Jerry Braunrtein Reactor Chemistrv Division I States, Indistinguishability, and the

Oak Ridge National laboratory Oak Ridge, Tennessee 37830 I Formula S = k h W in Thermodynamics

A recent series of articles in TEE JOURNAL entitled "Is A S Naught or Not (I, 11, and 111)" dealt with a supposed paradox in the evaluation of the en- tropy of a system from the number of quantum states available to the system (1-8). The basis of such an evaluation is the relation S = k in W, which embodies the inter-relationship of three developments of the past century having the greatest significance for chem- istry-quantum mechanics, statistical mechanics, and thermodynamics. It seemed worthwhile, therefore, to consider some of the simpler applications of this relation in order to clarify the use of terms, such as states (in different contexts) and distinguishability (of particles and of states), whose imprecise use can lead to am- biguities.

No originality is claimed for the ideas presented herein, which are based on much more thorough and lucid expositions by, e.g., Mayer and Mayer, (4), and Hill (6). Some recent textbooks of physical chemistry also present these ideas correctly, although (necessarily) in less detail (6). The discussion herein is not intended to provide the ultimate in rigor, but to summarize, per- haps heuristically, the above concepts as they apply to many chemical systems.

The Formula S = k In W

The entropy of a macroscopic system in some particu- lar thermodynamic state is given by the formula

S = k l n W (1)

where W is the number of quantum states of the macro- scopic system consistent with the thermodynamic state of the system. The thermodynamic state of the system is characterized by the specification of a small number of macroscopic physical quantities such as composition, temperature, pressure, etc; or, alternatively, composi-

Research sponsored by the U.S. Atomic Energy Commission under contract with the Union Carbide Corporation.

' I t should be noted that, fortunately, the loqarithm of W is required rather than W itself. Since W is frequently extremely large, e.g., of the order of NN, where N is Avagsdro's number, an error of a multipliative factor N in W causes an error of only 1 part in 6 X loz3 in in W [i.e., In N,NN = In N + N In N = (N + 1)lnNI.

2 Thus, opening a stopcock between two flasks of gas, initially at the same temperature and pressure, leads to an increase of ent,ropy if the particles of gas in one flask are distiuguishable from those in the other, but to no change of entropy if the par- ticles are indistinguishable.

tion, energy, volume, etc. An isolated system, to which eqn. (1) applies, is one in which all of the latter set of quantities are constant.

A quantum state of the macroscopic system is character- ized by the specification of one of the sets of quantum numbers or wave functions accessible to this system. If the particles (e.g., atoms and molecules) are inde- pendent of one another (i.e., non-interacting), a quan- tum state of the macroscopic system is characterized by the specification of all of the quantum numbers of each of the individual particles. Such a specification is obvi- ously impractical for a system containing of the order of Avogadro's number of particles. Furthermore, a great many specifications of the quantum numbers of the in- dividual particles may be consistent with a specified valueof the total energy (and volume) of the macroscopic system. Hence a single thermodynamic state may sam- ple many quantum states of the macroscopic system. It is nevertheless possible in some cases to determine the total number of quantum states of the macroscopic system, i.e., the total number of specifications of the quantum numbers of each of the individual particles.'

Example 1. Entropy of Mixing

As an example of the application of eqn. (I), consider the mixing of a crystal consisting of n~ particles A with another crystal consisting of ne particles B, where n~ and n~ are of the order of N (Avogadro's number) and A and B are distinguishable from one another experi- mentally, e.g., by elementary analysis or isotopic mass.

Particles A are indistinguishable from other particles A, in the sense that two atoms of &He are indistinguish- able or two molecules of Hz (or of DP) are indistinguish- able. Two particles of A in different quantum states are not distinguishable since there is no experiment which can determine, for example, whether A, is in some state or and Az in a different state 8, or vice versa. Any characteristic or label which permitted this would be a characteristic of the particle, rather than of the state, and A1 and A2 would no longer be identical particles. In writing a wave function for a collection of identical particles, correct results,-i.e., results which are consistent with spectroscopic measurements and thermodynamic measurements-are obtained only if the indistinguishability of the particles is incorporated in the wave functions2 (6). Thus, quantum mechanics must take into account the indistinguishability of iden-

Volume 46, Number I I , November 1969 / 719

tical particles in all states. The states must be distin- guishable, a t least in principle, in order for the concept of state to be meaningful. Two different thennodynamic states are distinguishable by, say, different values of temperature, or pressure, etc. Two different quantum states of an electron in a hydrogen atom are distinguish- able by the different values of the (orbital or spin) quantum numbers. Two different quantum states of a macroscopic system of identical particles may be distin- guishable by the differences in the numbers of particles having particular values of their quantum numbers (if the interactions among the particles are weak)-but not by which particles have particular values of the qnan- tum numbers, since this is not physically determinable. If the interactions among the particles are strong, it is not possible to characterize the quantum state of the macroscopic system in terms of the quantum states of the individual particles. For example, it is not possible to characterize the quantum states of an H1 molecule in terms of the quantum numbers of individual H atoms. Hz has its own set of quantum states. Similarly, it is not possible to characterize the quantum states of liquid helium in terms of the quantum numbers of independent helium atoms.

