Heat Capacity... Specifically

This post is to cover the various definitions of heat capacity. For a more thorough discussion of heat capacity, upcoming posts will examine some practical examples involving phase transitions, and finally delve into the components of heat capacity including a discussion about bond energies and degrees of freedom in molecular structure.

Before viewing this post, I recommend taking a look at my post entitled "Enthalpy Unleashed!" for a  comprehensive discussion about the First Law of Thermodynamics defined in terms of the Ideal Gas Law.

Formal Definitions of Heat Capacity

Before we jump into equations, let's define heat capacity by describing it in words.

Heat capacity defines a substance's ability to accept heat to raise its own temperature by a given amount. It is usually expressed in units Joule per Kelvin. This is an extensive property as it relies on sample size.

Also, you probably will run into something called specific heat capacity. This is given in units of mass times Joule per unit Kelvin, which causes it to be an intensive property, as it does not depend on the size of the substance of inquiry. Molar heat capacity is also an intensive property, as it is given in units of mole times Joule per Kelvin.

If this still seems too abstract, let's look at the term itself. Capacity is defined to be the maximum quantity that something can contain. For instance, if you fill your freezer with 200 bagels and discover you can't fit in anymore, your freezer has a capacity to hold 200 bagels. In contrast, if you slice your bagel for toasting and your toaster only has the space for two slices of bagel, we say the toaster has a capacity to hold 1 bagel. The capacity of the toaster is less than that of the freezer. Now, applying this idea to heat capacity, the heat capacity is the object's ability to hold heat. To quantify the heat accepted, we need to measure a change, which is done by examining the temperature change of the substance, since accepting heat will cause the molecules of the substance to vibrate more rapidly, invoking a temperature increase by the Kinetic Molecular Theory.

Now, with all that said and done, on with the integrals!

a) At constant volume, the internal energy is equal to the heat of the system. Integrating U with respect to temperature, we define the heat capacity to be

b) At constant pressure the enthalpy is equal to the heat of the system. Integrating H with respect to temperature, we define the heat capacity to be

Note that the heat capacity at constant volume is not equivalent to the heat capacity at constant pressure.

Enthalpy and the Ideal Gas in Terms of Heat Capacities

Recall the enthalpy equation written in terms of the Ideal Gas Law:

Integrating the above with respect to temperature gives:

And substituting in for the definitions of heat capacity expresses the equation as such: