# Do I understand why heat capacity is called 'heat *capacity*'?

The notion of heat capacity seemed a little odd to me. I thought that the word 'capacity' could be throwing me off. Do I understand the notion correctly?

I take the word 'capacity' to express, roughly, 'the maximum amount of something that a thing can take'. My first intuition is that the maximum temperature of a system equals whatever temperature corresponds to every molecule in that system moving at the limit of the speed of light. Accordingly, if 'heat' denotes a certain species of energy transfer, then the heat capacity (i.e. the maximum amount of energy that a system can receive) should be equal to the number calculated by, for each molecule in the system, taking the difference between the speed of light and the molecule's starting speed, summing those differences, and then dividing by the number of molecules in the system. Accordingly, if everything else is equal (i.e. pressure and volume), then the heat capacity of a system is a function of the starting temperature of the system and its volume. At first blush, that conception of heat capacity seems compatible with the equation $C_x = \frac {Q}{\Delta T_x}$.

• There is no maximum bound to how much energy a system (or even a single subatomic particle) can contain, in principle. Even though its speed is limited to the speed of light, kinetic energy increases infinitely as the speed barrier is approached. Whether there is a maximum temperature is a more subtle topic. Neither of these concepts are necessary to understand heat capacity, though. – Nicolau Saker Neto May 13 '15 at 23:59
• @NicolauSakerNeto Okay, so if energy increases infinitely, then heat capacity doesn't seem to have anything to do with capacity as I've understood: every system has an infinite capacity to receive energy. Clearly, I misunderstand something. How should I think of heat capacity instead? – Hal May 14 '15 at 0:04
• I don't think there's a particularly deep meaning for it. Heat capacity is just the proportionality constant relating how much a system's temperature increases/decreases when a certain amount of heat is introduced/extracted. Different materials will have different proportionality constants. – Nicolau Saker Neto May 14 '15 at 0:07

Expanding on Nicolau Saker Neto's comments above, the heat capacity dictates the (reversible) heat flux required to increase the system's temperature by one unit.

Definitions of the heat capacity ($C_V$ and $C_p$)

Another way to look at it is that the heat capacity dictates the response in internal energy ($U$)/enthalpy ($H$) of the system to temperature, as described below. Note that the heat capacity at constant volume ($C_V$) is not the same as the heat capacity at constant pressure ($C_p$) - and that both may be defined in several ways, the most common being:

$$C_V = \left(\frac{\partial U}{\partial T}\right)_{V,\mathbf{n}} = \left(\frac{\partial Q}{\partial T}\right)_{V,\mathbf{n}}$$

$$C_p = \left(\frac{\partial H}{\partial T}\right)_{p,\mathbf{n}} = \left(\frac{\partial Q}{\partial T}\right)_{p,\mathbf{n}}$$

Contributions to the heat capacity

In should be added that while translation - the kinetic energy of molecules in motion - are an important contribution to the heat capacity of a gas, both vibrations and rotations also contribute for all but the simplest molecules. For high temperatures (approx. $T \geq 3000\ \mathrm{K})$, the excitation of outer electrons will also contribute to the heat capacity. For even higher temperatures, the substance in question will behave as plasma.

In short

There are many complicating factors to the precise understanding of the origin of contributions to the heat capacity - both at the high end and low end of the temperature range. In all cases, however, the heat capacity will give you the answer to the question:

"How much energy do I have to add to this system at the given conditions in order to increase the temperature with one unit?"

Further reading on Wikipedia is recommended for anyone interested.