cooler air.

· Convection is called forced convection if the fluid is forced to flow in a tube or over a surface by external means such as a fan, pump, or the wind. In contrast, convection is called free (or natural) convection if the fluid motion is caused by buoyancy forces that are induced by density differences due to the variation of temperature in the fluid (see figure below).

Forced convection (left) and natural convection (right) For example, in the absence of a fan, heat transfer from the surface of the hot block will be by natural convection since any motion in the air in this case will be due to the rise of the warmer (and thus lighter) air near the surface and the fall of the cooler (and thus heavier) air to fill its place. Heat transfer between the block and the surrounding air will be by conduction if the temperature difference between the air and the block is not large enough to overcome the resistance of air to move and thus to initiate natural convection currents.

· Heat transfer processes that involve change of phase of a fluid are also considered to be convection because of the fluid motion induced during the process such as the rise of the vapor bubbles during boiling or the fall of the liquid droplets during condensation.

· The rate of heat transfer by convection Qconv is determined from Newton's law of cooling, which is expressed as

Qconv = hA (Ts – Tf) 2-9

· where h is the convection heat transfer coefficient, A is the surface area through which heat transfer takes place, Ts is the surface temperature, and Tf is bulk fluid temperature away from the surface. (At the surface, the fluid temperature equals the surface temperature of the solid.)

Note: The convection heat transfer coefficient h is not a property of the fluid. It is an experimentally determined parameter whose value depends on all the variables that influence convection such as the surface geometry, the nature of fluid motion, the properties of the fluid, and the bulk fluid velocity. Typical values of h, in W /(m2 . K), are 2-25 for the free convection of gases, 50-1000 for the free convection of liquids, 25-250 for the forced convection of gases, 50-20,000 for the forced convection of liquids, and 2500-100,000 for convection in boiling and condensation processes.

Radiation is the energy emitted by matter in the form of electromagnetic waves (or photons) as a result of the changes in the electronic configurations of the atoms or molecules. Unlike conduction and convection, the transfer of energy by radiation does not require the presence of an intervening medium. In fact, energy transfer by radiation is fastest (at the speed of light) and it suffers no attenuation in a vacuum. This is exactly how the energy of the sun reaches the earth.

· In heat transfer studies, we are interested in thermal radiation, which is the form of radiation emitted by bodies because of their temperature. It differs from other forms of electromagnetic radiation such as X-rays, gamma rays, microwaves, radio waves, and television waves which are not related to temperature. All bodies at a temperature above absolute zero emit thermal radiation.

· Radiation is a volumetric phenomena, and all solids, liquids, and gases emit, absorb, or transmit radiation to varying degrees. However, radiation is usually considered to be a surface phenomenon for solids that are opaque to thermal radiation such as metals, wood, and rocks since the radiation emitted by the interior regions of such material can never reach the surface, and the radiation incident on such bodies is usually absorbed within a few microns from the surface.

· The maximum rate of radiation that can be emitted from a surface at an absolute temperature Ts is given by the Stefan-Boltzmann, law as

Qemit.max=σATs4 2-10

where A is the surface area and σ = 5.67 x 10-8 W / (m2 . K4) is the Stefan-Boltzmann constant. The idealized surface which emits radiation at this maximum rate is called a blackbody, and the radiation emitted by a blackbody is called blackbody radiation. The radiation emitted by all real surfaces is less than the radiation emitted by a blackbody at the same temperatures and is expressed as

Qemit.max=εσATs 4 2-11

where ε is the emissivity of the surface. The property emissivity, whose value is in the range 0 ≤ ε ≤ 1, is a measure of how closely a surface approximates a blackbody for which ε = 1. The human skin has an emissivity of 0.95, and aluminum foil 0.07.

· Another important radiation property of a surface is its absorptivity, α, which is the fraction of the radiation energy incident on a surface that is absorbed by the surface. Like emissivity, its value is in the range 0 ≤ α ≤ 1. A blackbody absorbs the entire radiation incident on it. That is, a blackbody is a perfect absorber (α = 1) as well as a perfect emitter.

· In general, both ε and α of a surface depend on the temperature and the wavelength of the radiation. Kirchhoff's law of radiation states that the emissivity and the absorptivity of a surface are equal at the same temperature and wavelength. In most practical applications, the dependence of ε and α on the temperature and wavelength is ignored, and the average absorptivity of a surface is taken to be equal to its average emissivity. The rate at which a surface absorbs radiation (Qabs) is determined from (see figure)

Qabs=αQinc

where Qinc is the rate at which radiation is incident on the surface and α is the absorptivity of the surface. For opaque (nontransparent) surfaces, the portion of incident radiation that is not absorbed by the surface is reflected back.

· The difference between the rates of radiation emitted by the surface and the radiation absorbed is the net radiation heat transfer. If the rate of radiation absorption is greater than the rate of radiation emission, the surface is said to be gaining energy by radiation. Otherwise, the surface is said to be losing energy by radiation. In general, the determination of the net rate of heat transfer by radiation between two surfaces is a complicated matter since it depends on the properties of the surfaces, their orientation relative to each other, and the interaction of the medium between the surfaces with radiation. However, in the special case of a relatively small surface of emissivity ε and surface area A at absolute temperature Ts that is completely enclosed by a much larger surface at absolute temperature Tsurr separated by a gas (such as air) that does not interact with radiation (i.e., the amount of radiation emitted, absorbed, or scattered by the medium is negligible), the net rate of radiation heat transfer between these two surfaces is determined from

Qrad =εσA(Ts⁴–T⁴ surr)

Example 2-1: Consider a person standing in a breezy room at 20°C. Determine the total rate of heat transfer from this person if the exposed surface area and the average outer surface temperature of the person are 1.6 m2 and 29°C, respectively, and the convection heat transfer coefficient is 6 W/(m2.oC).

Solution 1. The heat transfer between the person and the air in the room will be by convection (instead of conduction) since it is conceivable that the air in the vicinity of the skin or clothing will warm up and rise as a result of heat transfer from the body,

initiating natural convection currents. It appears that the experimentally determined value for the rate of convection heat transfer in this case is 6 W per unit surface area (m2) per unit temperature difference (in K or oC) between the person and the air away from the person. Thus, the rate of convection heat transfer from the person to the air in the room is, from Eq. 2-9,

Qconv = hA (Ts – Tf) = 86.4 W

2. The person will also lose heat by radiation to the surrounding wall surfaces. We take the temperature of the surfaces of the walls, ceiling, and the floor to be equal to the air temperature in this case for simplicity, but we recognize that this does not need to be the case. These surfaces may be at a higher or lower temperature than the average temperature of the room air, depending on the outdoor conditions and the structure of the walls. Considering that air does not intervene with radiation and the person is completely enclosed by the surrounding surfaces, the net rate of radiation heat transfer from the person to the surrounding walls, ceiling, and the floor is, from Eq. 2-13

Qrad = εσA(Ts⁴ – T⁴surr) = 81.7 W

Note that we must use absolute temperatures in radiation calculations. Also note that we used the emissivity value for the skin and clothing at room temperature since the emissivity is not expected to change significantly at a slightly higher temperature. Then the rate of total heat transfer from the body is determined by adding thesetwo quantities to be

Qtotal = Qconv + Qrad = 168.1 W

Note: The heat transfer would be much higher if the person were not dressed since the exposed surface temperature would be higher. In the above calculations, heat transfer through the feet to the floor by conduction, which is usually very small, is neglected. Heat transfer from the skin by perspiration, which is the dominant mode of heat transfer in hot environments, is not considered here.

3. Work

Work, like heat, is an energy interaction between a system and its surroundings. As mentioned earlier, energy can cross the boundary of a closed system in the form of heat or work. Therefore, if the energy crossing the boundary of a closed system is not heat, it must be work.

· Heat is easy to recognize: Its driving force is a temperature difference between the system and its surroundings. Then we can simply say that an energy interaction which is not caused by a temperature difference between a system and its surroundings is work. More specifically, work is the energy transfer associated with a force acting through a distance. A rising piston, a rotating shaft, and an electric wire crossing the system boundaries are all associated with work interactions.

· The work done during a process between states 1 and 2 is denoted W12, or simply W. The work done per unit mass of a system is denoted w and is defined as

w = W/m 2-14

• The work done per unit time is called power and is denoted W The unit of power is kJ/s, or kW.

• The energy of a system decreases as it does work and increases as work is done on the system.

• Heat transfer and work are interactions between a system and its surroundings, and there are many similarities between the two:

1) Both are recognized at the boundaries of the system as they cross them. That is, both heat and work are boundary phenomena.

2) Systems possess energy, but not heat or work. That is, heat and work are transfer phenomena.

3) Both are associated with a process, not a state. Unlike properties, heat or work has no meaning at a state.

4) Both are path functions (i.e., their magnitudes depend on the path followed during a process as well as the end states).

Path functions have inexact differentials designated by the symbol δ. Therefore, a differential amount of heat or work is represented by δ Q or δW, respectively, instead of dQ or dW. Properties, however, are point functions (i.e., they depend on the state only, and not on how a system reaches that state), and they have exact differentials designated by the symbol d. A small change in volume, for example, is represented by dV and the total volume change during a process between states 1 and 2 is

2

1∫dV =V −V1 = ΔV

That is, the volume change during process 1-2 is always the volume at state 2 minus the volume at state 1, regardless of the path followed (see figure below). The total work done during process 1-2, however, is 2

1∫δW = W12

That is, the total work is obtained by following the process path and adding the differential amounts of work (δW) done along the way. The integral of δW is not W2 – W1 (i.e., the work at state 2 minus work at state 1), which is meaningless since work is not a property and systems do not possess work at a state.

Example 2-4 A well-insulated electric oven is being heated through its heating element. If the entire oven, including the heating element, is taken to be the system, determine whether this is a heat or work interaction.

