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KING FAHD UNIVERSITY OF PETROLEUM AND MINERALS
PHYSICS DEPARTMENT
PHYS-430: Thermal Physics
EXAM # 2
Name:
Please Solve all the Problems.
Problem 1.
Do Problem 5.14 (page 159) in the textbook.
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Problem 2.
Part A.
Do Problem 5.76 (page 205) in the textbook.
Part B.
Do Problem 5.85 (page 217) in the textbook.
Part C.
Do Problem 5.91 (page 217) in the textbook.
Problem 3
Part A.
(i)
(ii)
Derive a formula for the average speed and the most probable speed for a
particle obeying the two-dimensional Maxwell speed distribution.
Use the above distribution to derive a formula for the most probable kinetic
energy of a particle obeying the above distribution.
Part B.
A system consists of two weakly interacting particles each of which can be either in one
of two states with respective energies 1 and  2 , where 1   2 .
(i)
Calculate explicitly the mean (average) energy and the specific heat of the
system
(ii)
Estimate the values of the above quantities in the low ( T  0 ) and high
( T   ) temperatures limits.
(iii)
Estimate the temperature at which the average energy changes from the low to
the high temperature limit. (Plot of the average energy may help).
Problem 4
Part. A
Suppose that the following explicit formulas for the partition function were found
a. Z  V N (2 mkT )
5N
2
b. Z  (V  Nb) N (2 mkT )3 N 2 eaN
2
VkT
For both, calculate
(i)
Calculate the equation of state
(ii)
Calculate the specific heat at constant volume
(iii)
Identify the systems described by these partition functions
Part. B
Calculate
(i)
the average energy,
(ii)
the equation of state and
(iii)
the specific heat
of a gas of extremely relativistic particles satisfying   cp , where c is the speed of
light. (Use the momentum space for your calculations and assume classical particles).
Problem 5.
Part A.
The nuclei of atoms in a certain crystalline solid have spin one. According to quantum
theory, each nucleus can therefore be in any one of three quantum states labeled by the
quantum number m, where m=1, 0, or –1. Thus the nucleus has the same energy E   in
the states m=1 and m= -1 and zero in the state m=0.
(a) Find an expression, as a function of temperature T, of the nuclear contribution to
the molar internal energy of the solid (average nuclear energy multiplied by
Avogadro’s number).
(b) Find an expression, as a function of temperature T, of the nuclear contribution to
the molar internal entropy of the solid. Approximate the entropy in the low and
the high T limits.
(c) By directly counting the number of accessible states, calculate the nuclear entropy
at the low and the high temperature limits. Show that this agrees with the low T
and the high T limits in part (b).
(d) Derive the nuclear contribution to the specific heat of the solid. Plot it. And
estimate its dependence on temperature in the high T limit (need to do simple
mathematical approximations).
Part B.
When we talked about distinguishable and indistinguishable particles, we mentioned a
factor introduced by Gibbs to resolve an apparent paradox in some thermodynamical
quantities.
(a) Write in few lines what you know about this factor.
(b) How is it related to entropy and to the partition function
(c) Does the introduction of the factor affect the calculation of thermal averages?
Explain.
(d) Explain why there is no need to correct for the partition function of the
paramagnet and for systems represented by the Einstein Solid