Thermodynamics

volume. That is, P sat f (T sat ). Therefore, the partial derivative (P/T ) v canbe expressed as a total derivative (dP/dT ) sat , which is the slope of the saturationcurve on a P-T diagram at a specified saturation state (Fig. 12–9).This slope is independent of the specific volume, and thus it can be treatedas a constant during the integration of Eq. 12–18 between two saturationstates at the same temperature. For an isothermal liquid–vapor phase-changeprocess, for example, the integration yieldsors g s f a dPdT b 1v g v f 2sata dPdT b s fgsat v fg(12–20)(12–21)During this process the pressure also remains constant. Therefore, fromEq. 12–11,0 ggdh T ds v dP S dh T ds S h fg Ts fgSubstituting this result into Eq. 12–21, we obtaina dPdT b h fgsat Tv fg(12–22)which is called the Clapeyron equation after the French engineer andphysicist E. Clapeyron (1799–1864). This is an important thermodynamicrelation since it enables us to determine the enthalpy of vaporization h fg at agiven temperature by simply measuring the slope of the saturation curve ona P-T diagram and the specific volume of saturated liquid and saturatedvapor at the given temperature.The Clapeyron equation is applicable to any phase-change process thatoccurs at constant temperature and pressure. It can be expressed in a generalform asa dPdT b h 12sat Tv 12where the subscripts 1 and 2 indicate the two phases.→ff(12–23)PSOLIDChapter 12 | 659LIQUIDVAPORT∂P( –– ) = const.∂T satFIGURE 12–9The slope of the saturation curve on aP-T diagram is constant at a constantT or P.TEXAMPLE 12–5Evaluating the h fg of a Substance fromthe P-v-T DataUsing the Clapeyron equation, estimate the value of the enthalpy of vaporizationof refrigerant-134a at 20°C, and compare it with the tabulated value.Solution The h fg of refrigerant-134a is to be determined using the Clapeyronequation.Analysis From Eq. 12–22,h fg Tv fg a dPdT b sat