## Some problems from Kittel Thermal Physics, chapter II

Posted by peeterjoot on January 10, 2013

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# Motivation

Some review from the ancient second (or third?) year thermal physics course I took.

# Guts

## Energy and temperature [1] 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

Show that is negative.

## Answer

Temperature

We’ve got

or

Second derivative of entropy

From above

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 [1] 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.

## Answer

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 [1] problem 2.3

Entropy.

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 .

Planck Energy

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.

## Answer

Entropy

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 [3] much clearer. A similar diagram and argument can also be found in [2] section 3.8.

Taking logarithms and applying the Stirling approximation, our entropy is

Planck Energy

Now we make the replacement suggested in the problem (ie. assuming ), for

With , we have

or

A final rearrangement gives us the Planck result.

# References

[1] C. Kittel and H. Kroemer. *Thermal physics*. WH Freeman, 1980.

[2] RK Pathria. *Statistical mechanics*. Butterworth Heinemann, Oxford, UK, 1996.

[3] 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].

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