Solution For this problem, the interior surfaces of the oven form the system boundary. The energy content of the oven obviously increases during this process, as evidenced by a rise in temperature. This energy transfer to the oven is not caused by a temperature difference between the oven and the surrounding air. Instead, it is caused by the electrons crossing the system boundary and thus doing work. Therefore, this is a work interaction.

Example 2-5 Answer the question in Example 2-4 if the system is taken as only the air in the oven without the heating element.

5 -17

Solution This time, the system boundary will include the outer surface of the heating element and will not cut through it. Therefore, no electrons will be crossing the system boundary at any point. Instead, the energy generated in the interior of the heating element will be transferred to the air around it as a result of the temperature difference between the heating element and the air in the oven. Therefore, this is a heat transfer process. For both cases, the amount of energy transfer to the air is the same. These two examples show that the same interaction can be heat or work depending on how the system is selected.

Electrical Work

It was shown above that electrons crossing the system boundary do electrical work on the system. In an electric field, electrons in a wire move under the effect of electromotive forces, doing work. When N coulombs of electrons move through a potential difference V, the electrical work done is

We = VN

Which can also be expressed in the rate form

W e= VI 2-14

where W e is the electrical power and I the electric current. ( e W=VI = I2R =V2/R) In general, both V and I vary with time, and the electrical work done during a time interval Δt is expressed as

If both V and I remain constant during the time interval Δt, this equation will reduce to

We = VIΔt 2-16

Example 2-6 A small tank containing iced water at 0°C is placed in the middle of a large, wellinsulated tank filled with oil. The entire system is initially in thermal equilibrium at 0°C. The electric heater in the oil is now turned on, and 10 kJ of electrical work is done on the oil. After a while, it is noticed that the entire system is again at 0°C, but some ice in the small tank has melted. Considering the oil to be system A and the iced water to be system B, discuss the heat and work interactions for system A. system B, and the combined system (oil and iced water).

Solution The boundaries of each system are indicated by dashed lines in the figure. Notice that the boundary of system B also forms the inner part of the boundary of system A.

System A: When the heater is turned on, electrons cross the outer boundary of system A, doing electrical work. This work is done on the system, and therefore WA = 10 kJ. Because of this added energy, the temperature of the oil will rise, creating a temperature gradient, which results in a heat flow process from the oil to the iced water through their common boundary. Since the oil is restored to its initial temperature of 0 oC, the energy lost as heat must equal the energy gained as work. Therefore, QA = 10 kJ (or QA,out = 10 kJ).

System B: The only energy interaction at the boundaries of system B is the heat flow from system A. All the heat lost by the oil is gained by the iced water. Thus, WB = 0 and Qe = +10 kJ. Combined system: The outer boundary of system A forms the entire boundary of the combined system. The only energy interaction at this boundary is the electrical work. Since the tank is well insulated, no heat will cross this boundary. Therefore, Wcomb = 10 kJ and Qcomb = 0. Notice that the heat flow from the oil to the iced water is an internal process for the combined system and, therefore, is not recognized as heat. It is simply the redistribution of the internal energy.

Mechanical forms of Work

There are several different ways of doing work, each in some way related to a force acting through a distance. In elementary mechanics, the work done by a constant force F on a body that is displaced a distance s in the direction of the force is given by

W = Fs 2-17

If the force F is not constant, the work done is obtained by adding (i.e., integrating) the differential amounts of work (force times the differential displacement ds):

Obviously one needs to know how the force varies with displacement to perform this integration. Equations 2-17 and 2-18 give only the magnitude of the work. The sign is easily determined from physical considerations: The work done on a system by an external force acting in the direction of motion is positive, and work done by a system against an external force acting in the opposite direction to motion is negative.

There are two requirements for a work interaction between a system and its surroundings to exist:

1. there must be a force acting on the boundary, and

2. the boundary must move. Therefore, the presence of forces on the boundary

without any displacement of the boundary does not constitute a work

interaction. Likewise, the displacement of the boundary without any force to

oppose or drive this motion (such as the expansion of a gas into an evacuated

space) is not a work interaction.

In many thermodynamic problems, mechanical work is the only form of work involved. It is associated with the movement of the boundary of a system or with the movement of the entire system as a whole .Some common forms of mechanical work are discussed below.

Moving Boundary Work

One form of mechanical work frequently encountered in practice is associated with the expansion or compression of a gas in a piston-cylinder device. During this process, part of the boundary (the inner face of the piston) moves back and forth. Therefore, the expansion and compression work is often called moving boundary work, or simply boundary work. Some prefer to call it the P dV work for reasons explained below. Moving boundary work is the primary form of work involved in automobile engines. During their expansion, the combustion gases force the piston to move, which in turn forces the crank shaft to rotate.

The moving boundary work associated with real engines or compressors cannot be determined exactly from a thermodynamic analysis alone because the piston usually moves at very high speeds, making it difficult for the gas inside to maintain equilibrium.

Then the states that the system passes through during the process cannot be specified, and no process path can be drawn. Work, being a path function, cannot be determined analytically without knowledge of the path. Therefore, the boundary work in real engines or compressors is determined by direct measurements.

In this section, we analyze the moving boundary work for a quasi-equilibrium process, a process during which the system remains in equilibrium at all times. A quasi-equilibrium process, also called a quasi-static process, is closely approximated by real engines, especially when the piston moves at low velocities. Under identical conditions, the work output of the engines is found to be a maximum, and the work input to the compressors to be a minimum, when quasi-equilibrium processes are used in place of non-quasiequilibrium processes. Below, the work associated with a moving boundary is evaluated for a quasi-equilibrium process.

Consider the gas enclosed in the piston-cylinder device shown below.

The initial pressure of the gas is P, the total volume is V, and the cross-sectional area of the piston is A. If the piston is allowed to move a distance ds in a quasi-equilibrium manner, the differential work done during this process is

δW = Fds = PA ds = P dV 2-19

That is, the boundary work in the differential form is equal to the product of the absolute pressure P and the differential change in the volume dV of the system. This expression also explains why the moving boundary work is sometimes called the P dV work. In order to abide by the sign rule, for an expansion dV is positive, the pressure P is the absolute pressure which is always positive, thus, the work should be written as

δW = - P dV 2-20

Thus, the boundary work is negative during an expansion process and positive during a compression process, which is consistent with the sign convention adopted for work. The total boundary work done during the entire process as the piston moves is obtained by adding all the differential works from the initial state to the final state:

This integral can be evaluated only if we know the functional relationship between P and

V during the process. That is, P = f(V) should be available. Note that P = f(V) is simply the equation of the process path on a P-V diagram. The quasi-equilibrium expansion process described above is shown on a P-V diagram below. On this diagram, the differential area dA is equal to P dV, which is the differential work. The total area A under the process curve 1-2 is obtained by adding these

differential areas:

The area under the process curve on a P-V diagram is equal, in magnitude, to the

work done during a quasi-equilibrium expansion or compression process of a

closed system.

A gas can follow several different paths as it expands from state 1 to state 2. In general,each path will have a different area underneath it, and since this area represents the magnitude of the work, the work done will be different for each process.

This is expected, since work is a path function (i.e., it depends on the path followed as well as the end states). If work were not a path function, no cyclic devices (car engines, power plants) could operate as work-producing devices. The work produced by these devices during one part of the cycle would have to be consumed during another part, and there would be no net work output.

Note: If the relationship between P and V during an expansion or a compression process is given in terms of experimental data instead of in a functional form, obviously we cannot perform the integration analytically. But we can always plot the P-V diagram of the process, using these data points, and calculate the area underneath graphically to determine the work done.

Example 2-7A frictionless piston-cylinder device contains 0.1 lb of water vapor at 20 psi and 320 oF. Heat is now added to the steam until the temperature reaches 400°F. If the piston is not attached to a shaft and its mass is constant, determine the work done by the steam during this process.

Hint: Even though it is not explicitly stated, the pressure of the steam within the cylinder remains constant during this process since both the atmospheric pressure and the weight of the piston remain constant. Therefore, this is a constant-pressure process.

Example 2-8 A piston-cylinder device initially contains 0.4 m3 of air at 100 kPa and 80°C. The air is now compressed to 0.1 m3 in such a way that the temperature inside the cylinder remains constant. Determine the work done during this process.

Solution A sketch of the system and the P-V diagram of the process are shown below. At the specified conditions, air can be considered to be an ideal gas since it is at a high temperature and low pressure relative to its critical-point values (Tcr = -147°C, Pcr = 3390 kPa for nitrogen, the main constituent of air). For an ideal gas at constant temperature To,

P = C / V

where C is a constant. Eq. 2-21 becomes:

where P1 V1 = P2 V2 .

Polytropic Process

During expansion and compression processes of real gases, pressure and volume are often related by P Vn = C, where n and C are constants. A process of this kind is called a polytropic process. Below we develop a general expression for the work done during a polytropic process.

A sketch of the system and the P-V diagram of the process are shown below

The pressure for a polytropic process can be expressed as

n

The case for n = 1 is equivalent to the isothermal process already discussed.

Nonmechanical Forms of Work

Some work modes encountered in practice are not mechanical in nature. However, these nonmechanical work modes can be treated in a similar manner by identifying a generalized force F acting in the direction of a generalized displacement x. Then the work associated with the differential displacement under the influence of this force is determined from

δW = F.dx

Some examples of nonmechanical work modes are electrical work, where the generalized force is the voltage (the electrical potential) and the generalized displacement is the electrical charge as discussed in the last section; magnetic work, where the generalized force is the magnetic field strength and the generalized displacement is the total magnetic dipole moment; and electrical polarization work, where the generalized force is the electric field strength and the generalized displacement is the polarization of the medium (the sum of the electric dipole rotation moments of the molecules). Detailed consideration of these and other nonmechanical work modes can be found in specialized books on these topics.

4. Energy

Chemical thermodynamics deals with the internal energy U, energy possessed by the system by virtue of the mass and motion of the molecules, intermolecular forces, chemical composition, etc. Any energy that the system possesses because of other considerations is ignored. The internal energy is relative. ΔU in total internal energy is the difference between the energy in the final state and that in the initial state. The most significant aspect of this kind of relation is that the energy change depends only on the initial and final states and is independent of the path linking these states.

