For simplicity let's think about the heat capacity for a single component system, for example methane, CH₄, gas. What happens when we add energy to it by heating it? Well, it gets hotter - its temperature increases. But for a given amount of heat, the methane will not necessarily increase in temperature by the same amount as will other chemicals, for example ozone, O₃, gas. When energy is added to an atom or molecule in the form of heat, it increases the thermal motion of the atom or molecule. The particles move faster as their kinetic energy is increased. The thermal energy that comes from free movement along x, y or z in the air or in a container is called translational energy. There are other kinds of molecular motions that can take in heat. These are molecular rotations, in which the molecules tumble in space -

For linear molecules, rotation around the bond between the atoms does not involve a change in the position of the atoms in space. Consequently, this mode has no significant contribution to the heat capacity in linear molecules. Theoreticians eliminate this insignificant mode by assuming that the atoms are points. For all nonlinear molecules even including small planar bent molecules like water, atoms change position in space during rotation in all 3 directions.

The amount of energy needed to raise the temperature of a substance by one degree in temperature will depend upon how many different ways the atom or molecule can store the energy in the form of these thermal motions and how much energy each kind of thermal motion involves.In general, for molecules with similar bonding the bigger the molecule the more heat has to be added to raise its temperature by one degree. For example, the heat capacity at 1 bar (C_p) and 298.15 K for one mole of methane gas (CH₄) is 35.3 Joules/K-mole, for ethane gas (C₂H₆) it is 52.6 Joules/K-mole and for propane gas (C₃H₈) it is 73.51 Joules/K-mole.

One of the simplest models of heat capacity uses the equipartion of energy theorem to calculate the energy that a gas gains as its temperature increases. In this model, energy is "partitioned" among all modes of energy available to the gas, with each mode being equally likely to gain energy as the temperature is increased. In order to describe a molecule in three dimensions, we will need 3N coordinates, or degrees of freedom, where N is the total number of atoms comprising the molecule and the 3 results from the three possible directions - x, y and z in cartesian space. The molecular motions couple these coordinates, however, and it is most convenient to separate them into translational, rotational and vibrational modes.

Translational Energy: The equipartition of energy theory assumes that all molecules will have the same kinetic energy E = 1/2 kT per translational mode. Since there are three translational modes (along x, y and z), every gas molecule whether it is composed of a single atom or it is polyatomic will have a translational kinetic energy of E_trans = 3/2 kT.

Rotational Energy: In separating out rotational modes of motion, we assume that the atoms comprising the molecule are zero dimensional; they are points. Monatomic gases, therefore, cannot possess rotational energy since there is nothing around which to rotate. This is actually a good deal, since N = 1 for a monatomic gas and we have already used up all our 3N degrees of freedom on its translational motion. All other gases (N > 2) will have either two or three rotational modes, depending whether the molecule is linear or nonlinear. Rotational motion for a linear molecule is two dimensional and thus there are two degrees of freedom associated with it. Nonlinear molecules need all three dimensions to describe their rotational motion and, correspondingly, they need three degrees of freedom. Like translational energy, rotational energy is purely kinetic. There will, therefore, be 1/2 kT of energy associated with each rotational mode. Atomic gases will have E_rot = 0 kT, diatomic and linear polyatomic gases will have E_rot = kT and nonlinear gases will have E_rot = 3/2 kT.

Vibrational Energy: All degrees of freedom not associated with either a translational or a rotational mode must be vibrational in nature. Atoms are points and, as such, they do not vibrate. Thus they cannot raise their temperature by adding energy to vibrational motion. After subtracting out degrees of freedom used for tranlational and rotational modes, linear molecules have 3N-5 degrees of freedom left for vibrational motion and nonlinear molecules have 3N-6. Because vibrational motion has both a kinetic energy and a potential energy component, there is kT associated with each vibrational degree of freedom. The vibrational energy contribution to the total energy of the gas is: monatomic gases E_vib = 0 kT, diatomic and linear polyatomic gases E_vib = (3N-5)kT and nolinear molecules E_vib = (3N-6)kT.

Heat Capacities: Finally, we can calculate our heat capacities. Since

we merely need to total up all components to the energy of the system and differentiate with respect to T.

• monatomics C_V = 3/2k
• linear C_V = 5/2k (3N-5)k
• nonlinear C_V = 3k (3N-6)k

The values above are for a single molecule. To make this a molar heat capacity, you need merely multiply by Avogardro's number, N_A. Since N_Ak = R, the ideal gas constant, you can get molar heat capacities simply by replacing all the "k"s in the above equation by R.

If you want to work at constant pressure, as you would for a chemical reaction on a bench top, you can convert these values to C_p, if you assume that your gas is an ideal gas. For an ideal gas, C_p = C_v R (these are molar quantities). For molar heat capacities at constant pressure, the equations that predict heat capacities by equipartition of energy theory are

• monatomics C_P = 5/2R
• linear C_P = 7/2R (3N-5)R
• nonlinear C_P = 4R (3N-6)R

The equipartion of energy theory applies only to gases and, even in that restriction, is much too simple to be very accurate. Its usefulness is not in its accuracy but rather in demonstrating how molecules take in heat. One complication is that vibrational modes generally take much more energy to excite and at room temperature might not be excited to any significant extent at all. So at low temperatures, these modes will not contribute very much to the heat capacity. What a "low temperature" is, depends on the type of bond. For example, I₂ has a fairly weak single bond and can be approximated at a high temperature limit at several hundred degrees Kelvin, whereas for the triple-bonded N₂ the high temperature limit requires the gas to be at several thousand Kelvin. At low temperatures, we might therefore try to get around this by simply not including the vibrational component at all. The molar heat capacities at the low temperature limit can therefore be estimated by the equipartition of energy theory.

• monatomics C_P = 5/2R
• linear C_P = 7/2R
• nonlinear C_P = 4R

The heat capacity of solids and liquids are a bit more difficult to model. At low temperature, heat capacities of solids depend on what kind of solid you have, whether it is magnetic or a conductor for example. To see how solids are modeled, check out the Law of Dulong and Petit. Liquids are the most complicated to model of all because their "translation" - moving the molecules through the condensed phase involves overcoming intermolecular interactions. Water, which has very strong hydrogen bonding, has an anomalously high heat capacity.

All heat capacities depend on the temperature and tend to increase in value as the temperature increases. To calculate the heat that is transferred to the chemical at constant volume very accurately, we need to evaluate an integral

At constant pressure, the integral becomes

However, over a small temperature range, we can approximate the heat capacities as being constant, which gives us a very simple set of equations

where ΔT = T₂-T₁. It is these simpler equations that you will use in most high school chemistry applications.