Page 165 - Advanced thermodynamics for engineers

7.3 Tds RELATIONSHIPS 151

and hence Eqn (7.46) becomes

vp vu
c v dT þ T dv pdv ¼ c v dT þ dv: (7.49)
vT vv
v T
or

vu vp
¼ T p (7.50)
vv T vT v
Equation (7.50) can be used to evaluate the variation in internal energy with volume for both ideal
and van der Waals gases.
Ideal gas
The equation of state of an ideal gas is pv ¼ RT and hence

vp R
¼ (7.51)
vT v
v
Thus

vu RT
¼ p ¼ 0: (7.52)
vv T v
Hence, the specific internal energy of an ideal gas is not a function of its specific volume
(or density). This is in agreement with Joule’s experiment that u s f(v) at constant temperature.
van der Waals gas
The equation of state of a van der Waals gas is

RT a
p ¼ 2 ;
v b v
and hence

vp R
¼ (7.53)
vT v b
v
which gives the change of internal energy with volume as
vu RT a

¼ p ¼ : (7.54)
vv T v b v 2
This means that the internal energy of a van der Waals gas is a function of its specific volume or
density. This is not surprising because density is a measure of the closeness of the molecules of the
substance (see Section 8.2), and the internal energy variation is related to the force of attraction be-
tween the molecules. This means that some of the internal energy in a van der Waals gas is stored in the
attraction forces between the molecules, and not all of the thermal energy is due to molecular motion,
as was the case for the ideal gas.