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2 Macroscopic Thermodynamics
U VdW (T , V ; N) = U ideal gas (T ; N) −
aN 2
V
.
For an atomic Van der Waals gas, whose particles possess only translational motion,
for example, the internal energy is thus given as
U VdW (T , V ; N) =
3
2 Nk B T −
aN 2
V
.
This result is consistent with the association of the second term in expression
(2.8.9b) with the existence of an attraction between Van der Waals atoms or
molecules. Such an attraction must necessarily depend upon the average separation
between the Van der Waals particles. It should thus be no surprise that (isothermal)
volume changes will cause changes in the internal energy of the Van der Waals fluid
that will be manifested as a volume dependence of the internal energy.
Example 2.9 Enthalpy, entropy, and Helmholtz energy for a Van der Waals fluid.
We have already obtained an expression for the internal energy for a Van der
Waals fluid of structureless particles in Example 2.8 above. The enthalpy for a Van
der Waals fluid is then given as
H VdW (T , V ; N) = U VdW (T , V ; N) + (P V ) VdW
or, using the expression for U VdW (T , V ; N) obtained in Example 2.8 plus the Van
der Waals equation of state, as
H VdW (T , V ; N) = H ideal gas (T ; N) + Nk B T
Nb
V − Nb
−
2aN 2
V
.
The absolute entropy for a Van der Waals fluid may be determined from
Eq. (2.3.12a) in the form
dS VdW =
C V
T
dT +
Nk B
V − Nb
dV ,
as
S VdW (T , V ; N) = S 0 + C V ln T + Nk B ln(V − Nb) ,
which has the same form as the absolute entropy for an ideal gas, but with the
volume V replaced by the free volume V − Nb for the Van der Waals fluid.
The Helmholtz energy may now be obtained as
A VdW (T , V ; N) = U VdW (T , V ; N) − T S VdW (T , V ; N) ,