Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 7.4-23

FIRST-ORDER PHASE TRANSITIONS

Physics — Statistical Mechanics · heat-transfer-straight-line-process

Problem

A system is expanded along a straight line in the

P

v

plane, from the initial state

(P_0, v_0)

to the final state

(P_f, v_f)

. Calculate the heat transfer per mole to the system in this process. It is to be assumed that

\alpha

\kappa_T

, and

c_p

are known only along the isochore

v = v_0

and the isobar

P = P_f

; in fact it is sufficient to specify that the quantity

(c_v \kappa_T/\alpha)

has the value

AP

on the isochore

v = v_0

, and the quantity

(c_p/v\alpha)

has the value

Bv

on the isobar

P = P_f

, where

A

and

B

are known constants. That is

\frac{c_v \kappa_T}{\alpha} = AP \qquad (\text{for } v = v_0)

\frac{c_p}{v\alpha} = Bv \qquad (\text{for } P = P_f)

Answer:

Q = \tfrac{1}{2}A(P_f^2 - P_0^2) + \tfrac{1}{2}B(v_f^2 - v_0^2) + \tfrac{1}{2}(P_0 - P_f)(v_f - v_0)