into the pressure derivative given above (), we obtain This result (also known as the Clapeyron equation) equates the slope of the coexistence curve to the function of the specific latent heat , the temperature , and the change in specific volume . Instead of the specific, corresponding molar values may also be used. Derivation from Gibbs–Duhem relation Suppose two phases, and , are in contact and at equilibrium with each other. Their chemical potentials are related by Furthermore, along the coexistence curve, One may therefore use the Gibbs–Duhem relation (where is the specific entropy, is the specific volume, and is the molar mass) to obtain Rearrangement gives from which the derivation of the Clapeyron equation continues as in the previous section. Ideal gas approximation at low temperatures When the phase transition of a substance is between a gas phase and a condensed phase (liquid or solid), and occurs at temperatures much lower than the critical temperature of that substance, the specific volume of the gas phase greatly exceeds that of the condensed phase . Therefore, one may approximate at low temperatures. If pressure is also low, the gas may be approximated by the ideal gas law, so that where is the pressure, is the specific gas constant, and is the temperature. Substituting into the Clapeyron equation we can obtain the Clausius–Clapeyron equation for low temperatures and pressures, where is the specific latent heat of the substance. Instead of the specific, corresponding molar values (i.e. in kJ/mol and = 8.31 J/(mol⋅K)) may also be used. Let and be any two points along the coexistence curve between two phases and . In general, varies between any two such points, as a function of temperature. But if is approximated as constant, or These last equations are useful because they relate equilibrium or saturation vapor pressure and temperature to the latent heat of the phase change without requiring specific-volume data. For instance, for water near its normal boiling point, with a molar enthalpy of vaporization of 40.7 kJ/mol and = 8.31 J/(mol⋅K), . Clapeyron's derivation In the original work by Clapeyron, the following argument is advanced. Clapeyron considered a Carnot process of saturated water vapor with horizontal isobars. As the pressure is a function of temperature alone, the isobars are also isotherms. If the process involves an infinitesimal amount of water, , and an infinitesimal difference in temperature , the heat absorbed is