36.9 at 2500 °C, and 37.5 at 3500 °C. The last value corresponds almost exactly to the predicted value for f = 7. The following is a table of some constant-pressure molar heat capacities cP,m of various diatomic gases at standard temperature (25 °C = 298 K), at 500 °C, and at 5000 °C, and the apparent number of degrees of freedom f* estimated by the formula f* = 2cP,m/R − 2: (*) At 59 C (boiling point) The quantum harmonic oscillator approximation implies that the spacing of energy levels of vibrational modes are inversely proportional to the square root of the reduced mass of the atoms composing the diatomic molecule. This fact explains why the vibrational modes of heavier molecules like are active at lower temperatures. The molar heat capacity of at room temperature is consistent with f = 7 degrees of freedom, the maximum for a diatomic molecule. At high enough temperatures, all diatomic gases approach this value. Rotational modes of single atoms Quantum mechanics also explains why the specific heat of monatomic gases is well predicted by the ideal gas theory with the assumption that each molecule is a point mass that has only the f = 3 translational degrees of freedom. According to classical mechanics, since atoms have non-zero size, they should also have three rotational degrees of freedom, or f = 6 in total. Likewise, the diatomic nitrogen molecule should have an additional rotation mode, namely about the line of the two atoms; and thus have f = 6 too. In the classical view, each of these modes should store an equal share of the heat energy. However, according to quantum mechanics, the energy difference between the allowed (quantized) rotation states is inversely proportional to the moment of inertia about the corresponding axis of rotation. Because the moment of inertia of a single atom is exceedingly small, the activation temperature for its rotational modes is extremely high. The same applies to the moment of inertia of a diatomic molecule (or a linear polyatomic one) about the internuclear axis, which is why that mode of rotation is not active in general. On the other hand, electrons and nuclei can exist in excited states and, in a few exceptional cases, they may be active even at room temperature, or even at cryogenic temperatures. Polyatomic gases The set of all possible ways to infinitesimally displace the n atoms