In the case of the crystal of pure A a t O°K, each of the narticles will be in its lowest vibrational energy level. -- If this energy level is non-degenerate, i.e., if there is only one quantum state in the lowest energy level, there will be only one vibrational quantum state available to the crystal. The sites of the crystal lattice are distinguish- able, however, by their position, even if a particle on any site has the same energy as a particle on any other site. Hence the particles are all in different configurational states. But if all of the sites are occupied by particles A, i t is impossible to distinguish which A-particle is on which site. Thus there is only one quantum state available to the macroscopic crystal A and S A = k In 1 = 0. Similarly SB = k In 1 = 0.

If A and B are mixed on nA + ns sites of a crystal lattice, however, it is possible to distinguish whether a particular site is occupied by A or by B. The number of distinguishable configurations of the mixed crystal of A and B is the number of distinguishable ways in which n~ indistinguishable A particles and ns indistinguish- able B particles can be placed on nA + ne distinguish- able sites. For a random (ideal) mixture, this is equal to the number of distinguishable arrangements of nA white balls and n~ black balls in nA + ne numbered boxes and is given by

Interchange of an A-particle with a B-particle between two lattice sites gives rise to a new (i.e., distinguishable) configurational quantum state of the mixed crystal. Interchange of 2 A-particles or of 2 B-particles does not give a new (i.e., physically distinguishable) state.

Equation (1) may be applied to calculate the entropy of the mixed crystal

Making use of Stirling's approximation for the factorials of large numbers

In Nil G N; In Nt - N,

eqn. (3) becomes, for the mixed crystal Sas = k [ ( n ~ + n ~ ) In (n* + ns) - n* Inn* - ns In n~ -

(n* + nd + n* + n ~ 1

N is Avogadro's number, N A and NB are the numbers of moles of particles A and B, and X A and X B are the mole fractions of A and B in the mixture.

The entropy of mixing of A and B, AS, is given by

AS = SAB - SA - S B = -kN [PA In XA + NB In Xsl (5)

Equation (5) gives the correct thermodynamic formula for the entropy of ideal mixing of A with B, provided the constant kin eqn. (1) is Boltzmann's constant (kN = R, the gas constant).

The residual entropy a t O°K of an isotopic mixture corresponds to Example I. This residual entropy must be taken into account in thermodynamic calculations for chemical reactions in which different isotopic ratios of the elements may occur for reactants and products. This residual entropy may be neglected where isotopic fractionation does not occur.

Example II. Entropy of Orientalional Disorder

In the case of a crystal of N particles of type A, where each of the particles can be in either of two states of (nearly) equal energy, e.g., with its spin oriented parallel or anti-parallel to an applied magnetic field

W = 2 N

and eqn. (1) gives S = k In ZN = R in 2. From these two examples, it should be clear that one cannot always write the entropy (as was done in articles I and 111) a3 S = R In W', with W' the number of states of a particle. This latter formulation would be correct only if, as in Example 11, all of the states of each particle could be assigned independently of the states of the other parti- cles. This is not the case in the first example, where occupany of a site by an A-particle precludes the oc- cupancy of that site by a B-parti~le.~

The entropy of orientational disorder of nuclear spins may be neglected except when coupling nuclear spin states with other states (e.g., rotational, as in o- andp-hydrogen) is significant.

The now explained discrepancy of R in 2 between the statistical entropy and the third law entropy of compounds such as CO and N20 also is an example of the entropy of orientational disorder.

Conclusions

The point to be emphasized is that states are dis-

a However, the particles in Example I1 are independent in the sense of not interacting with one another energetically.

The proposed restatement of the Third Law of Thermo- dynamics in articles I and I11 in terms of indistinguishability of particles would imply, incorrectly, that the entropy of a mole of gaseous 4 H ~ is zero, because the atoms are indistinguishable. (Although many different translational energy slates are repre- sented, there is no way (without a Maxwell demon) to distinguish which atom is in which state.)

6 Use of the thermodynamic relation for an isothermal reversible process, AS = q.,,/T, without demonstrating the existence of an isothermal reversible process between the two states in question, as in I, is not justified. May one write this expression for the isothermal transformation of water to ice at 272"K7

720 / Journol of Chemical Edumtion

tinguishable and similar particles are not. Similar particles do not become distinguishable by virtue of oc- cupying different states. It is therefore misleading to refer, as in (I), to "states in which particles are in- distinguishable" (implying the existence of some states in which these same particles become distinguishable). The entropy is zero in the thermodynamic state of a macroscopic system if there is only one quantum state accessible to that system in thermodynamic equilib- r i u m - ~ expressed by eqn. (l)."any "paradoxes" disappear with proper usage of the concept^.^

Literature Cited

(1) CAMPBELL, J. A,, J. CHEM. EDUC., 45,9 (1968). (2) CAMPBELL, J. A., J. CHEM. EDUC., 45,244 (1968). (3) CAMPBELL, J. A,, J. CHEM. EDUC., 45,340 (1968). (4) MAYER, J. E. AND MAYER, M. G., "Sta,tistical Mechanics,"

John Wiley & Sons, New York, 1940. See particularly pp. 63-67 and 92.

(5) HILL, T. L., "Introduction to Statistical Thermodynamics," Addison-Wesley Publishing Co., Reading, Mass., 1960, - -. Chapters 1-2.

(6) See especially EGa~ns, D. F., GREGORY, N. W., HALSEY, G. D.. AND RABINOVITCH. B. S.. "Phvsied Chemistrv." John ~ i l e y & Sans, New 'iork, 1964, 323-7, 337-47:'

Volume 46, Number 1 1 , November 1969 / 721