Thus:

· The change Δ.U in the internal energy of a system depends only on the initial and final states of the system and not on the path connecting those states. Although both Q and W depend on the path, Q + W = ΔU is independent of the path.

· The energy is an extensive state property of the system. Under the same conditions of temperature and pressure, 10 mol of the substance composing the system has ten times the energy of 1 mol of the substance. The energy per mole is an intensive state property of the system.

· Energy is conserved in all transformations. A perpetual-motion machine is a machine which by its action creates energy by some transformation of a system. The first law of thermodynamics asserts that it is impossible to construct such a machine.

5. The First Law of Thermodynamics

So far, we have considered various forms of energy such as heat Q, work W, and total energy E individually, and no attempt has been made to relate them to each other during a process. The first law of thermodynamics, also known as the conservation of energy principle, provides a sound basis for studying the relationships among the various forms of energy and energy interactions.

· Based on experimental observations, the first law of thermodynamics states that energy can be neither created nor destroyed; it can only change forms. Therefore, every bit of energy should be accounted for during a process.

· The first law cannot be proved mathematically, but no process in nature is known to have violated the first law, and this should be taken as sufficient proof.

· Consider a system undergoing a series of adiabatic processes from a specified state 1 to another specified state 2. Being adiabatic, these processes obviously cannot involve any heat transfer but they may involve several kinds of work interactions. Careful measurements during these experiments indicate the following:

For all adiabatic processes between two specified states of a closed system, the net work done is the same regardless of the nature of the closed system and the details of the process.

Considering that there are an infinite number of ways to perform work interactions under adiabatic conditions, the statement above appears to be very powerful, with a potential for far-reaching implications. This statement, which is largely based on the experiments of Joule in the first half of the nineteenth century, cannot be drawn from any other known physical principle, and is recognized as a fundamental principle. This principle is called the first law of thermodynamics or just the first law.

· A major consequence of the first law is the existence and the definition of the property total energy E. Considering that the net work is the same for all adiabatic processes of a closed system between two specified states, the value of the net work must depend on the end states of the system only, and thus it must correspond to a change in a property of the system. This property is the total energy. Note that the first law makes no reference to the value of the total energy of a closed system at a state. It simply states that the change in the total energy during an adiabatic process must be equal to the net work done. Therefore, any convenient arbitrary value can be assigned to total energy at a specified state to serve as a reference point.

· Implicit in the first law statement is the conservation of energy. Although the essence of the first law is the existence of the property total energy, the first law is often viewed as a statement of the conservation of energy principle. Below we develop the first law or the conservation of energy relation for closed systems with the help of some familiar examples using intuitive arguments.

A. Let us consider first some processes that involve heat transfer but no work interactions. Take the example with the potato in the oven. As a result of heat transfer to the potato, the energy of the potato will increase. If we disregard any mass transfer (moisture loss from the potato), the increase in the total energy of the potato becomes equal to the amount of heat transfer. This and other similar examples can be summarized as follows:

In the absence of any work interactions between a system and its surroundings, the amount of net heat transfer is equal to the change in energy of a closed system

That is, Q = ΔE when W = 0 B.

B. Now consider a well-insulated (i.e., adiabatic) room heated by an electric heater as the system. As a result of electrical work done, the energy of the system will increase. Since the system is adiabatic and cannot have any heat interactions with the surroundings (Q = 0), the conservation of energy principle dictates that the electrical work done on the system must equal the increase in energy of the system. That is,

We = ΔE.

We know that the temperature of air rises when it is compressed (think of the example of the air in a cylinder with a piston). This is because energy is added to the air in the form of boundary work. In the absence of any heat transfer (Q = 0), the entire boundary work will be stored in the air as part of its total energy. The conservation of energy principle again requires that Wb = ΔE.

Thus, for adiabatic processes, the amount of work done is equal to the change in the energy of a closed system. That is,

W = ΔE when Q = 0

C. Now we are in a position to consider simultaneous heat and work interactions. When a system involves both heat and work interactions during a process, their contributions are simply added. That is, if a system receives 12 kJ of heat while a paddle wheel does 6 kJ of work on the system, the net increase in energy of the system for this process will be 18 kJ.

To generalize our conclusions, the first law of thermodynamics, or the conservation of energy principle for a closed system or a fixed mass, may be expressed as follows:

Net energy transfer to (or from) the system as heat and work = net change in the total energy of the system:

Q + W = ΔE (in kJ)

Here Q = net heat transfer across system boundaries ( = ΣQin - ΣQout )

W = net work done in all forms ( = ΣWin - ΣWout )

ΔE = net change in total energy of the system E2 - E1

ΔE = ΔU + ΔKE + ΔPE

Most closed systems encountered in practice are stationary, i.e., they do not involve any changes in their velocity or the elevation of their center of gravity during a process. Thus, for stationary closed systems, the changes in kinetic and potential energies are negligible, and the first-law relation reduces to

Q + W = ΔU (in kJ)

Sometimes it is convenient to consider the work term in two parts: Wother and Wb, where Wother represents all forms of work except the boundary work. (This distinction has important bearings with regard to the second law of thermodynamics, as is discussed in later chapters.) Then the first law takes the following form:

Q + Wother + Wb = ΔE (in kJ) 2-31

Note: It is extremely important that the sign convention be observed for heat and work interactions. Heat flow into and work done to a system are positive, and heat flow from a system and work done by a system are negative. A system may involve more than one form of work during a process. The only form of work whose sign we do not need to be concerned with is the boundary work Wb as defined by Eq. 2-21. Boundary work calculated by using Eq. 2-21 will always have the correct sign. The signs of other forms of work are determined by inspection.

The first-law relation for closed systems can be written in various forms. Dividing Eq. 2- 28 by the mass of the system, for example, gives the first-law relation on a unit-mass basis as

q + w = Δe 2-32

The rate form of the first law is obtained by dividing Eq. 2-28 by the time interval Δt and taking the limit as Δt→0. This yields to

where dQ/dt is the rate of net heat transfer, dW/dt is the power, and dE/dt is the rate of change of total energy. Equation 2-28 can be expressed in differential form as

δQ +δW = dE 2-34

For a cyclic process, the initial and final states are identical, and therefore ΔE = E2 – E1 = 0. Then the first-law relation for a cycle simplifies to

Q + W = 0

That is, the net heat transfer and the net work done during a cycle must be equal.

Note: As energy quantities, heat and work are not that different, and you probably wonder why we keep distinguishing them. After all, the change in the energy content of a system is equal to the amount of energy that crosses the system boundaries, and it makes no difference whether the energy crosses the boundary as heat or work. It seems as if the first-law relations would be much simpler if we had just one quantity which we could call energy interaction to represent both heat and work. Well, from the first-law point of view, heat and work are not different at all, and the first law can simply be expressed as

Ein – Eout = ΔE 2-35

where Ein and Eout are the total energy that enters and leaves the system, respectively, during a process. But from the second-law point of view, heat and work are very different, as you will see later.

Example 2-10 Consider the quasi-equilibrium expansion or compression of a gas in piston cylinder device. Show that the boundary work Wb and the change internal energy ΔU in the first law relation can be combined into one term, ΔH, for such a system undergoing a constant-pressure process.

Solution Neglecting the changes in kinetic and potential energies and expressing the work as in Eq. 2-31,

Q + Wother + Wb = U2 – U1

For a constant-pressure process, the boundary work is given by Wb =- P0 (V2 – V1). The sign preserves the sign convention that when the surroundings do work on the system, the final volume is less than the initial and the net work is positive. Substituting this into the above relation gives

Q + Wother - P0 (V2 – V1) = U2 – U1

But P0 = P2 = P1 thus Q + Wother = (U2 +P2V2) – (U1 + P1V1)

Taking

H = U + PV,

leads to

Q + Wother = H2 – H1 2-36

which is the desired relation. This equation is very convenient to use in the analysis of closed systems undergoing a constant-pressure quasi-equilibrium process since the boundary work is automatically taken care of by the enthalpy terms, and one no longer needs to determine it separately

CHAPTER 4

SECOND LAW OF THERMODYNAMIC

1. Introduction

In the preceding lectures, we applied the first law of thermodynamics,or the conservation of energy principle, to processes involving closed systems. Energy is a conserved property, and no process is known to have taken place in violation of the first law of thermodynamics. Therefore, it is reasonable to conclude that a process must satisfy the first law to occur. However, as explained below, satisfying the first law alone does not ensure that the process will actually take place.

It is common experience that a cup of hot coffee left in a cooler room eventually cools off. This process satisfies the first law of thermodynamics since the amount of energy lost by the coffee is equal to the amount gained by the surrounding air. Now let us consider the reverse process-the hot coffee getting even hotter in a cooler room as a result of heat transfer from the room air. We all know that this process never takes place. Yet, doing so would not violate the first law as long as the amount of energy lost by the air is equal to the amount gained by the coffee.

It is clear from the above that processes proceed in a certain direction and not in the reverse direction. The first law places no restriction on the directionof a process, but satisfying the first law does not ensure that that process will actually occur. This inadequacy of the first law to identify whether a process can take place is remedied by introducing another general principle, the second law of thermodynamics. We show later in Part III that the reverse processes discussed above violate the second law of thermodynamics. This violation is easily detected with the help of a property, called entropy, defined in the next part. A process will not occur unless it satisfies both the first and the second laws of thermodynamics.

