Page 39 - Bruno Linder Elementary Physical Chemistry

August 18, 2010 11:36 9in x 6in b985-ch03 Elementary Physical Chemistry

24 Elementary Physical Chemistry

3.3.1. Reversible Process

Using an ideal gas as the work producing system, we can write for the high
temperature isothermal transition

dU =0 = dq 2 − PdV =dq 2 − RT 2 dV/V (3.9)

keeping in mind that an ideal gas varies only with temperature. Integrating
from V 1 to V 2 at constant T 2 gives (see Fig. 3.1)

q 2 = RT 2 ln V 2 /V 1 (3.10a)
Similarly, the low temperature transition at T 1 gives

q 1 = RT 1 ln V /V 2 (3.10b)

1
Adiabatic changes of ideal gases obviously obey the rule

dq =dU − RT dV/V = 0 (3.11a)

Recall that at constant volume ∆U = q V and so
(dU/dT ) V =dq V /dT = C V and therefore, dU = C V dT.

Accordingly, we can express (3.11a) as

RT
dq = C V dT + dV = 0 (3.11b)
V
Dividing by T ,gives

C V dT/T + RdV/V = 0 (3.11c)

Recalling that C P − C V = R for an ideal gas, and introducing the symbol
γ = C P /C V, we get, upon dividing by C V ,

dT/T +[R/C V ]dV/V =dT/T +[(C P − C V )/C V ]dV/V
=dT/T +(γ − 1)dV/V

Integration along the adiabat from the high isotherm (T 2 ,V 2 )to the
low isotherm (T 1 ,V ) (see Fig. 3.1) gives

2

ln(T 1 /T 2 )+(γ − 1) ln(V /V 2 ) = 0 (3.12a)
2