[SOLVED] Phys 362 statistical and thermal physics: homework 12

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
.a) Express dH in terms of dP, dS and dN and use the result to express T,V, 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.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.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 207 106 K1 and the isothermal compressibility is 3.57 1010 Pa1. Determine the heat capacity at constant volume, cV , per mole under these conditions. Does it
differ by much from cP ?