The second law has been stated in several ways.

v The principle of Thomson (Lord Kelvin) states: 'It is impossible by a cyclic process to take heat from a reservoir and to convert it into work without simultaneously transferring heat from a hot to a cold reservoir.' This statement of the second law is related to equilibrium, i.e. work can be obtained from a system only when the system is not already at equilibrium. If a system is at equilibrium, no spontaneous process occurs and no work is produced. Evidently, Kelvin's principle indicates that the spontaneous process is the heat flow from a higher to a lower temperature, and that only from such a spontaneous process can work be obtained.

v The principle of Clausius states: 'It is impossible to devise an engine which, working in a cycle, shall produce no effect other than the transfer of heat from a colder to a hotter body.' A good example of this principle is the operation of a refrigerator.

v The principle of Planck states: 'It is impossible to construct an engine which, working in a complete cycle, will produce no effect other than raising of a weight and the cooling of a heat reservoir.'

v The Kelvin-Planck principle may be obtained by combining the principles of Kelvin and of Planck into one equivalent statement as the Kelvin-Planck statement of the second law. It states: 'No process is possible whose sole result is the absorption of heat from a reservoir and the conversion of this heat into work.'

The second law of thermodynamics is also used in determining the theoretical limits for the performance of commonly used engineering systems, such as heat engines and refrigerators, as well as predicting the degree of completion of chemical reactions.

The second law is not a deduction from the first law but a separate law of nature, referring to an aspect of nature different from that contemplated by the first law. The first law denies the possibility of creating or destroying energy, whereas the second law denies the possibility of utilizing energy in a particular way. The continuous operation of a machine that creates its own energy and thus violates the first law is called perpetual motion of the first kind. A cyclic device which would continuously absorb heat from a single reservoir and convert that heat completely to mechanical work is called a perpetual-motion machine of the second kind. Such a machine would not violate the first law (the principle of conservation of energy), since it would not create energy, but economically it would be just as valuable as if it did so. Hence, the second law is sometimes stated as follows: 'A perpetual motion machine of the second kind is impossible.'

2. Thermal Energy Reservoir

In the development of the second law of thermodynamics, it is very convenient to have a hypothetical body with a relatively large thermal energy capacity (mass x specific heat) that can supply or absorb finite amounts of heat without undergoing any change in temperature. Such a body is called a thermal energy reservoir, or just a reservoir.

3. Heat Engines

Work can easily be converted to other forms of energy, but converting other forms of energy to work is not that easy. The mechanical work done by a propeller placed in a bucket of water, for example, is first converted to the internal energy of the water. This energy may then leave the water as heat. We know from experience that any attempt to reverse this process will fail. That is, transferring heat to the water will not cause the shaft to rotate. From this and other observations, we conclude that

work can be converted to heat directly

and completely, but converting heat to work requires the use of some special

devices. These devices are called heat engines.

Heat engines differ considerably from one another, but all can be characterized by the following:

1. They receive heat from a high-temperature source (solar energy, oil furnace, nuclear reactor, etc.).

2. They convert part of this heat to work (usually in the form of a rotating shaft).

3. They reject the remaining waste heat to a low-temperature sink (theatmosphere, rivers, etc.).

4. They operate on a cycle Heat engines and other cyclic devices usually involve a fluid to and from which heat is transferred while undergoing a cycle. This fluid is called the working fluid.

The work-producing device that best fits into the definition of a heat engine is the steam power plant, which is an external-combustion engine. That is, the combustion process takes place outside the engine, and the thermal energy released during this process is transferred to the steam as heat. The schematic of a basic steam power plant is shown below

The various quantities shown on this figure are as follows

Qin = amount of heat supplied to steam in boiler from a high temperature source (furnace)

Qout = amount of heat rejected from steam in condenser to a low temperature sink (the atmosphere, a river, etc.)

Wout = amount of work delivered by steam as it expands in turbine

Win = amount of work required to compress water to boiler pressure

Above the quantities are indicated with in and out, and they are all positive The net work output of this power plant is simply the difference between the total work output of the plant and the total work input.

Wnet out = Wout – W in 3-1

The net work can also be determined from the heat transfer data alone. For a closed system undergoing a cycle, the change in internal energy ΔU is zero, and therefore the net work output of the system is also equal to the net heat transfer to the system:

Wnet out = Q in – Qout 3-2

4. Thermal Efficiency

In Eq. 3-2, Qout represents the magnitude of the energy wasted in order to complete the cycle. But Qout is never zero; thus, the net work output of a heat engine is always less than the amount of heat input. That is, only part of the heat transferred to the heat engine is converted to work. The fraction of the heat input that is converted to net work output is a measure of the performance of a heat engine and is called the thermal efficiency. Performance or efficiency, in general, can be expressed in terms of the desired output and the required input as

Performance = desired output / required input 3-3

or,

η = Wnet out / Q in 3-4a

or

η = 1 - Qout / Q in 3-4b

Cyclic devices such as heat engines, refrigerators, and heat pumps operate between a high-temperature medium (or reservoir) at temperature TH and a low-temperature medium (or reservoir) at temperature TL. To bring uniformity to the treatment of heat engines, refrigerators, and heat pumps, we define the following two quantities QH = magnitude of heat transfer between cyclic device and high temperature medium at temperature TH

QL = magnitude of heat transfer between cyclic device and low temperature medium at temperature TL

5. The Second Law of Thermodynamics: Kelvin-Planck Statement

No heat engine can convert all the heat it receives to useful work. This limitation on the thermal efficiency of heat engines forms the basis for the Kelvin-Planck statement of the second law of thermodynamics, which is expressed as follows:

• It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work.

The Kelvin-Planck statement can also be expressed as follows:

No heat engine can have a thermal efficiency of 100 percent, or for a power plant to operate, the working fluid must exchange heat with the environment as well as the furnace.

Note: The impossibility of having a 100 percent efficient heat engine is not due to friction or other dissipative effects. It is a limitation that applies to both the idealized and the actual heat engines.

a. Refrigerators and Heat Pumps

We all know from experience that heat flows in the direction of decreasing temperature, i.e., from high-temperature mediums to low temperature ones. This heat transfer process occurs in nature without requiring any devices. The reverse process, however, cannot occur by itself. The transfer of heat from a low-temperature medium to a high-temperature one requires special devices called refrigerators.

Refrigerators, like heat engines, are cyclic devices. The working fluid used

in the refrigeration cycle is called a refrigerant. The most frequently used

refrigeration cycle is the vapor-compression refrigeration cycle which involves

four main components: a compressor, a condenser, an expansion valve, and an

evaporator, as shown below

The refrigerant enters the compressor as a vapor and is compressed to the condenser pressure. It leaves the compressor at a relatively high temperature and cools down and condenses as it flows through the coils of the condenser by rejecting heat to the surrounding medium. It then enters a capillary tube where its pressure and temperature drop drastically due to the throttling effect. The low-temperature refrigerant then enters the evaporator, where it evaporates by absorbing heat from the refrigerated space. The cycle is completed as the refrigerant leaves the evaporator and reenters the compressor.

In a household refrigerator, the freezer compartment where heat is picked up by the refrigerant serves as the evaporator and the coils behind the refrigerator where heat is dissipated to the kitchen air as the condenser.

A refrigerator is shown schematically below. Here QL is the magnitude of the heat removed from the refrigerated space at temperature TL,, QH is the magnitude of the heat rejected to the warm environment at temperature TH, and Wnet,in is the net work input to the refrigerator. As discussed before, QL and QH represent magnitudes and so are positive quantities.

b. Coefficient of Performance

The efficiency of a refrigerator is expressed in terms of the coefficient of performance (COP), denoted by COPR. The objective of a refrigerator is to remove heat (QL) from the refrigerated space. To accomplish this objective, it requires a work input of W net,in Then the COP of a refrigerator can be expressed as

COPR = desired output /required input = QL / W net,in 3-7a

or

Note: The value of COPR can be greater than unity. That is, the amount of heat removed from the refrigerated space can be greater than the amount of work input. This is in contrast to the thermal efficiency, which can never be greater than 1. In fact, one reason for expressing the efficiency of a refrigerator by another term-the coefficient of performance- is the desire to avoid the oddity of having efficiencies greater than unity.

c. Heat Pumps

Another device that transfers heat from a low-temperature medium to a high-temperature one is the heat pump. Refrigerators and heat pumps operate on the same cycle but differ in their objectives. The objective of a refrigerator is to maintain the refrigerated space at a low temperature by removing heat from it. Discharging this heat to a higher-temperature medium is merely a necessary part of the operation, not the purpose. The objective of a heat pump, however, is to maintain a heated space at a high temperature. This is accomplished by absorbing heat from a low-temperature source, such as well water or cold outside air in winter, and supplying this heat to the high-temperature medium such as a house.

An ordinary refrigerator that is placed in the window of a house with its door open to the cold outside air in winter will function as a heat pump since it will try to cool the outside by absorbing heat from it and rejecting this heat into the house through the coils behind it. The measure of performance of a heat pump is also expressed in terms of the coefficient of performance COPHP, defined as

3-8a

The relation implies that the coefficient of performance of a heat pump is always greater than unity since COPR is a positive quantity. That is, a heat pump will function, at worst, as a resistance heater, supplying as much energy to the house as it consumes. In reality, however, part of QH is lost to the outside air through piping and other devices, and COPHP may drop below unity when the outside air temperature is too low. When this happens, the system usually switches to a resistance heating mode. Most heat pumps in operation today have seasonally averaged COP of 2 to 3.

6. The Second Law of Thermodynamics: Clausius Statement

Clausius statement is related to refrigerators or heat pumps. The Clausius statement is expressed as follows:

It is impossible to construct a device that operates in a cycle and produces no effect other than the transfer of heat from a lower temperature body to a higher-temperature body

It is common knowledge that heat does not, spontaneously, flow from a cold medium to a warmer one. The Clausius statement does not imply that a cyclic device that transfers heat from a cold medium to a warmer one is impossible to construct. In fact, this is precisely what a common household refrigerator does. It simply states that a refrigerator will not operate unless its compressor is driven by an external power source, such as an electric motor. This way, the net effect on the surroundings involves the consumption of some energy in the form of work, in addition to the transfer of heat from a colder body to a warmer one. That is, it leaves a trace in the surroundings. Therefore, a household refrigerator is in complete compliance with the Clausius statement of the second law. Both the Kelvin-Planck and the Clausius statements of the second law are negative statements, and a negative statement cannot be proved. Like any other physical law, the second law of thermodynamics is based on experimental observations. To date, no experiment has been conducted that contradicts the second law, and this should be taken as sufficient evidence of its validity.

