Description
1 Gibbs free energy and thermodynamic derivatives
a) Starting with G = E − T S − PV, express dG in terms of dT and dP and use the result
to express S and V in terms of derivatives of G (remember to indicate variables in the
parentheses subscripts).
b) Use the second derivative rule to show that
! ∂S
∂P
”
T
= −
!∂V
∂T
”
P
.
2 Enthalpy and thermodynamic variables
a) Express dH in terms of dP, dS and dN and use the result to express T,V,µ in terms of
relevant derivatives of H (remember to indicate variables in the parenthesis subscript).
b) Show that
!∂T
∂P
”
S,N
=
!∂V
∂S
”
P,N
for any system.
3 Enthalpy for an ideal gas
The enthalpy of a system is
H = E + PV.
a) Show that
!∂H
∂P
”
T
= T
! ∂S
∂P
”
T
+ V.
b) Use the previous result plus one of the Maxwell relations to show that for an ideal gas
!∂H
∂P
”
T
= 0.
4 Energy of a van der Waals gas
In general
!∂E
∂T
”
V
= cV
and
!∂E
∂V
”
T
= T
!∂P
∂T
”
V
− P.
1
a) Show that
!∂cV
∂V
”
T
= T ∂2P
∂T2 .
b) Starting with the equation of state for a van der Waals gas, show that
!∂E
∂V
”
T
= N2
V 2 a
and also that
!∂cV
∂V
”
T
= 0.
c) Suppose that cV is independent of temperature for a van der Waals gas. Use the previous
results to determine an expression for the energy of the gas E = E(V,T) in terms of
cV ,N,V and a.
5 Heat capacities for water
Consider water at standard temperature and pressure (298 K and 1.01 × 105 Pa). The heat
capacity at constant pressure per mole is cP = 73 J/mol K. The (volume) thermal expansion
coefficient is 207 × 10−6 K−1 and the isothermal compressibility is 3.57 × 10−10 Pa−1. Determine the heat capacity at constant volume, cV , per mole under these conditions. Does it
differ by much from cP ?
