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Maxwell Construction

The equation of state of an ideal gas of [Graphics:../Images/index_gr_62.gif] noninteracting classical particles reads [5, Section 1.8]

[Graphics:../Images/index_gr_63.gif]

where [Graphics:../Images/index_gr_64.gif] is the Boltzmann constant. A simple approximation, taking into account the interaction between particles, is the van der Waals equation of state,

[Graphics:../Images/index_gr_65.gif]

with parameters [Graphics:../Images/index_gr_66.gif] and [Graphics:../Images/index_gr_67.gif]. It turns out that there exists a critical temperature [Graphics:../Images/index_gr_68.gif] below which loops appear in the p-v diagram. These loops are forbidden for thermodynamic reasons. For [Graphics:../Images/index_gr_69.gif], a phase transition between gas and liquid takes place; at [Graphics:../Images/index_gr_70.gif] the difference between the liquid and gaseous phases vanishes. [Graphics:../Images/index_gr_71.gif] and the corresponding critical volume [Graphics:../Images/index_gr_72.gif] are determined by the condition that the first and second derivatives of [Graphics:../Images/index_gr_73.gif] with respect to [Graphics:../Images/index_gr_74.gif] vanish (i.e., [Graphics:../Images/index_gr_75.gif] and [Graphics:../Images/index_gr_76.gif]). We define [Graphics:../Images/index_gr_77.gif] by solving for [Graphics:../Images/index_gr_78.gif] in the van der Waals equation (2),

[Graphics:../Images/index_gr_79.gif]
[Graphics:../Images/index_gr_80.gif]

and determine the critical values [Graphics:../Images/index_gr_81.gif] and [Graphics:../Images/index_gr_82.gif].

[Graphics:../Images/index_gr_83.gif]
[Graphics:../Images/index_gr_84.gif]

The critical pressure [Graphics:../Images/index_gr_85.gif] is defined by [Graphics:../Images/index_gr_86.gif].

[Graphics:../Images/index_gr_87.gif]
[Graphics:../Images/index_gr_88.gif]

We rescale [Graphics:../Images/index_gr_89.gif], [Graphics:../Images/index_gr_90.gif], and [Graphics:../Images/index_gr_91.gif] by [Graphics:../Images/index_gr_92.gif], [Graphics:../Images/index_gr_93.gif], and [Graphics:../Images/index_gr_94.gif], respectively.

[Graphics:../Images/index_gr_95.gif]
[Graphics:../Images/index_gr_96.gif]

Then we denote the dimensionless parameters using [Graphics:../Images/index_gr_97.gif] and obtain a dimensionless form of the van der Waals equation (2).

[Graphics:../Images/index_gr_98.gif]
[Graphics:../Images/index_gr_99.gif]

Plotting a set of [Graphics:../Images/index_gr_100.gif] curves for [Graphics:../Images/index_gr_101.gif] over [Graphics:../Images/index_gr_102.gif] displays the unphysical van der Waals loops which arise for [Graphics:../Images/index_gr_103.gif].

[Graphics:../Images/index_gr_104.gif]

[Graphics:../Images/index_gr_105.gif]

Over a certain temperature range, three volumes [Graphics:../Images/index_gr_106.gif] correspond to each pressure [Graphics:../Images/index_gr_107.gif]. For thermodynamic reasons, the transition from the gas with large volume [Graphics:../Images/index_gr_108.gif], to the liquid with small volume [Graphics:../Images/index_gr_109.gif], takes place at the pressure [Graphics:../Images/index_gr_110.gif] for which

[Graphics:../Images/index_gr_111.gif]

At [Graphics:../Images/index_gr_112.gif], gas and liquid are simultaneously present for all volumes [Graphics:../Images/index_gr_113.gif] with [Graphics:../Images/index_gr_114.gif]; one observes a two-phase mixture. Geometrically, equation (3) means that the area between the curve [Graphics:../Images/index_gr_115.gif] and the straight line [Graphics:../Images/index_gr_116.gif] for [Graphics:../Images/index_gr_117.gif] is the same as the corresponding area for [Graphics:../Images/index_gr_118.gif]. This is known as the Maxwell construction. We want to construct the line [Graphics:../Images/index_gr_119.gif], i.e., the two-phase line. For the two-phase line we need two equations in order to determine the two unknown volumes [Graphics:../Images/index_gr_120.gif] and [Graphics:../Images/index_gr_121.gif]:

[Graphics:../Images/index_gr_111.gif]

There is no simple analytical solution to equations (4) so, for each temperature [Graphics:../Images/index_gr_124.gif], we determine [Graphics:../Images/index_gr_125.gif] and [Graphics:../Images/index_gr_126.gif] using FindRoot. Fairly accurate initial guesses are required, and these can be obtained from the plots of the unphysical van der Waals [Graphics:../Images/index_gr_127.gif] equation.

The two equations read

[Graphics:../Images/index_gr_128.gif]
[Graphics:../Images/index_gr_129.gif]

and

[Graphics:../Images/index_gr_130.gif]
[Graphics:../Images/index_gr_131.gif]

An automatic way to find initial guesses follows.

[Graphics:../Images/index_gr_132.gif]

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

[Graphics:../Images/index_gr_133.gif]

Here are [Graphics:../Images/index_gr_134.gif] and [Graphics:../Images/index_gr_135.gif] for each temperature [Graphics:../Images/index_gr_136.gif].

[Graphics:../Images/index_gr_137.gif]
[Graphics:../Images/index_gr_138.gif]

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

[Graphics:../Images/index_gr_139.gif]

and

[Graphics:../Images/index_gr_140.gif]

here are plots of the Maxwell construction.

[Graphics:../Images/index_gr_141.gif]
[Graphics:../Images/index_gr_142.gif]

[Graphics:../Images/index_gr_143.gif]

The three lowest curves (i.e., [Graphics:../Images/index_gr_144.gif]) show the separation into liquid (small volume) and gaseous phases. Along each two-phase line, liquid and gas coexist in thermal equilibrium.


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