Because entropy is such an important state function, it is natural to seek a description of its meaning on the microscopic level.
Entropy is sometimes said to be a measure of “disorder.” According to this idea, the entropy increases whenever a closed system becomes more disordered on a microscopic
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4.8 THESTATISTICALINTERPRETATION OFENTROPY 130
state1 state2
reversible Rảq=TbDS
irreversible Rảq=Tb< S
Figure 4.12 Reversible and irreversible paths between the same initial and final equi- librium states of a closed system. The value ofSis the same for both paths, but the values of the integralR
.ảq=Tb/are different.
scale. This description of entropy as a measure of disorder is highly misleading. It does not explain why entropy is increased by reversible heating at constant volume or pressure, or why it increases during the reversible isothermal expansion of an ideal gas. Nor does it seem to agree with the freezing of a supercooled liquid or the formation of crystalline solute in a supersaturated solution; these processes can take place spontaneously in an isolated system, yet are accompanied by an apparentdecreaseof disorder.
Thus we should not interpret entropy as a measure of disorder. We must look elsewhere for a satisfactory microscopic interpretation of entropy.
A rigorous interpretation is provided by the discipline ofstatistical mechanics, which derives a precise expression for entropy based on the behavior of macroscopic amounts of microscopic particles. Suppose we focus our attention on a particular macroscopic equilib- rium state. Over a period of time, while the system is in this equilibrium state, the system at each instant is in a microstate, or stationary quantum state, with a definite energy. The microstate is one that isaccessibleto the system—that is, one whose wave function is com- patible with the system’s volume and with any other conditions and constraints imposed on the system. The system, while in the equilibrium state, continually jumps from one ac- cessible microstate to another, and the macroscopic state functions described by classical thermodynamics are time averages of these microstates.
The fundamental assumption of statistical mechanics is that accessible microstates of equal energy are equally probable, so that the system while in an equilibrium state spends an equal fraction of its time in each such microstate. The statistical entropy of the equilibrium state then turns out to be given by the equation
SstatDklnW CC (4.8.1)
wherekis the Boltzmann constantkDR=NA,W is the number of accessible microstates, andC is a constant.
In the case of an equilibrium state of a perfectly-isolated system of constant internal energyU, the accessible microstates are the ones that are compatible with the constraints and whose energies all have the same value, equal to the value ofU.
It is more realistic to treat an equilibrium state with the assumption the system is in ther- mal equilibrium with an external constant-temperature heat reservoir. The internal energy then fluctuates over time with extremely small deviations from the average valueU, and the accessible microstates are the ones with energies close to this average value. In the language
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CHAPTER 4 THE SECOND LAW
4.8 THESTATISTICALINTERPRETATION OFENTROPY 131
of statistical mechanics, the results for an isolated system are derived with a microcanonical ensemble, and for a system of constant temperature with a canonical ensemble.
A changeSstatof the statistical entropy function given by Eq. 4.8.1 is the same as the change S of the macroscopic second-law entropy, because the derivation of Eq. 4.8.1 is based on the macroscopic relation dSstatDảq=T D.dU ảw/=T with dU andảwgiven by statistical theory. If the integration constant C is set equal to zero, Sstat becomes the third-law entropyS to be described in Chap. 6.
Equation 4.8.1 shows that a reversible process in which entropy increases is accom- panied by an increase in the number of accessible microstates of equal, or nearly equal, internal energies. This interpretation of entropy increase has been described as the spread- ing and sharing of energy11 and as the dispersal of energy.12 It has even been proposed that entropy should be thought of as a “spreading function” with its symbol S suggesting spreading.13,14
11Ref. [100].
12Ref. [96].
13Ref. [97].
14The symbolSfor entropy seems originally to have been an arbitrary choice by Clausius; see Ref. [82].
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PROBLEMS 132
PROBLEMS
An underlined problem number or problem-part letter indicates that the numerical answer appears in Appendix I.
4.1 Explain why an electric refrigerator, which transfers energy by means of heat from the cold food storage compartment to the warmer air in the room, is not an impossible “Clausius de- vice.”
4.2 A system consisting of a fixed amount of an ideal gas is maintained in thermal equilibrium with a heat reservoir at temperatureT. The system is subjected to the following isothermal cycle:
1. The gas, initially in an equilibrium state with volume V0, is allowed to expand into a vacuum and reach a new equilibrium state of volumeV0.
2. The gas is reversibly compressed fromV0toV0. For this cycle, find expressions or values forw,H
ảq=T, andH dS.
4.3 In an irreversible isothermal process of a closed system:
(a) Is it possible forSto be negative?
(b) Is it possible forSto be less thanq=T?
4.4 Suppose you have two blocks of copper, each of heat capacityCV D200:0J K 1. Initially one block has a uniform temperature of300:00K and the other310:00K. Calculate the entropy change that occurs when you place the two blocks in thermal contact with one another and surround them with perfect thermal insulation. Is the sign ofS consistent with the second law? (Assume the process occurs at constant volume.)
4.5 Refer to the apparatus shown in Figs. 3.22 on page 97 and 3.25 on page 99 and described in Probs. 3.3 and 3.8. For both systems, evaluateSfor the process that results from opening the stopcock. Also evaluateR
ảq=Textfor both processes (for the apparatus in Fig. 3.25, assume the vessels have adiabatic walls). Are your results consistent with the mathematical statement of the second law?
water
b
m
Figure 4.13
4.6 Figure 4.13 shows the walls of a rigid thermally-insulated box (cross hatching). Thesystem is the contents of this box. In the box is a paddle wheel immersed in a container of water, connected by a cord and pulley to a weight of mass m. The weight rests on a stop located a distanceh above the bottom of the box. Assume the heat capacity of the system,CV, is
Thermodynamics and Chemistry, 2nd edition, version 7a © 2015 by Howard DeVoe. Latest version:www.chem.umd.edu/thermobook
CHAPTER 4 THE SECOND LAW
PROBLEMS 133
independent of temperature. Initially the system is in an equilibrium state at temperatureT1. When the stop is removed, the weight irreversibly sinks to the bottom of the box, causing the paddle wheel to rotate in the water. Eventually the system reaches a final equilibrium state with thermal equilibrium. Describe areversibleprocess with the same entropy change as this irreversible process, and derive a formula forSin terms ofm,h,CV, andT1.
CHAPTER 5