Maxwells Construction and the Lever Rule

The reader might have noticed that the isotherms obtained from an equation of state, such as the van der Waals equation, do not coincide with the isotherms shown in Fig 7.2 at the part of the curve that is flat, i.e. where the liquid and vapor states coexist. The flat part of the curve represents what is physically realized when a gas is compressed at a temperature below the critical temperature. Using the condition that the chemical potential of the liquid and the vapor phases must be equal at equilibrium. Maxwell was able to specify where the flat part of the curve would occur. 

Let us consider a van der Waals isotherm for T (Fig. 7.10). Imagine a steady decrease in volume starting at the point Q. Let the point P be such that, at this pressure, the chemical potentials of the liquid and vapor phases are equal. At this point the vapor will begin to condense and the volume can be decreased with no change in the pressure. This decrease in volume can continue until all the vapor condenses to a liquid at the point L. If the volume is maintained at some value between P and L, liquid and vapor coexist. Along line PL the chemical potentials of the liquid and the vapor are equal. Thus the total change in the chemical potential along curve LMNOP must be equal to zero  

Now since the chemical potential is a function of T and p, and since the path is an isotherm, it follows from the Gibbs-Duhem relation that d i = V dp. Using this relation we may write the Aove integral as 

This condition specifies the flat line on which the chemical potentials of the liquid and the vapor are equal, the line that is physically realized. It is called the Maxwell construction. 

At the point P the substance is entirely in the vapor phase with volume V at the point L it is entirely in the liquid phase with volume Vi. At any point S on the line LP, if a mole fraction x of the substance is in the vapor phase, the total volume Vs of the system is 

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