Deriving S = MnQ Melvyn P. Melrose, University of London, King's College, Strand, London WC2R ZLS, United Kingdom
Most physical chemistry texts do not try to prove that the entrowv .. as defined in statistical thermodvnamics is the same as the entropy in classical thermodynamics. Instead, an inductive justification is usually given: the statistical entropy has certain properties in common with the classical entropy (for example, extension); and the statistical formula gives the same results for ideal gases as calorimetric calculations based on the Third Law. I t must he said that no derivation of Boltzmann's formula can he completely deductive, since it will he necessary for any "proof" to make an assumption about the relationship between statistical mechanics and classical thermodynamics. Nevertheless, not all assumptions are equally plausible: and simply introducing S = hlnSl as a posit leaves the student to wonder whv the entrowv should he a function onlv of 12. a that the formula is generally vaiid. A quite different approach is adopted in some texts of statistical physics (1,2). Here the need to justify S = klnSl is obviated by simply treating the formula as a definition, which is then related tothermalmeasurements without any reference to classical thermodynamics, apart from the First Law. In this approach is a conventional definition of a property S of a closed system which has Sl accessible microstate when its energy lies between E and E 6E. (For a macroscopic system 1nCl is insensitive to the magnitude of 6E, and differs negligibly from in&, where Sld is the number of microstates compatible with the most probable distrihution of energy.) The temperature, T, of the system is defined hy another convention:
+
(2)
The general thermodynamic force in the system (example: pressure) is then defined as the (negative) derivative of the total energy with respect to the appropriate outer variable (example: volume) when inn is constant:
This result requires some justification hecause the operational definition of pressure refers to work done in quasi-static wrocesses. Anweal is made to the Ehrenfest (or adiabatic)
Direct comparison of eqn. (5) with the First Law of thermodynamics identifies T d S with the heat absorbed in a quasistatic process. An appropriate choice of h in eqn. (2) renders T the same as ideal gas temperature, and when this has been demonstrated, the need for a classical treatment of entropy and temperature has been eliminated. Thus, the Second Law and the conditions for equilibrium are seen as consequences of the basic (entropic) postulate of statistical mechanics (1). has encountered temperature and pressure in terms of mac-
is a demonstration that the S in eqn. (5) is the same as the S in eqn. (1) without assuming eqn. (2) and eqn. (3). Of these two assumptions eqn. (3) is the more ubiquitous, since in the context of classical thermodynamics, where P = -(dEldVin. eon. (3) alreadv imwlies that S is a function of Sl alone. lndeed,given eqns. (i), and (3) it is easy to deduce both eon. (2) and e m . . (1). . This is essentiallv the derivation of ~ a y eand r Mayer ( 3 ) ,who used the adiabatic principle to justify eqn. (3). This principle is also tacitly involved in those derivations ( 4 , 5 ) which use the method of the canonical enit is necessary to semble: in order to Drove S = -kZR;lnR; .~.~ assume that reversible work does not affect the canonical distribution numbers pi. What one would prefer is a derivation which did not involve the adiabatic principle. The derivation which follows makes what is possibly the minimum assumption for there to be a relationship between statistical mechanics and thermodynamics: that a svstem in statistical eauilibrium is also in thermodynamic equilibrium. That is, when 1nQhas attained a maximum value in an isolated svstem., all warts . of the svstem which can interact thermally have the same temperature and all arts which can interact mechanicallv have the same pressure. The derivation begins in a manner similar to that of Maver and Maver hut then identifies P (or P I T ) by methods which are different.
(41,
The Derivation
An isolated system is divided into two parts, A and B, by a heat conducting wall. Since the two parts are large enough to he regarded as independent, the energy of the whole system, E,, is given h j E,
E,
+ El, = a constant
(6)
We suppose that when the energy of A is (to within specified limits E,,there are Sl, microstates accessible to A and QI, microstates accessible to B. The number of microstates of the whole system compatible with this particular distrihution of the total energy is given hy
the exact differential
evidently hecomes d E = TdS - P d V
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(5)
so that
Since the volume and the other outer variables of each subsystem are fixed, a change in the value of inn, depends only on changes in the energies of A and B, so that
Since 6E,
=
-6Eh, we can write eqn. (9) as
At equilibrium inn, is maximal with respect to E, and 61nR, vanishes, giving
The conditions for thermal equilibrium expressed by eqn. (11) could easily he generalized by considering an isolated system partitioned into three or more subsystems. Each part of a system in thermal equilihrium will have the same value of (dlnR1dE)v regardless of its volume, composition, and the equilibrium value of its energy. The requirement in classical thermodynamics for thermal equilibrium is that each suhsystem has the same temperature, T. Therefore fixing (alnRIaE), fixes T and no other thermodynamic variable.
