Clausius Clapeyron Calculator

Along any two-phase coexistence line the chemical potentials of the two phases are equal, and requiring that equality to hold as temperature changes gives the exact Clapeyron relation dP/dT = deltaH / (T * deltaV). For a liquid-vapor or solid-vapor boundary two approximations collapse this into a form that can be integrated: the molar volume of the condensed phase is dropped because it is typically three orders of magnitude smaller than the vapor volume, and the vapor is treated as ideal so that V = RT/P. Substituting and integrating with deltaH_vap held constant produces ln(P2/P1) = (deltaH_vap/R) * (1/T1 - 1/T2), the working equation. Temperatures must be absolute, and R is 8.314 J per mol per K. Typical enthalpies of vaporization at the normal boiling point are 40.66 kJ/mol for water, 38.56 for ethanol, 30.8 for benzene and 5.57 for nitrogen. Plotting ln P against 1/T yields a straight line whose slope is minus deltaH_vap/R, which is the standard laboratory route to an enthalpy of vaporization. Accuracy is best within roughly 30 to 50 K of the reference point. It degrades as the critical point is approached, because deltaH_vap falls toward zero there while the vapor becomes markedly non-ideal and the liquid volume is no longer negligible. Trouton rule offers a sanity check: for non-associating liquids deltaH_vap divided by the normal boiling point clusters near 88 J per mol per K, while hydrogen bonded liquids such as water and alcohols run well above that value.

Benoit Paul Emile Clapeyron, a French engineer trained at the Ecole Polytechnique, published his memoir on the motive power of heat in 1834. Its lasting contribution was to restate the largely verbal reasoning of Sadi Carnot in calculus and graphical form, and in doing so he derived the relation between the slope of a phase boundary, the latent heat of the transition and the volume change. At that date latent heat was still understood through caloric theory, so the relation was correct in form but not yet grounded in energy conservation. Rudolf Clausius rebuilt the derivation on thermodynamic foundations. In his 1850 paper on the motive power of heat he reconciled Carnot and Clapeyron with the conservation principle of Mayer and Joule, and in the work that followed through 1865, in which he named entropy, the phase boundary relation emerged as a consequence of the second law rather than an isolated result. The integrated exponential form used today, obtained by assuming an ideal vapor and a temperature independent latent heat, became a fixture of physical chemistry teaching in the late nineteenth century. It remains central to meteorology, where the roughly seven percent increase in saturation vapor pressure per degree Celsius underpins projections of atmospheric moisture, and to vacuum distillation, freeze drying and pressure cooking design.