Download Derivation of the Clausius-Clapeyron Equation for Saturated H2O Vapor/Liquid Mixtures - Pr and more Assignments Meteorology in PDF only on Docsity! The temperature (T) /Specific volume (v) projection: Phase Descriptions are indicated by the following subscripts: - Compressed liquid w - Saturated liquid v - Saturated vapor Specific volume (v) Temperature (T) v / w mixture v The Clausius-Clapeyron equation - A relationship between pressure and temperature for saturated vapor/liquid H2O mixtures. This relationship is tabulated in the Smithsonian Tables and in the appendix of Iribarne and Godson. An ingredient of the proof is the Maxwell relations (See pp. 40 - 43 of Iribarne and Godson). By starting with the combined 1st and 2nd law statements, you should be able to arrive at the Maxwell relations. When finished, we will arrive at the following analytic forms: )(Tfew (saturation vapor pressure over liquid water) and )(Tfei (saturation vapor pressure over ice) We start the proof by considering a closed system containing a saturated vapor/liquid H2O mixture: Consider the T/v diagram shown on the previous page The “saturated states” are on the “dome” or below it. The derivation of the Clausius-Clapeyron equation starts with a generalized Maxwell relationship (see page 40 in Iribarne and Godson): TV V S T P (1) For our particular problem, we have Tv w v s T e (2) For a vapor-liquid mixture, it can be shown that the left-hand side of (2) is a total derivative, i.e., dT de T e w v w (3) Also for a vapor-liquid mixture, the term on the right hand side (RHS) of (2) can be simplified via the relationships developed on the previous page Tv s = Twv wv dvv dss )( )( = Twv wv vv ss Combining the LHS and the RHS of the Maxwell relation (2) we get the differential form of the Clausius-Clapeyron equation: Twv wvw vv ss dT de (4) Equation (4) is the differential form of the Clausius-Clapeyron Equation. A few more relationships are needed before integrating (4): wv gg Specifies chemical equilibrium, mathematically sThg This is a general definition Combining these two relationships, and remembering that the temperature is fixed, we have, TwvTwv ssThh )()( or, Twvv ssTTl )()( (5) Then there is Kirchhoff’s thermodynamic law which relates latent heat to the heat capacities (for liquid and vapor) and to the latent heat at a reference temperature (To). We will prove Kirchhoff’s law at a later time. )()()()( opvwovv TTccTlTl (6) Also the ideal gas law for the vapor, TRve vvw (7) and the assumption (a good one for the vapor/liquid system) that the specific volume of the vapor is large relative to the specific volume of the liquid (see denominator of (4)). Equations (4), (5), (6) and (7) are the ingredients going into the integration, plus the fact that vv >> wv Tv wvw v ss dT de Twvv ssTTl )()( )()()()( opvwovv TTccTlTl TRve vvw The result is called the integrated form of the Clausius-Clapeyron Equation )ln()() 11 ())(( 1 exp)( ,,,, o vpw o ovpwov v oww T T cc TT TccL R eTe Some things to think about: 1. Explain why dT de T e w v w for vapor/liquid mixtures. 2. Using the four equations shown at the top of this page, and thermodynamic data, integrate to obtain the Clausius-Clapeyron Equation