The derivation may now be concluded as in Mayer and Mayer (3). Equation (3) having been justified, comparison of eqn. (5) with eqn. (4) yields dS = dlnRl(0T)
(21)
Now d S and dln0 are both exact differentials, so P T must be a constant, or a function of R. Since d S and dlnR are both extensive quantities the latter possibility is eliminated, leaving and Integration then yields the Boltzmann formula, apart from an additive constant which is independent of E and V. The derivation given above does not establish the numerical value of k . This can be achieved hy the standard method of calculating the pressure of an ideal gas in terms of /3 and V, or by an alternative derivation of the Boltzmann formula which identifies P I T rather than P. Alternative Derivation From Eqn. ( 13) Here we consider that the piston sruarating A and B can transfer heat so that both thermal and mechanical equilibrium are attained. The terms on the right-hand side of eqn. (13) both vanish independently, giving
The two sub-systems A and B are now considered to be real gases or fluids separated by a frictionless piston. For asystem not initially in equilibrium, lnR, will change as a result of changes in both the energy and volume of A (6V, = -6Vh):
Since 1nR is a function of E and V we note that, in general,
and V,. ~ lsystems l in thermal and mxhanical equilibrium will have the same values of ( d l n 0 1 a E ) ~and (alnRlaV)~. From classical thermodynamics we know that such systems will have the same temperature and pressure. The most general novel implication of eqn. (24) is
whereupon eqn. (13) can be rewritten as
Now, in any circumstances, we have SE. - SEb 6V, SVb but at equilibrium, when 61nQ = 0, imposing the mathematical condition Inn, = constant upon eqn. (16) is equivalent to requiring that Innbis constant. Hence the conditions for equilibrium are
Since y ( P , T ) is the same function for all fluids, no generality will he lost if an ideal gas is used to identify it. Now the quantum-statistical model for an ideal Boltzmann gas is an assembly of N independent particles occupying a three dimensional 'box' whose volume. V. is ereat enough for the classical approximation to be &lid. a his approximation requires (1)that the one- article enerrv -.states lie in a contin;urn, so that the number of one-particle states, d(r),between 0 and e is proportional to V, and (2) that d(e) >> N when r is of the order of a thermal energy (RTIN). Under these conditions Reif (2) has shown that for the whole assembly of specified total energy E ,
and Consequently We now consider that the piston cannot transfer heat so that the system attains mechanical equilibrium but is restrained from reaching thermal equilibrium. Since (9. z &, the right-hand side of eqn. (18) vanishes only if
But under reversible, adiabatic conditions the left-hand side of eqn. (19) is known from classical thermodynamics to be negative pressure. Hence
However, for a mole of an ideal gas the thermodynamic equation of state is known to he
where k = R I N Comparison of eqn. (29) with eqn. (27) then immediately identifies y(P,T):
and this result identifies P for any system Volume 60 Number 2
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From classical thermodynamics, however, we know that
which immediately leads to
Integration of eqn. (32) with respect to V with E a n d N fixed now yields: Our goal will be achieved if we can show that the constant of integration is actually independent of E . Differentiation of eqn. (33) with respect to E with V and N fixed gives
Now that the constant is seen to he independent of thermodynamic state its empirical significanceis nil, and no harm will come from setting it to zero. In this step of discarding the integration constant the Third Law receives some theoretical justification-a bonus which is missed if S = klnR is presented as a posit. An areument in favor of the axiomatic aovroach (eans. must at some stage give equal priority to classical thermodynamics. The derivations given here assume the conditions for equilibrium and the classical definition of dS;but the Second Law (i.e., the statement that d S cannot he negative for an isolated system) has not been assumed, and the Third Law has been given some justification. Literature Cited
Now from classical thermodynamics the left-hand side of eqn. (34) is known to be l/T, while (alnn/aE), is known to be function of T alone from eqn. (12). Therefore, aconstlaE must vanish for all E and N. This obtains in general only if the constant is independent of E. Hence
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(1) Mandl.F.,"StatisticalPhysics~ Wiley,London. 1971. (2) Reif, F., "Statistical Physics? MeCrsw-Hill, New York,London, 1965. (3) M a w , J. E., and Mayer,M. G., "Statiitical Mechanics." Wiley, New York, 1966. (4) Hill, T. L., "An InLroduetiun to StetisticalThermodymami~s,"Addison-Wesley, Reading,
MA, 1960. (51 Schmedinger, E., "Statistical Thermodynamics: Ithscs, NY, 1967.
2nd ed., Cornell University Pr~s,
