Reference: Daniel V. Schroeder, *An Introduction to Thermal Physics*, (Addison-Wesley, 2000) – Problems 3.22 – 3.23.

We can use the formulas we obtained for the two-state paramagnet to derive a formula for entropy as a function of temperature. We start with

which gives the fraction of dipoles, each with magnetic moment , aligned parallel to the field at temperature . The entropy can be written in terms of as

From 1 we have

From 2 we have

In what follows, we’ll define

We then have

Also

Therefore

As , and

Thus the entropy drops to zero at as required by the third law of thermodynamics.

At the other extreme, when , and

This is the maximum value of which occurs when so that half the dipoles are parallel and half are antiparallel.

A plot of versus (effectively a plot of versus ) looks like this:

If we increase , the dip becomes broader, since a stronger field means that we need a higher temperature to cause significant disruption of the dipoles. Conversely, if we decrease the field, then the dip becomes narrower, since we need very low temperatures to freeze the dipoles into alignment.

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