Some problems from Kittel Thermal Physics, chapter II
Posted by peeterjoot on January 10, 2013
Some review from the ancient second (or third?) year thermal physics course I took.
Energy and temperature  problem 2.1
Suppose , where is a constant and is the number of particles. This form of actually applies to an ideal gas.
Show that is negative.
Second derivative of entropy
This doesn’t seem particularly suprising if we look at the plots. For example for and we have (Fig 1)
The rate of change of entropy with energy decreases monotonically and is always positive, but always has a negative slope.
Paramagnetism  problem 2.2
Find the equilibrium value at temperrature of the fractional magnetization
of the system of spins each of magnetic moment in a magnetic field . The spin excess is . Take the entropy as the logarithm of the multiplicity as given in (1.35):
for . Hint: Show that in this approximation
with . Further, show that , where denotes , the thermal average energy.
I found this problem very hard to interpret. What exactly is being asked for? Equation (1.35) in the text was
from which we find the entropy 1.2.5 directly after taking logarithms. The temperature is found directly
The magnetization, for a system that has spin excess was defined as
and we can substitute that for
and take derivatives for the temperature
This gives us a relation between temperature and the energy of the system with spin excess , and we could write
Is this the relation that this problem was asking for?
Two things I don’t understand from this problem:
- Where does come from? If we calculate the expectatation of the spin excess, we find that it is zero
- If has a non-zero value, then doesn’t that make also zero? It seems to me that in 1.0.10 is the energy of a system with spin excess , and not any sort of average energy?
Quantum harmonic oscillator  problem 2.3
Find the entropy of a set of oscillators of frequency as a function of the total quantum number . Use the multiplicity function (1.55) and make the Stirling approximation . Replace by .
Let denote the total energy of the oscillators. Express the entropy as . Show that the total energy at temperature is
This is the Planck result; it is derived again in Chapter by a powerful method that does not require us to find the multiplicity function.
The multiplicity was found in the text to be
I wasn’t actually able to follow the argument in the text, and found the purely combinatoric wikipedia argument  much clearer. A similar diagram and argument can also be found in  section 3.8.
Taking logarithms and applying the Stirling approximation, our entropy is
Now we make the replacement suggested in the problem (ie. assuming ), for
With , we have
A final rearrangement gives us the Planck result.
 C. Kittel and H. Kroemer. Thermal physics. WH Freeman, 1980.
 RK Pathria. Statistical mechanics. Butterworth Heinemann, Oxford, UK, 1996.
 Wikipedia. Einstein solid — Wikipedia, The Free Encyclopedia, 2012. URL http://en.wikipedia.org/w/index.php?title=Einstein_solid&oldid=530449869. [Online; accessed 2-January-2013].