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So, today’s BMC post is a bit unusual, in that we are focusing on a particular bit of physics history along with a fairly narrow subject: the heat capacity in metals.

### What is heat capacity?

For this post to have any meaning, I must first explain what heat capacity is.  It’s pretty simple to understand, as it is simply the ratio of how much energy you dump into something to how much the temperature of that thing changes.  What can make it tricky, though, is when the heat capacity of a system is related to its temperature.  This causes some trickiness, which is exactly the kind of shenanigans we seen when we look at metals.

### A history of trying to figure out heat capacity

In simple, ideal systems (like non-interacting gas particles in a nice, uniform box), heat capacity can be rather simple to calculate.  However, for most real physical systems, more interactions and effects need to be considered.  The issue with figuring things out in thermodynamic systems is that things tend to change their behavior at really low temperatures, and these have only become available to us in the last century.  Initially, measurements of the heat capacity of metals, and some theoretical calculation (read: assumption), led to the Dulong-Petit expression for the heat capacity of metals.  This is the simplest possible form: it has nothing to do with temperature!  Just a nice constant value, depending only on the amount of material.  However, as smaller temperatures were probed, this was found to be lacking, as the heat capacity dropped below the predicted value.

### Einstein, Debye, and Freezing out

If you were to plot the actual specific heat of a metal, as you got to lower and lower temperatures, you would find a figure like this:

The two curves, the Einstein and Debye models, both match pretty closely to each other.  The Einstein model fits an an exponential function to the heat capacity curve, while the Debye model uses a cubic function.  What this means, is that as the temperature drops, there the heat capacity drops off, until it gets pretty close to zero, when it levels off.  The physical reason this is happening is as the system drops in temperature, degrees of freedom (vibration modes primarily) “freeze out”, and as the equipartition theorem tells us, that means there is less energy stored in the system in proportion to the temperature.  So, as you heat a system up from absolute zero, it is initially very easy to heat, then ramps up towards a certain value, and then the heat capacity stays constant, as all the available degrees of freedom are accessible.