7. Equivalence of the Two Statements

The Kelvin-Planck and the Clausius statements are equivalent in their consequences, and either statement can be used as the expression of the second law of thermodynamics. Any device that violates the Kelvin-Planck statement also violates the Clausius statement, and vice versa. This can be demonstrated as follows:

• Kelvin-Planck statement: No heat engine can have a thermal efficiency of 100 percent, or for a power plant to operate, the working fluid must exchange heat with the environment as well as the furnace.

• Clausius statement: It is impossible to construct a device that operates in a cycle and produces no effect other than the transfer of heat from a lower temperature body to a higher-temperature body

Consider the heat-engine-refrigerator combination shown in figure below left, operating between the same two reservoirs.

The heat engine is assumed to have, in violation of the Kelvin-Planck statement, a thermal efficiency of 100 percent, and therefore it converts all the heat QH it receives to work W. This work is now supplied to a refrigerator that removes heat in the amount of QL from the low-temperature reservoir and rejects heat in the amount of QL + QH to the hightemperature reservoir. During this process, the high-temperature reservoir receives a net amount of heat QL (the difference between QL + QH and QH). Thus the combination of these two devices can be viewed as a refrigerator, as shown in figure at right that transfers heat in an amount of QL from a cooler body to a warmer one without requiring any input from outside. This is clearly a violation of the Clausius statement. Therefore, a violation of the Kelvin-Planck statement results in the violation of the Clausius statement.

It can also be shown in a similar manner that a violation of the Clausius statement leads to the violation of the Kelvin-Planck statement. Therefore, the Clausius and the Kelvin- Planck statements are two equivalent expressions of the second law of thermodynamics.

8. Reversible and Irreversible Processes

The second law of thermodynamics states that no heat engine can have an efficiency of 100 percent. Then what is the highest efficiency that a heat engine can possibly have? Before we can answer this question, we need to define an idealized process first, which is called the reversible process.

The processes that were discussed above occurred in a certain direction. Once having taken place, these processes cannot reverse themselves spontaneously and restore the system to its initial state. For this reason, they are classified as irreversible processes. Once a cup of hot coffee cools, it will not heat up retrieving the heat it lost from the surroundings. If it could, the surroundings, as well as the system (coffee), would be restored to their original condition, and this would be a reversible process.

A reversible process is defined as a process that can be reversed without leaving any trace on the surroundings. That is, both the system and the surroundings are returned to their initial states at the end of the reverse process. This is possible only if the net heat and net work exchanged between the system and the surroundings is zero for the combined (original and reverse) process. Processes that are not reversible are called irreversible processes.

It should be pointed out that a system can be restored to its initial state following a process, regardless of whether the process is reversible or irreversible. But for reversible processes, this restoration is made without leaving any net change on the surroundings, whereas for irreversible processes, the surroundings usually do some work on the system and therefore will not return to their original state. Reversible processes actually do not occur in nature. They are merely idealizations of actual processes. Reversible processes can be approximated by actual devices, but they can never be achieved. That is, all the processes occurring in nature are irreversible. You may be wondering, then, why we are bothering with such fictitious processes. There are two reasons. First, they are easy to analyze, since a system passes through a series of equilibrium states during a reversible process; second, they serve as idealized models to which actual processes can be compared.

Engineers are interested in reversible processes because work-producing devices such as car engines and gas or steam turbines deliver the most work, and work-consuming devices such as compressors, fans, and pumps require least work when reversible processes are used instead of irreversible ones. Reversible processes can be viewed as theoretical limits for the corresponding irreversible ones. Some processes are more irreversible than others. We may never be able to have a reversible process, but we may certainly approach it. The more closely we approximate a reversible process, the more work delivered by a work-producing device or the less work required by a work-consuming device.

The concept of reversible processes leads to the definition of second-law efficiency for actual processes, which is the degree of approximation to the corresponding reversible processes. This enables us to compare the performance of different devices that are designed to do the same task on the basis of their efficiencies. The better the design, the

lower the irreversibilities and the higher the second-law efficiency.

a. Irreversibilities

The factors that cause a process to be irreversible are called irreversibilities. They include

• friction,

• unrestrained expansion,

• mixing of two gases,

• heat transfer across a finite temperature difference,

• electric resistance,

• inelastic deformation of solids, and

• chemical reactions.

The presence of any of these effects renders a process irreversible. A reversible process involves none of these. Some of the frequently encountered irreversibilities are discussed briefly below.

b. Friction

Friction is a familiar form of irreversibility associated with bodies in motion. When two bodies in contact are forced to move relative to each other, a friction force that opposes the motion develops at the interface of these two bodies, and some work is needed to overcome this friction force. The energy supplied as work is eventually converted to heat during the process and is transferred to the bodies in contact, as evidenced by a temperature rise at the interface. When the direction of the motion is reversed, the bodies will be restored to their original position, but the interface will not cool (during the process), and heat will not be converted back to work. Instead, more of the work will be converted to heat while overcoming the friction forces which also oppose the reverse motion. Since the system (the moving bodies) and the surroundings cannot be returned to their original states, this process is irreversible. Therefore, any process that involves friction is irreversible. The larger the friction forces involved, the more irreversible the process is.

Friction does not always involve two solid bodies in contact. It is also encountered between a fluid and solid and even between the layers of a fluid moving at different velocities. A considerable fraction of the power produced by a car engine is used to overcome the friction (the drag force) between the air and the external surfaces of the car, and it eventually becomes part of the internal energy of the air. It is not possible to reverse this process and recover that lost power, even though doing so would not violate the conservation of energy principle.

c. Non-Quasi-Equilibrium Expansion and Compression

In Part 1 we defined a quasi-equilibrium process as one during which the system remains infinitesimally close to a state of equilibrium at all times. Consider a frictionless adiabatic piston-cylinder device that contains a gas. Now the piston is pushed into the cylinder, compressing the gas. If the piston velocity is not very high, the pressure and the temperature will increase uniformly throughout the gas. Since the system is always maintained at a state close to equilibrium, this is a quasi-equilibrium process.

Now the external force on the piston is slightly decreased, allowing the gas to expand.The expansion process will also be quasi-equilibrium if the gas is allowed to expand slowly. When the piston returns to its original position, all the boundary (P dV) work done on the gas during compression is returned to the surroundings during expansion. That is, the net work for the combined process is zero. Also, there has been no heat transfer involved during this process, and thus both the system and the surroundings will return to their initial states at the end of the reverse process. Therefore, the slow frictionless adiabatic expansion or compression of a gas is a reversible process.

Now let us repeat this adiabatic process in a non-quasi-equilibrium manner, as shown below.

1. If the piston is pushed in very rapidly, the gas molecules near the piston face will not have sufficient time to escape, and they will pile up in front of the piston. This will raise the pressure near the piston face, and as a result, the pressure there will be higher than the pressure in other parts of the cylinder. The non-uniformity of pressure will render this process non-quasi-equilibrium. The actual boundary work is a function of pressure, as measured at the piston face. Because of this higher pressure value at the piston face, a non-quasi-equilibrium compression process will require a larger work input than the corresponding quasi-equilibrium one.

2. When the process is reversed by letting the gas expand rapidly, the gas molecules in the cylinder will not be able to follow the piston as fast, thus creating a lowpressure region before the piston face. Because of this low-pressure value at the piston face, a non-quasi-equilibrium process will deliver less work than a corresponding reversible one.

3. Consequently, the work done by the gas during expansion is less than the work done by the surroundings on the gas during compression, and thus the surroundings have a net work deficit. When the piston returns to its initial position, the gas will have excess internal energy, equal in magnitude to the work deficit of the surroundings.

4. The system can easily be returned to its initial state by transferring this excess internal energy to the surroundings as heat. But the only way the surroundings can be returned to their initial condition is by completely converting this heat to work, which can only be done by a heat engine that has an efficiency of 100 percent. This, however, is impossible to do, even theoretically, since it would violate the second law of thermodynamics.

5. Since only the system, not both the system and the surroundings can be returned to its initial state, we conclude that the adiabatic non-quasi-equilibrium expansion or compression of a gas is irreversible.

Another example of non-quasi-equilibrium expansion processes is the unrestrained expansion of a gas separated from a vacuum by a membrane, as shown below.

When the membrane is punctured, the gas fills the entire tank. The only way to restore the system to its original state is to compress it to its initial volume, while transferring heat from the gas until it reaches its initial temperature. From the conservation of energy considerations, it can easily be shown that the amount of heat transferred from the gas equals the amount of work done on the gas by the surroundings. The restoration of the surroundings involves conversion of this heat completely to work, which would violate the second law. Therefore, unrestrained expansion of a gas is an irreversible process.

d. Heat Transfer

Another form of irreversibility familiar to us all is heat transfer through a finite temperature difference. Consider a can of cold soda left in a warm room. Heat will flow from the warmer room air to the cooler soda. The only way this process can be reversed and the soda restored to its original temperature is to provide refrigeration, which requires some work input. At the end of the reverse process, the soda will be restored to its initial state, but the surroundings will not be.

The internal energy of the surroundings will increase by an amount equal in magnitude to the work supplied to the refrigerator. The restoration of the surroundings to its initial state can be done only by converting this excess internal energy completely to work, which is impossible to do without violating the second law. Since only the system, not both the system and the surroundings, can be restored to its initial condition, heat transfer through a finite temperature difference is an irreversible process.

Heat transfer can occur only when there is a temperature difference between a system and its surroundings. Therefore, it is physically impossible to have a reversible heat transfer process. But a heat transfer process becomes less and less irreversible as the temperature difference between the two bodies approaches zero. Then heat transfer through a differential temperature difference dT can be considered to be reversible. As dT approaches zero, the process can be reversed in direction (at least theoretically) without requiring any refrigeration. Notice that reversible heat transfer is a conceptual process and cannot be duplicated in the laboratory.

