### Maxwell Construction

where is the Boltzmann constant. A simple approximation, taking into account the interaction between particles, is the van der Waals equation of state,

with parameters and . It turns out that there exists a critical temperature below which loops appear in the p-v diagram. These loops are forbidden for thermodynamic reasons. For , a phase transition between gas and liquid takes place; at the difference between the liquid and gaseous phases vanishes. and the corresponding critical volume are determined by the condition that the first and second derivatives of with respect to vanish (i.e., and ). We define by solving for in the van der Waals equation (2),

and determine the critical values and .

The critical pressure is defined by .

We rescale , , and by , , and , respectively.

Then we denote the dimensionless parameters using and obtain a dimensionless form of the van der Waals equation (2).

Plotting a set of curves for over displays the unphysical van der Waals loops which arise for .

Over a certain temperature range, three volumes correspond to each pressure . For thermodynamic reasons, the transition from the gas with large volume , to the liquid with small volume , takes place at the pressure for which

At , gas and liquid are simultaneously present for all volumes with ; one observes a two-phase mixture. Geometrically, equation (3) means that the area between the curve and the straight line for is the same as the corresponding area for . This is known as the Maxwell construction. We want to construct the line , i.e., the two-phase line. For the two-phase line we need two equations in order to determine the two unknown volumes and :

There is no simple analytical solution to equations (4) so, for each temperature , we determine and using FindRoot. Fairly accurate initial guesses are required, and these can be obtained from the plots of the unphysical van der Waals equation.

and

An automatic way to find initial guesses follows.

These initial guesses allow the volumes to be determined using FindRoot.

Here are and for each temperature .

Now we display the Maxwell construction for the van der Waals equation. Defining

and

here are plots of the Maxwell construction.

The three lowest curves (i.e., ) show the separation into liquid (small volume) and gaseous phases. Along each two-phase line, liquid and gas coexist in thermal equilibrium.

Converted by Mathematica

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