sometimes also known as the isentropic expansion factor. Ideal gas For an ideal gas, evaluating the partial derivatives above according to the equation of state, where R is the gas constant, for an ideal gas Substituting this equation reduces simply to Mayer's relation: The differences in heat capacities as defined by the above Mayer relation is only exact for an ideal gas and would be different for any real gas. 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 heat capacity of a body made of the material in question, is the mass of the body, is the volume of the body, is the density of the material. For gases, and also for other materials under high pressures, there is need to distinguish between different boundary conditions for the processes under consideration (since values differ significantly between different conditions). Typical processes for which a heat capacity may be defined include isobaric (constant pressure, ) or isochoric (constant volume, ) processes. The corresponding specific heat capacities are expressed as From the results of the previous section, dividing through by the mass gives the relation A related parameter to is , the volumetric heat capacity. In engineering practice, for solids or liquids often signifies a volumetric heat capacity, rather than a constant-volume one. In such cases, the specific heat capacity is often explicitly written with the subscript , as . Of course, from the above relationships, for solids one writes For pure homogeneous chemical compounds with established molecular or molar mass, or a molar quantity, heat capacity as an intensive property can be expressed on a per-mole basis instead of a per-mass basis by the following equations analogous to the per mass equations: where n is the number of moles in the body or thermodynamic system. One may refer to such a per-mole quantity as molar heat capacity to distinguish it from specific heat capacity on a per-mass basis. Polytropic heat capacity The polytropic heat capacity is calculated at processes if all the thermodynamic properties (pressure, volume, temperature) change: The most important polytropic processes run between the adiabatic and the isotherm functions, the polytropic index is between 1 and the adiabatic exponent (γ or κ). Dimensionless heat capacity The dimensionless heat capacity of a material is where is the heat capacity of