and the corresponding work is where is the difference between the volumes of in the liquid phase and vapor phases. The ratio is the efficiency of the Carnot engine, . Substituting and rearranging gives where lowercase denotes the change in specific volume during the transition. Applications Chemistry and chemical engineering For transitions between a gas and a condensed phase with the approximations described above, the expression may be rewritten as where is the pressure, is the specific gas constant (i.e., the gas constant divided by the molar mass), is the absolute temperature, and is a constant. For a liquid–gas transition, is the specific latent heat (or specific enthalpy) of vaporization; for a solid–gas transition, is the specific latent heat of sublimation. If the latent heat is known, then knowledge of one point on the coexistence curve, for instance (1 bar, 373 K) for water, determines the rest of the curve. Conversely, the relationship between and is linear, and so linear regression is used to estimate the latent heat. Meteorology and climatology Atmospheric water vapor drives many important meteorologic phenomena (notably, precipitation), motivating interest in its dynamics. The Clausius–Clapeyron equation for water vapor under typical atmospheric conditions (near standard temperature and pressure) is where is saturation vapor pressure, is temperature, is the specific latent heat of evaporation of water, is the gas constant of water vapor. The temperature dependence of the latent heat (and of the saturation vapor pressure ) cannot be neglected in this application. Fortunately, the August–Roche–Magnus formula provides a very good approximation: where is in hPa, and is in degrees Celsius (whereas everywhere else on this page, is an absolute temperature, e.g. in kelvins). This is also sometimes called the Magnus or Magnus–Tetens approximation, though this attribution is historically inaccurate. But see also the discussion of the accuracy of different approximating formulae for saturation vapour pressure of water. Under typical atmospheric conditions, the denominator of the exponent depends weakly on (for which the unit is degree Celsius). Therefore, the August–Roche–Magnus equation implies that saturation water vapor pressure changes approximately exponentially with temperature under typical atmospheric conditions, and hence the water-holding capacity of the atmosphere increases by about 7% for every 1 °C rise in temperature. Example One of the uses of this equation is to determine if a phase transition will occur in a given situation. Consider the question of how much pressure is needed to melt ice