The smaller the temperature difference between two bodies, the smaller the heat transfer rate will be. When the temperature difference is small, any significant heat transfer will require a very large surface area and a very long time. Therefore, even though approaching reversible heat transfer is desirable from a thermodynamic point of view, it is impractical and not economically feasible.

e. Internally and Externally Reversible Processes

A process is an interaction between a system and its surroundings, and a reversible process involves no irreversibilities associated with either of them. A process is called internally reversible if no irreversibilities occur within the boundaries of the system during the process. During an internally reversible process, a system proceeds through a series of equilibrium states, and when the process is

reversed, the system passes through exactly the same equilibrium states while returning to its initial state. That is, the paths of the forward and reverse processes coincide for an internally reversible process. The quasi-equilibrium process discussed earlier is an example of an internally reversible process.

A process is called externally reversible if no irreversibilities occur outside the system boundaries during the process. Heat transfer between a reservoir and a system is an externally reversible process if the surface of contact between the system and the reservoir is at the temperature of the reservoir.

A process is called totally reversible, or simply reversible, if it involves no irreversibilities within the system or its surroundings.A totally reversible process involves no heat transfer through a finite temperature difference, no non-quasi-equilibrium changes, and no friction or other dissipative effects.

As an example, consider the transfer of heat to two identical systems that are undergoing a constant-pressure (thus constant-temperature) phase-change process, as shown in figure below. Both processes are internally reversible, since both take place isothermally and both pass through exactly the same equilibrium states. The first process shown is externally reversible also, since heat transfer for this process takes place through an infinitesimal temperature difference dT. The second process, however, is externally irreversible, since it involves heat transfer through a finite temperature difference ΔT.

9. THE CARNOT CYCLE

We mentioned earlier that heat engines are cyclic devices and that the working fluid of a heat engine returns to its initial state at the end of each cycle. Work is done by the working fluid during one part of the cycle and on the working fluid during another part. The difference between these two is the net work delivered by the heat engine. The efficiency of a heat-engine cycle greatly depends on how the individual processes that make up the cycle are executed. The net work, thus the cycle efficiency, can be maximized by using processes that require the least amount of work and deliver the most, that is, by using reversible processes.

Therefore, it is no surprise that the most efficient cycles are reversible cycles, i.e., cycles that consis entirely of reversible processes.

Reversible cycles cannot be achieved in practice because the irreversibilities associated with each process cannot be eliminated. However, reversible cycles provide upper limits on the performance of real cycles. Heat engines and refrigerators that work on reversible cycles serve as models to which actual heat engines and refrigerators can be compared. Reversible cycles also serve as starting points in the development of actual cycles and are modified as needed to meet certain requirements. Probably the best known reversible cycle is the Carnot cycle, first proposed in 1824 by a French engineer Sadi Carnot. The theoretical heat engine that operates on the Carnot cycle is called the Carnot heat engine.

The Carnot cycle is composed of four reversible processes-two isothermal and two adiabatic. Consider a closed system that consists of a gas contained in an adiabatic pistoncylinder device, as shown in figure below. The insulation of the cylinder head is such that it may be removed to bring the cylinder into contact with reservoirs to provide heat transfer.

The four reversible processes that make up the Carnot cycle are as follows:

Reversible isothermal expansion (process 1-2, TH = constant). Initially (state 1) the temperature of the gas is TH, and the cylinder head is in close contact with a source at temperature TH. The gas is allowed to expand slowly, doing work on the surroundings. As the gas expands, the temperature of the gas tends to decrease. But as soon as the temperature drops by an infinitesimal amount dT, some heat flows from the reservoir into the gas, raising the gas temperature to Tll .Thus, the gas temperature is kept constant at TH. Since the temperature difference between the gas and the reservoir never exceeds a differential amount dT, this is a reversible heat transfer process. It continues until the piston reaches

position 2. The amount of total heat transferred to the gas during this process is QH. For an ideal gas,

3-9

• Reversible adiabatic expansion (process 2-3, temperature drops from TH to TL).At state 2, the reservoir that was in contact with the cylinder head is removed and replaced by insulation so that the system becomes adiabatic. The gas continues to expand slowly, doing work on the surroundings until its temperature drops from TH to TL (state 3). The piston is assumed to be frictionless and the process to be quasi-equilibrium, so the process is reversible as well as adiabatic. Since the gas is ideal and the process is adiabatic, Q = 0, So, by the first law

W23 = ΔU = nCV (TL – TH) 3-10

• Reversible isothermal compression (process 3-4, TL = constant). At state 3, the insulation at the cylinder head is removed, and the cylinder is brought into contact with a sink at temperature TL. Now the piston is pushed inward by an external force, doing work on the gas. As the gas is compressed, its temperature tends to rise. But as soon as it rises by an infinitesimal amount dT, heat flows from the gas to the sink, causing the gas temperature to drop to TL. Thus, the gas temperature is maintained constant at TL. Since the temperature difference between the gas and the sink never exceeds a differential amount dT, this is a reversible heat transfer process. It continues until the piston reaches position 4. The amount of heat rejected from the gas during this process is QL.

3-11

• Reversible adiabatic compression (process 4-1, temperature rises from TL to TH). State 4 is such that when the low-temperature reservoir is removed and the insulation is put back on the cylinder head and the gas is compressed in a reversible manner, the gas returns to its initial state (state 1). The temperature rises from TL to TH during this reversible adiabatic compression process, which completes the cycle.

W41 = ΔU = nCV (TH – TL) 3-12

The P-V diagram of this cycle is shown below.

Remembering that on a P-V diagram the area under the process curve represents the boundary work for quasi-equilibrium (internally reversible) processes, we see that the area under curve 1-2-3 is the work done by the gas during the expansion part of the cycle, and the area under curve 3-4-1 is the work done on the gas during the compression part of the cycle. The area enclosed by the path of the cycle (area 1-2-3-4-1) is the difference between these two and represents the net work done during the cycle.The total work produced during the cycle is the sum of individual for in each process.

By adding the above four equations we get

3-13

From the first law,

ΔU = Q + W = 0

or

Q = -W,

Then

3-14

Thus, the work output of the engine is equal to the heat absorbed by the system. Since V1 and V4 lie on one adiabat, and V2 and V3 on the other we can write

which gives

Dividing Eq 3-15 by 3-9 we get

The efficiency of a Carnot engine operating in reversible cycles is therefore

It should be noted that no heat engine is 100% efficient. If the rejected heat were included as part of its output, the efficiency of every engine would be 100%.

The above definition of efficiency applies to every type of heat engine; it is not

restricted to the Carnot engine only.

Since every step in this cycle is carried out reversibly, the maximum possible work is obtained for the particular working substance and temperature considered. It can be shown that no other engine working between the same two temperatures can convert thermal energy to mechanical energy with a greater efficiency than does the Carnot engine. Other reversible engines, in fact, will have the same efficiency as the Carnot engine.

Notice that if we acted stingily and compressed the gas at state 3 adiabatically instead of isothermally in an effort to save QL, we would end up back at state 2, retracing the process path 3-2. By doing so we would save QL, but we would not be able to obtain any net work output from this engine. This illustrates once more the necessity of a heat engine exchanging heat with at least two reservoirs at different temperatures to operate in a cycle and produce a net amount of work.

a. The Reversed Carnot Cycle

The Carnot heat-engine cycle described above is a totally reversible cycle. Therefore, all the processes that comprise it can be reversed, in which case it becomes the Carnot refrigeration cycle. This time, the cycle remains exactly the same, except that the directions of any heat and work interactions are reversed: Heat in the amount of QL is absorbed from the low-temperature reservoir, heat in the amount of QH is rejected to a high-temperature reservoir, and a work input of Wnet,in is required to accomplish all this.

Remember, for a refrigerator, the efficiency is termed Coefficient of Performance (see Eq. 3-6) and can be larger than 100%. Being a reversible cycle, the Carnot cycle is the most efficient cycle operating between two specified temperature limits. Even though the Carnot cycle cannot be achieved in reality, the efficiency of actual cycles can be improved by attempting to approximate the Carnot cycle more closely.

b. THE CARNOT PRINCIPLES

The second law of thermodynamics places limitations on the operation of cyclic devices as expressed by the Kelvin-Planck and Clausius statements. A heat engine cannot operate by exchanging heat with a single reservoir, and a refrigerator cannot operate without a net work input from an external source. We can draw valuable conclusions from these statements. Two conclusions pertain to the thermal efficiency of reversible and irreversible (i.e., actual) heat engines, and they are known as the Carnot principles.

They are expressed as follows:

· The efficiency of an irreversible heat engine is always less than the efficiency of a reversible one operating between the same two reservoirs.

· The efficiencies of all reversible heat engines operating between the same two reservoirs are the same.

These two statements can be proved by demonstrating that the violation of either statement results in the violation of the second law of thermodynamics. To prove the first statement, consider two heat engines operating between the same reservoirs, as shown below.

A. A reversible and an irreversible heat engine operating between the same two reservoirs (the reversible heat engine is then reversed to run as a refrigerator)

B. The equivalent combined system

One engine is reversible, and the other is irreversible. Now each engine is supplied with the same amount of heat QH. The amount of work produced by the reversible heat engine is Wrev, and the amount produced by the irreversible one is Wirrev.

In violation of the first Carnot principle, we assume that the irreversible heat engine is more efficient than the reversible one (that is, ηirrev > ηrev) and thus delivers more work than the reversible one. Now let the reversible heat engine be reversed and operate as a refrigerator. This refrigerator will receive a work input of Wrev and reject heat to the hightemperature reservoir. Since the refrigerator is rejecting heat in the amount of QH to the high-temperature reservoir and the irreversible heat engine is receiving the same amount of heat from this reservoir, the net heat exchange for this reservoir is zero. Thus it could be eliminated by having the refrigerator discharge QH directly into the irreversible heat engine.

