discrete amount (quanta). Depending on the temperature, the average heat energy per molecule may be too small compared to the quanta needed to activate some of those degrees of freedom. Those modes are said to be "frozen out". In that case, the specific heat capacity of the substance is going to increase with temperature, sometimes in a step-like fashion, as more modes become unfrozen and start absorbing part of the input heat energy. For example, the molar heat capacity of nitrogen at constant volume is (at 15 °C, 1 atm), which is . That is the value expected from theory if each molecule had 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. Because of those two extra degrees of freedom, the specific heat capacity of (736 J⋅K−1⋅kg−1) is greater than that of an hypothetical monatomic gas with the same molecular mass 28 (445 J⋅K−1⋅kg−1), by a factor of . This value for the specific heat capacity of nitrogen is practically constant from below −150 °C to about 300 °C. In that temperature range, the two additional degrees of freedom that correspond to vibrations of the atoms, stretching and compressing the bond, are still "frozen out". At about that temperature, those modes begin to "un-freeze", and as a result starts to increase rapidly at first, then slower as it tends to another constant value. It is 35.5 J⋅K−1⋅mol−1 at 1500 °C, 36.9 at 2500 °C, and 37.5 at 3500 °C. The last value corresponds almost exactly to the predicted value for 7 degrees of freedom per molecule. Derivations of heat capacity Relation between specific heat capacities Starting from the fundamental thermodynamic relation one can show, where, is the coefficient of thermal expansion, is the isothermal compressibility, and is density. A derivation is discussed in the article Relations between specific heats. For an ideal gas, if is expressed as molar density in the above equation, this equation reduces simply to Mayer's relation, where and are intensive property heat capacities expressed on a per mole basis at constant pressure and constant volume, respectively. Specific heat capacity The specific heat capacity of a material on a per mass basis is which in the absence of phase transitions is equivalent to where is the