Re: Entropy Again

[palmer@sfu.ca (Leigh Palmer)]

> In review: do I sense correctly that the list feels that the equivalence
> between S=klnW and dS=dQ/ has not been presented in sufficiently clear terms
> AND that it would be worth while to do so.

I will tersely go through my intuitive connection, assuming you all can
infer the definitions of symbols, etc. Questions welcome after lecture.

For simplicity let's consider a simple homogeneous hydrostatic system.
Chemical and other complications can be added later, but the features
of the demonstration will be present here. Start with *two* definitions:

                               1      1  / dS \             / dE \
  S(E,V) = k ln W   and   b = ---- = --- |----|   (or   T = |----|  )
                               kT     k  \ dE /             \ dS /
                                               V                  V

(That's a partial derivative.) Clearly you can't make the leap without
the second definition since you need a statistical equivalent to the
thermodynamic temperature as well as one for the entropy.

Cross multiplying the latter definition one obtains an expression for
the energy change in an infinitessimal isochoric process:

  dE  = T dS
    V       V

Since no work is done on the system in an isochoric process, by the
first law of thermodynamics (neatly bypassing the requirement to
assign a new symbol for work) we have:

                           dQ
  dE  = dQ    =>    dS  = ----                                     (1)
    V                 V    T

To get to a *general* reversible infinitessimal process we must next
perform reversible adiabatic work on the system. In such a process the
accessible microstates of the system mutate continuously from their
initial form to their final form, but no new states become accessible,
otherwise the hypothesis of reversibility is violated, since on
reversal the number of accessible states W would have to diminish.
Expressing this change (or lack thereof) we obtain:

                                          k
  dW          = 0    =>    dS          = --- dW          = 0       (2)
    adiabatic                adiabatic    W    adiabatic

For our general reversible infinitessimal process the entropy change
is given by combining (1) and (2):
                             dQ
  dS = dS   + dS          = ---- + 0
         V      adiabatic    T

Since by the statistical definition the entropy is manifestly a state
function, any process which takes the system from an initial to a final
state changes the entropy by the same amount as any other process which
connects the same initial and final states. The equivalence holds for
any reversible process whatever, since one can build any reversible
process from a series of alternating isochoric and adiabatic reversible
infinitessimal processes connecting intemediate states.

            /      /            dQ
  delta S = | dS = |           ----
            /      /            T
                    reversible

The conventional way to approach this is to teach the classical form
(which was discovered first) before the statistical form. In the
preceding argument I have appealed to the quantum intuition regarding
the adiabatic evolution of system microstates, but it is important to
realize that Gibbs and Boltzmann did this all before there was even a
hint of quantum mechanics!

> BTW I have looked in both Reif books and don't see such a clear presentation
> there -- maybe I need a more specific citation.

I think (big) Reif does this as well as any text. I think that you may
have to read a little more than what you did.

Leigh