Now considering the refrigerator and the irreversible engine together, we have an engine that produces a net work in the amount of Wirrev – Wrev while exchanging heat with a single reservoir-a violation of the Kelvin-Planck statement of the second law. Therefore, our initial assumption that ηirrev > ηrev is incorrect. Then we conclude that no heat engine can be more efficient than a reversible heat engine operating between the same reservoirs.

· Note: The second Carnot principle can also be proved in a similar manner. This time,let us replace the irreversible engine by another reversible engine that is more efficient and thus delivers more work than the first reversible engine. By following through the same reasoning as above, we will end up having an engine that produces a new amount of work while exchanging heat with a single reservoir, which is a violation of the second law. Therefore we conclude that no reversible heat engine can be more efficient than another reversible heat engine operating between the same two reservoirs, regardless of how the cycle is completed or the kind of working fluid used.

10. THERMODYNAMIC EFFICIENCY

We have shown that

all engines operating in a reversible and cyclic manner between the same two temperatures will possess the same thermodynamic efficiency, whatever the working substance

. Eq. 3-18 gives the efficiency of a reversible engine. The efficiency of a reversible heat engine or any reversible machine is thus determined by the temperature of the heat introduced and the temperature of the heat discharged. It appears that, in order to have an engine of 100% efficiency, either the temperature TH must be infinite or the temperature TL1 must be very small. None of these hoices are feasible though. From Equation 3-18 we have

This shows that the absolute magnitudes of the quantities of heat absorbed and rejected are proportional to the temperatures of the heat reservoirs in either cycle. We may generalize Equation 3-9 as

(3-20)

If the ideal gas in the Carnot cycle is considered as being carried through a series of infinitesimally small steps throughout the cycle, then Eq. 3-20 becomes

(3-21)

we will see later that by definition,

(3-22)

is the entropy of the system,

Thus

∫dS =0 (3-23)

11. OTTO CYCLE THE IDEAL CYCLE FOR SI ENGINES

The Otto cycle is the ideal cycle for spark-ignition reciprocating engines. It is named after Nikolaus A. Otto, who built a successful four-stroke engine in 1876 in Germany using the cycle proposed by Frenchman Beau de Rochas in 1862. In most spark-ignition engines, the piston executes four complete strokes (two mechanical cycles) within the cylinder, and the crankshaft completes 2 revolutions for each thermodynamic cycle. These engines are called four-stroke internal combustion engines. A schematic of each stroke as well as a P-V diagram for an actual four-stroke spark-ignition engine is shown below (part a).

· Initially, both the intake and the exhaust valves are closed, and the piston is at its lowest position (BDC). During the compression stroke, the piston moves upward, compressing the air-fuel mixture

· Shortly before the piston reaches its highest position (TCD), the spark plug fires and the mixture ignites, increasing the pressure and temperature of the system.

· The high-pressure gases force the piston down, which in turn forces the crankshaft to rotate, producing a useful work output during the expansion or power stroke. At the end of this stroke, the piston is at its lowest position (the completion of the first mechanical cycle), and the cylinder is filled with combustion products.

· Now the piston moves upward one more time, purging the exhaust gases through the exhaust valve (the exhaust stroke), and down a second time, drawing in fresh air-fuel mixture through the intake valve (the intake stroke). Notice that the pressure in the cylinder is slightly above the atmospheric value during the exhaust stroke and slightly below during the intake stroke.

In two-stroke engines, all four functions described above are executed in just two strokes: the power stroke and the compression stroke. In these engines, the crankcase is sealed, and the outward motion of the piston is used to slightly pressurize the air-fuel mixture in the crankcase, as shown below.

Also, the intake and exhaust valves are replaced by openings in the lower portion of the cylinder wall. During the latter part of the power stroke, the piston uncovers first the exhaust port, allowing the exhaust gases to be partially expelled, and then the intake port, allowing the fresh air-fuel mixture to rush in and drive most of the remaining exhaust gases out of the cylinder. This mixture is then compressed as the piston moves upward during the compression stroke and is subsequently ignited by a spark plug.

The two-stroke engines are generally less efficient than their four stroke counterparts because of the incomplete expulsion of the exhaust gases and the partial expulsion of the fresh air-fuel mixture with the exhaust gases. However, they are relatively simple and inexpensive, and they have high power-to-weight and power-to-volume ratios, which make them suitable for applications requiring small size and weight such as formotorcycles, chain saws, and lawn mowers.

Note: Advances in several technologies-such as direct fuel injection, stratified charge combustion, and electronic controls-brought about a renewed interest in two-stroke engines, which can offer high performance and fuel economy while satisfying the future stringent emission requirements. For a given weight and displacement, a well-designed two-stroke engine can provide significantly more power than its four-stroke counterpart because two-stroke engines produce power on every engine revolution instead of every other one. In the new two-stroke engines under development, the highly atomized fuel spray that is injected with compressed air into the combustion chamber towards the end of the compression stroke burns much more completely. The fuel is sprayed after the exhaust valve is closed, which prevents unburned fuel from being ejected into the atmosphere.

With stratified combustion, the flame, which is initiated by igniting a small amount of rich fuel/air mixture near the spark plug, propagates through the combustion chamber filled with much leaner mixture, and this results in much cleaner combustion. Also, the advances in electronics made it possible to ensure the optimum operation under varying engine load and speed conditions. Major car companies have research programs underway on twostroke engines which are expected to make a comeback in the near future.

The thermodynamic analysis of the actual four-stroke or two-stroke cycles described above is not a simple task. However, the analysis can be simplified significantly if the airstandard assumptions are utilized. The resulting cycle which closely resembles the actual

operating conditions is the ideal Otto cycle. It consists of four internally reversible processes:

· 1-2 adiabatic (isentropic) compression

· 2-3 V = constant heat addition

· 3-4 adiabatic (isentropic) expansion

· 4-1 V = constant heat rejection

Note: As we will see later, the entropy of a fixed mass will not change during an internally reversible, adiabatic process, which is called isentropic (constant entropy) process. An isentropic process appears as a vertical line on a T-S diagram. The execution of the Otto cycle in a piston-cylinder device together with a P-V diagram is illustrated in the figure above (part b). The Otto cycle is executed in a closed system, and thus the first-law relation for any of the processes is expressed as

ΔU = Q + W 3-31

No work is involved during the two heat transfer processes since both take place at constant volume. Therefore, heat transfer to and from the working fluid can be expressed, under the cold-air-standard assumptions, as

Qin = Q23 = U3 – U2 = CV (T3 – T2) 3-32

Qout = - Q41 = - (U1 – U4) = CV (T4 – T1)

Then the thermal efficiency of the ideal-air-standard Otto cycle becomes

Processes 1-2 and 3-4 are adiabatic and V2 = V3 and V4 = V1. Thus

3-33

Substituting these equations into the thermal efficiency relation and simplifying we get

3-34

where r is the compression ratio

r = V1/V2

Equation 3-34 shows that under the cold-air-standard assumptions,

· the thermal efficiency of an ideal Otto cycle depends on the compression ratio of the engine and the specific heat ratio (γ) of the working fluid (if different from air).

· The thermal efficiency of the ideal Otto cycle increases with both the compression ratio and the specific heat ratio. This is also true for actual sparkignition internal combustion engines. A plot of thermal efficiency versus the compression ratio is given below for γ = 1.4, which is the specific-heat-ratio value of air at room temperature.

· For a given compression ratio, the thermal efficiency of an actual spark-ignition engine will be less than that of an ideal Otto cycle because of the irreversibilities, such as friction, and other factors such as incomplete combustion.

We can observe from the graph that the thermal efficiency curve is rather steep at low compression ratios but flattens out starting with a compression ratio value of about 8.

· Therefore, the increase in thermal efficiency with the compression ratio is not that pronounced at high compression ratios. Also,

· when high compression ratios are used, the temperature of the air-fuel mixture rises above the autoignition temperature of the fuel (the temperature at which the fuel ignites without the help of a spark) during the combustion process, causing an early and rapid burn of the fuel at some point or points ahead of the flame front, followed by almost instantaneous inflammation of the end gas. This premature ignition of the fuel, called autoignition, produces an audible noise, which is called engine knock.

Note (optional): Autoignition in spark- ignition engines cannot be tolerated because it hurts performance and can cause engine damage. The requirement that autoignition not be allowed places an upper limit on the compression ratios that can be used in spark-ignition internal combustion engines.

Improvement of the thermal efficiency of gasoline engines by utilizing higher compression ratios (up to about 12) without facing the autoignition problem has been made possible by using gasoline blends that have good antiknock characteristics, such as gasoline mixed with tetraethyl lead. Tetraethyl lead has been added to gasoline since the 1920s because it is the cheapest method of raising the octane rating, which is a measure of the engine knock resistance of a fuel. Leaded gasoline, however, has a very undesirable side effect: it forms compounds during the combustion process that are hazardous to health and pollute the environment. In an effort to combat air pollution, the government adopted a policy in the mid-1970s that resulted in the eventual phase-out of the leaded gasoline. Unable to use lead, the refiners developed other, more elaborate techniques to improve the antiknock characteristics of the gasoline. Most cars made since 1975 have been designed to use unleaded gasoline, and the compression ratios had to be lowered to avoid engine knock.

The thermal efficiency of car engines has decreased somewhat as a result of decreased compression ratios. But, owing to the improvements in other areas (reduction in overall automobile weight, improved aerodynamic design, etc.), today's cars have better fuel economy and consequently get more miles per gallon of fuel. This is an example of how engineering decisions involve compromises, and efficiency is only one of the considerations in reaching a final decision. The second parameter affecting the thermal efficiency of an ideal Otto cycle is the specific heat ratio γ. For a given compression ratio, an ideal Otto cycle using a monatomic gas (such as argon or helium, γ = 1.667) as the working fluid will have the highest thermal efficiency. The specific heat ratio γ, and thus the thermal efficiency of the ideal Otto cycle, decreases as the molecules of the working fluid get larger. In the figure γ is shown as k.

