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ENTROPY AND PROBABILITY 71

give a deduction rather than a hypothetical assertion' [E13]. Since Einstein had

by then already reinvented Boltzmann's first definition, it appears safe to assume

that he was referring to the counting of complexions. Not only did he regard that

definition as artificial. More than that, he believed that one could dispense with

such countings altogether: 'In this way, [I] hope to eliminate a logical difficulty

which still hampers the implementation of Boltzmann's principle' [El3]. In order

to illustrate what he had in mind, he gave a new derivation of a well-known for-

mula for the change of entropy S of an ideal gas when, at constant temperature

T, the volume changes reversibly from F 0 to V:

where n is the number of molecules in the gas, R is the gas constant, and TV is

Avogadro's number. As we shall see later, this equation played a crucial role in

Einstein's discovery of the light-quantum. (To avoid any confusion, I remind the

reader that this relation has nothing to do with any subtleties of statistical mechan-

ics, since it is a consequence of the second law of thermodynamics for reversible

processes and of the ideal gas law.*) Einstein derived Eq. 4.15 by the following

reasoning. Boltzmann's principle (Eq. 4.3), which he wrote in the form

(it took until 1909 before Einstein would write k instead of R/N) implies that a

reversible change from a state 'a' to a state 'b' satisfies

Let the system consist of subsystems 1,2,..., which do not interact and therefore

are statistically independent. Then

*For an infinitesimal reversible change, the second law can be written (p = pressure)

where cv, the specific heat at constant volume, S, and U, the internal energy, all are in general
functions of V and T. From

For a classical ideal gas, this last relation reduces to dU/d V = 0 since in this case NpV = nRT.
In turn, dU/d V = 0 implies that cv is a function of T only. (Actually, for an ideal gas, cv does not
depend on T either, but we do not need that here.) Hence TdS(V,T) = c,(T)dT + nRTdV/NV.
For a finite reversible change, this yields Eq. 4.15 by integration with respect to the volume.

and from Eq. 4.16 it follows that