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1. Starting with
dS =
1
T
dE +
P
T
dV −
µ
T
dN
show that the entropy can be derived from the Helmholtz free energy, defined
as F ≡ E − TS, to be
S = −
∂F
∂T
N,V
2. Consider a system in which the entropy S(N, V, E) is an extensive quantity.
(a) Show that
S =
∂S
∂N
V,E
N +
∂S
∂V
N,E
V +
∂S
∂E
N,V
E
(b) Show that this in turn results in
E = TS − PV + µN
3. Derive the Maxwell relations
∂T
∂µ
N,P
= −
∂N
∂S
T,P
and
∂P
∂T
S,N
=
∂S
∂V
P,N
4. A substance has the following properties:
(i) At constant temperature T0 the work done by it expanding from V0 to V
is
W = T0 ln V
V0
(1)
(ii) The entropy of the substance is give by
S =
V
V0
T
T0
a
(2)
where V0, T0, and a are fixed constants.
1
(a) Calculate the Helmholtz free energy (relative to the Helmholtz free energy at (V0, T0)).
(b) Find the equation of state.
(c) Find the work done by an arbitrary expansion at an arbitrary constant
temperature.
5. Consider a Carnot cycle where the working substance is an ideal gas with the
equation of state PV = NT, energy E = NT/(γ − 1), and entropy given by
S = N ln “
E
E0
1
γ−1 V
V0
#
+ NS(N), (3)
where γ is a constant. The cycle operates between temperatures T1 and T2
(T1 > T2) and decompresses at T1 from volume Va to volume Vb
.
(a) Explicitly calculate the work done and heat gained or lost in each step of
the cycle.
(b) Explicitly show this cycle has an efficiency
η = 1 −
T2
T1
2
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