1. Art and Diseño

INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI
Department of Physics
Statistical Mechanics: PH404
Tutorial-II
1. One of the equations of state of black body radiation in a box of volume V and at
temperature T is given by
E
P =
3V
where E is the internal energy and P is the pressure. Using thermodynamic principles,
obtain the Stefan-Boltzmann law, the other equation of state. Knowing two equations
of state, find the fundamental equation of the system.
2. Compute the Helmholtz free energy for a van der Waals gas. The equation of state is
P =
NkB T
aN 2
− 2
V − Nb
V
where a and b are constants which depend on the type of gas and N is the number of
molecules. Assume that heat capacity is CV = 3NkB /2.
3. Consider the following pair of equations
3
E = PV
2
and P = a
V Tn
N
where a is a constant and n is an arbitrary integer. For which value of n, these
equations would represent a compatible pair of thermodynamic equations of state?
For that value of n calculate the Helmholtz free energy of the system.
4. A thermodynamic system is described by the fundamental equation
S
−a
N
4
=b
V E4
,
N3
where a and b are constants. Obtain an expression for the Gibbs free energy G(N, P, T )
of the system.
5. Compute the entropy, enthalpy, Helmholtz free energy, and Gibbs free energy for a
paramagnetic substance and write them explicitly in terms of their natural variables
if possible. Assume that mechanical equation of state is M = cH/T where M is the
total magnetization, H is the external magnetic field, c is the Curie constant, and T
is the temperature. The heat capacity at constant magnetization CM can be taken as
constant,
6. Show that for a magnetic system
2
χT (CH − CM ) = T αH
and CH /CM = χT /χS
where αH is the coefficient of magnetization expansion and χT , χS are the isothermal
and isentropic susceptibilities respectively.
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7. Show that
∂CP
∂P
= −T V αP2 +
T
∂αP
∂T
.
P
8. Compute the heat capacity CP , the compressibilities, κT and κS , and the thermal
expansion coefficient αP of a monatomic van der Waals gas for which the mechanical
equation of state is
NkB T
aN 2
P =
− 2
V − Nb
V
and the heat capacity is CV = 3NkB /2, where V is the volume. Is this gas stable for
all values of P and V ?
9. The internal energy of a model paramagnet is given by
E = NkB T0 exp
S
M2
+ 2 2
NkB N M0
where T0 and M0 are positive constants. Calculate χT and χS , the isothermal and
isentropic magnetic susceptibilities, of the system. Obtain χT as a function of T and
χS as a function of T and H.
10. The following two equations are assumed to be fundamental equations of physical
systems
N 5/2 T 1/2
S 3/2 V 2
(ii) F = b
(i) E = a 5/2 ,
N
V 3/2
where a and b are positive constants. Do they satisfy stability criteria of thermodynamic systems?
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