Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 5.3-8

THE EXTREMUM PRINCIPLE IN THE LEGENDRE TRANSFORMED REPRESENTATIONS

Physics — Statistical Mechanics · two-level-system-entropy-representation

Problem

In Chapter 15 it is shown by a statistical mechanical calculation that the fundamental equation of a system of

\tilde{N}

``atoms'' each of which can exist in an atomic state with energy

\varepsilon_u

or in an atomic state with energy

\varepsilon_d

(and in no other state) is

F = -\tilde{N}k_B T \ln\!\left(e^{-\beta\varepsilon_u} + e^{-\beta\varepsilon_d}\right)

Here

k_B

is Boltzmann's constant and

\beta = 1/k_BT

. Show that the fundamental equation of this system, in entropy representation, is

S = NR\ln\!\left(\frac{1 + Y^{\varepsilon_d/\varepsilon_u}}{Y^\gamma}\right)

where

Y \equiv \frac{U - \tilde{N}\varepsilon_u}{\tilde{N}\varepsilon_d - U}

Hint: Introduce

\beta = (k_BT)^{-1}

, and show first that

U = F + \beta\partial F/\partial\beta = \partial(\beta F)/\partial\beta

. Also, for definiteness, assume

\varepsilon_u < \varepsilon_d

, and note that

\tilde{N}k_B = NR

where

\tilde{N}

is the number of atoms and

N

is the number of moles.