the temperature , as allowed by the equation of state, when the specific volume of the material is forcefully kept constant at . Analogously, for any pressure , let be the function that describes how the specific volume varies with the temperature, when the pressure is kept constant at . Namely, those functions are such that and for any values of . In other words, the graphs of and are slices of the surface defined by the state equation, cut by planes of constant and constant , respectively. Then, from the fundamental thermodynamic relation it follows that This equation can be rewritten as where is the coefficient of thermal expansion, is the isothermal compressibility, both depending on the state . The heat capacity ratio, or adiabatic index, is the ratio of the heat capacity at constant pressure to heat capacity at constant volume. It is sometimes also known as the isentropic expansion factor. Calculation from first principles The path integral Monte Carlo method is a numerical approach for determining the values of heat capacity, based on quantum dynamical principles. However, good approximations can be made for gases in many states using simpler methods outlined below. For many solids composed of relatively heavy atoms (atomic number > iron), at non-cryogenic temperatures, the heat capacity at room temperature approaches 3R = 24.94 joules per kelvin per mole of atoms (Dulong–Petit law, R is the gas constant). Low temperature approximations for both gases and solids at temperatures less than their characteristic Einstein temperatures or Debye temperatures can be made by the methods of Einstein and Debye discussed below. However, attention should be made for the consistency of such ab-initio considerations when used along with an equation of state for the considered material. 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. See also Specific heat of melting (Enthalpy of fusion) Specific heat of vaporization (Enthalpy of vaporization) Frenkel line Heat capacity ratio Heat equation Heat transfer coefficient History of thermodynamics Joback method (Estimation of heat capacities) Latent heat Material properties (thermodynamics) Quantum statistical mechanics R-value (insulation) Specific heat of vaporization