At room temperature it is 1.4 for air, 1.3 for carbon dioxide, and 1.2 for ethane. The working fluid in actual engines contains larger molecules such as carbon dioxide, and the specific heat ratio decreases with temperature, which is one of the reasons that the actual cycles have lower thermal efficiencies than the ideal Otto cycle. The thermal efficiencies of actual spark-ignition engines range from about 25 to 30 percent.

CHAPTER 5

THIRD LAW OF THERMODYNAMIC

Two types of arguments found in the literature should be addressed: those that attempt to show that the attainment of zero absolute temperature is not prohibited by the second law and those that attempt to show that existence of a reservoir at zero absolute temperature does not threaten the second law.

By applying the mathematical formalism of thermodynamics down to and including zero absolute temperature, it has been shown that this temperature can be reached in a finite number of steps15 or that the work required to reduce a system16 to this temperature is finite.17 As shown in Appendix 3A, the mathematical formalism is such that the use of T = 0 as a lower limit of integration gives the appearance of being permissible. A similar condition probably obtains for these arguments which, despite their apparent cogency, are incomplete because the possibility that the existence of a reservoir at zero absolute temperature might pose a threat to the second law was not examined.

Nernst's proof that the unattainability principle is required by the second law was based on the argument that if a reservoir at zero absolute temperature existed, it would be possible to operate a Carnot engine using this reservoir to convert heat taken in at a higher temperature completely into work. This is essentially the argument presented here. The two most damaging objections against this position were based on possible operating difficulties associated with the Carnot cycle.

The first objection calls into question the possibility of carrying out an isothermal process at zero absolute temperature because the effects of heat leaks and frictional heat are much more pronounced at this extreme. This is an objection of degree rather than principle and should carry no weight when it is recognized that the logical structure constituting thermodynamics rests on such idealizations as reversibility, isothermality, and adiabaticity. As these idealizations can never be realized in the physical world, it seems pointless to argue that they would be more difficult to achieve at low temperature.

The second objection points to the ambiguity associated with the isothermal step in the Carnot cycle presumed to occur at zero absolute temperature. Because no heat is rejected, this step would be adiabatic as well as isothermal but it would not necessarily be isentropic for it can only be said that the entropy change is 0/0. It has been argued that when a system attempting to follow a Carnot cycle reaches zero absolute temperature the second law is not threatened because there is no assurance that the system would take the isothermal path and complete the cycle rather than take the adiabatic path and return to a previous state. The emphasis here is misplaced! Because a single violation would vitiate the second law, concern should be directed to the possibility, no matter how small, that the system would take the isothermal path. There is no assurance that this would not occur and therefore the unattainability principle is needed.

Both of these inoperability objections seem to deman a premature reality check. It could be argued that the Carnot engine is simply a concept that is part of the logical, mathematical formalism of thermodynamics and it is rather the final result of the argument which should be subjected to a reality check. In this regard it should be noted that the observed conformance to Eq. (3-5) may be taken as proof that the concept of a Carnot engine is viable in the limit as T approaches zero. This is because the Maxwell relation, Eq. (3-4), can be derived through the agency of a Carnot cycle as was originally shown by Maxwell himself.21

3.5 THE STATUS AND INTERPRETATION OF THE THIRD LAW

We have seen that Eq. (3-1) can be understood only in a logical sense and to that end a derivation showing its descent from the second law has been presented. As this derivation is essentially an elaboration of Nernst's original derivation which was never fully accepted, it is reasonable to expect that it could suffer the same fate. However, whether Eq. (3-1) is regarded as deriving from the second law or whether it is regarded as an additional statement required to save the second law, it is still possible to see it as a logical requirement. It could be stated that Eq. (3-1) is necessary to define the limiting entropy change which we have seen would have the indeterminate form 0/0 otherwise. By reversing the argument presented in Sec. 3.3, it is easily seen that the unattainability principle follows from Eq. (3-1).

Although Eq. (3-1) has now been given thermodynamic justification, its exceptions seem uncomfortably numerous for a thermodynamic relationship and it is therefore appropriate to examine its applicability. This problem has beenaddressed by Simon22 and resolved by the following statement:

“At absolute zero the entropy differences vanish between all those states of a system between which a reversible transition is possible in principle even at the lowest temperatures.”

Simon's statement is completely general, however, in regard to the behavior of glasses the statement of Fowler and Guggenheim23 is more specific.

"For any isothermal process involving only phases in internal equilibrium or, alternatively, if any phase is in frozen metastable equilibrium, provided the process does not disturb this frozen equilibrium, lim T® 0DS = 0."

Simon assigned unquestioned thermodynamic status to Eq. (3-1) and pointed out that the restrictions made explicit in his statement are implicitly made in applying any other thermodynamic relationship. The question is not whether Eq. (3-1) is valid, but rather whether the application of thermodynamics to a particular system is valid. Valid thermodynamic systems must exist in equilibrium states and thus be capable of undergoing reversible processes. As Eq. (3-1) is applied only under the most stringent conditions where "frozen-in," nonequilibrium states are not unexpected, it is natural that it will not seem to possess the unexceptional status accorded to the other laws and relations of thermodynamics. However, this is a problem in the application of thermodynamics and should not call the validity of Eq. (3-1) into question

.Because of the widespread use of the Lewis and Randall convention leading to the convenience of "absolute" entropies and because of the remarkable success in calculating these values via the methods of quantum statistical mechanics, we are tempted to regard entropy as an intrinsic property of matter and thereby seek a physical microscopic interpretation. However, we have seen for the case of liquids, gases, and glasses that this is not a fruitful approach. Equation (3-1) is the most general statement and has been shown to be simply a necessary logical statement. This suggests the view that entropy is merely a defined state function embedded in the logical-mathematical structure of thermodynamics. Thus, it seems appropriate that quantum statistical mechanics yields a representation of entropy in logical rather than physical terms. Because classical thermodynamics neither provides nor requires physical visualization of its functions, entropy is no less useful for want of a microscopic physical interpretation. While this view does not provide the physical insight available through a microscopic interpretation, it is at least free of contradictions.

3.6 ABSOLUTE ENTROPIES

If the requisite calorimetric data are available, the entropy of an ideal gas can, in general, be calculated from

where CP(c), CP(l), and CP(g) are the heat capacities of the crystalline, liquid, and gas phases, DhF and DhV are the latent heats of fusion and vaporization, and TF and TV are the corresponding phase transition temperatures. We have already seen that when elements are assigned zero entropy at zero absolute temperature, the third law specifies the entropy of any substance, So, to be zero at this condition. This convention allows values of "absolute" entropy to be calculated from calorimetric data via the above equation. This is a convenience and not a necessity. In fact, in using this convention the isotope effect has been ignored.

Most elements consist of a mixture of isotopes. From the microscopic point of view, there will be many possible arrangements of these sets of distinguishable particles on the crystal lattice and therefore Wo will be significantly greater than unity. From a macroscopic point of view, we would simply say that there is an entropy of mixing to be considered. Because the isotopic composition of the elements is now well known, the isotopic effect could be dealt with quantitatively. However, as a practical matter it is not necessary to do so because the isotopic composition is expected to remain constant for processes of thermodynamic interest and therefore any correction for this effect would cancel on determining an entropy change.

Using quantum statistical mechanics the Sackur-Tetrode equation, Eq. (2- 21), for the entropy of a monatomic ideal gas was derived in Sec. 2.3.2

This equation yields absolute entropy values in excellent agreement with calorimetric values. Since most of the noble gases are isotopic mixtures, this situation must be considered in applying Eq. (2-21). As Eq. (2-21) shows a dependence of entropy on particle mass, the entropy of each isotope can be calculated separately and a weighted average determined for the mixture. Consistent with the convention used for the calorimetric absolute entropy, the

entropy of mixing the isotopes is ignored. Thus, "absolute" entropies are not absolute.

While the concept of absolute entropy derives from the third law and the Sackur-Tetrode equation yields excellent values of absolute entropy,it will be noticed that it does not show the expected behavior as absolute zero temperature is approached. Instead of approaching zero, the Sackur-Tetrode entropy becomes negatively infinite in this limit. This discrepancy is resolved by the quantum statistical mechanical treatment of ideal gases at extremely low temperatures24 where it is necessary to use different types of statistics. The rules for determining probabilities of quantum states as given in Sec. 2.1 characterize Maxwell-Boltzmann statistics. While this statistical scheme works well for most problems of thermodynamic interest, it fails at very low temperatures where it becomes necessary to use different schemes for determining the probability of quantum states. There are two of these schemes: Fermi-Dirac statistics and Bose-Einstein statistics. Both of these statistics when applied to ideal gases yield expressions showing vanishing entropy at zero absolute temperature.

These entropy expressions are relatively complex, but reduce to the Sackur- Tetrode equation at higher temperatures. In this sense, the Sackur-Tetrode equation can be considered consistent with the third law. As the Sackur-Tetrode equation applies to monatomic gases whose energy can be manifested only as translational energy, a different approach is needed

for polyatomic gases. In accounting for the energy of molecules, it is necessary to include contributions due to the rotation of the molecule and to vibrations between atoms within the molecule as well as translational energy. Each mode of energy is quantized and the energy levels for the rotational and vibrational modes can be calculated from spectroscopic data. Because it is assumed that translational, rotational, and vibrational energy modes are independent, the total energy is additive and the partition function can be written as a product of the translational, rotational, and vibrational partition functions. The thermodynamic functions are expressed in terms of the logarithm of the partition function and therefore the contributions to them from the various energy modes are additive.25 Absolute entropies calculated in this manner are in excellent agreement with calorimetric values and are actually considered to be more reliable. Most ideal-gas thermodynamic property tabulations are based on calculation from spectroscopic data.