high atomic count per molecule, like octane, can therefore have a very large heat capacity per mole, and yet a relatively small specific heat (per unit mass). If the molecule could be entirely described using classical mechanics, then the theorem of equipartition of energy could be used to predict that each degree of freedom would have an average energy in the amount of kT, where k is the Boltzmann constant, and T is the temperature. If the number of degrees of freedom of the molecule is f, then each molecule would be holding, on average, a total energy equal to fkT. Then the molar heat capacity (at constant volume) would be cV,m = fR where R is the ideal gas constant. According to Mayer's relation, the molar heat capacity at constant pressure would be cP,m = cV,m + R = fR + R = (f + 2)R Thus, each additional degree of freedom will contribute R to the molar heat capacity of the gas (both cV,m and cP,m). In particular, each molecule of a monatomic gas has only f = 3 degrees of freedom, namely the components of its velocity vector; therefore cV,m = R and cP,m = R. Rotational modes of a diatomic molecule For example, the molar heat capacity of nitrogen at constant volume is 20.6 J⋅K−1⋅mol−1 (at 15 °C, 1 atm), which is 2.49 R. From the theoretical equation cV,m = fR, one concludes that each molecule has f = 5 degrees of freedom. These turn out to be three degrees of the molecule's velocity vector, plus two degrees from its rotation about an axis through the center of mass and perpendicular to the line of the two atoms. The degrees of freedom due to translations and rotations are called the rigid degrees of freedom, since they do not involve any deformation of the molecule. Because of those two extra degrees of freedom, the molar heat capacity cV,m of (20.6 J⋅K−1⋅mol−1) is greater than that of an hypothetical monatomic gas (12.5 J⋅K−1⋅mol−1) by a factor of . Frozen and active degrees of freedom According to classical mechanics, a diatomic molecule like nitrogen should have more degrees of internal freedom, corresponding to vibration of the two atoms that stretch and compress the bond between them. For thermodynamic purposes, each direction in which an atom can independently vibrate relative to the rest of the molecule introduces two degrees of