to cause a piston to move). applies only if , that is, no work is done. Consider the difference between adding heat to the gas with a locked piston and adding heat with a piston free to move, so that pressure remains constant. In the second case, the gas will both heat and expand, causing the piston to do mechanical work on the atmosphere. The heat that is added to the gas goes only partly into heating the gas, while the rest is transformed into the mechanical work performed by the piston. In the first, constant-volume case (locked piston), there is no external motion, and thus no mechanical work is done on the atmosphere; is used. In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. Ideal-gas relations For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., , where is the amount of substance in moles. In thermodynamic terms, this is a consequence of the fact that the internal pressure of an ideal gas vanishes. Mayer's relation allows us to deduce the value of from the more easily measured (and more commonly tabulated) value of : This relation may be used to show the heat capacities may be expressed in terms of the heat capacity ratio () and the gas constant (): Relation with degrees of freedom The classical equipartition theorem predicts that the heat capacity ratio () for an ideal gas can be related to the thermally accessible degrees of freedom () of a molecule by Thus we observe that for a monatomic gas, with 3 translational degrees of freedom per atom: As an example of this behavior, at 273 K (0 °C) the noble gases He, Ne, and Ar all have nearly the same value of , equal to 1.664. For a diatomic gas, often 5 degrees of freedom are assumed to contribute at room temperature since each molecule has 3 translational and 2 rotational degrees of freedom, and the single vibrational degree of freedom is often not included since vibrations are often not thermally active except at high temperatures, as predicted by quantum statistical mechanics. Thus we have For example, terrestrial air is