5.11. Derive the equation for the moist adiabatic lapse rate. Begin by assuming that the system of 1mol of air and Xmol of water vapor is raised adiabatically a distance dz. As it is raised, METEOROLOGY FOR AIR POLLUTION CONTROL ENGINEERS 117 P,T, and X will all change. Assume that the volume of the liquid formed by condensing some of the water is negligible, so that V=(1+X)PRT By an energy balance on this closed system we have dU=dQdW=0PdV and dU=TUdT+XUdX+PUdP=CVdT+lcondensation,waterdX+PUdP Noting that dP=0, we eliminate dU between equations (B) and (C). Differentiate Eq. (A), substitute dV into the resulting equation, and simplify to RT(1+X)(TdT+PdP+1+XdX)=(CVaiir+XCVwaser)dT+ucondensation,waterdX Replace dP with its value from Eq. (5.6) and rearrange to yield dzdT=(CV,air+XCV,waler)+R(1+X)gMavg(CV,air+XCV,water)+R(1+X)(ucondensation,water+RT)dzdX Check by setting X equal to zero and seeing if the result is the same as Eq. (5.9). Equation (E) is the complete equation. For most situations outside the tropics, X1, so that we may ignore the X in the (1+X) expressions, but not in the dX/dz. Make that simplification and several thermodynamic simplifications, then use Eq. (5.9) to get dzdT(dzdT)adiahacic,dryCP,airhcondensation,waterdzdX