Van der Waals coexistence locus

The coexistence locus for liquid vapor equilibrium of a van der Waals fluid can be obtained exactly. The wxMaxima code for obtaining it is shown below the figure.

The traditional vapor-liquid coexistence locus for the van der Waals fluid.

reset;
vg(x) := -1/6*(4*x*exp(2*x) - exp(4*x) + 1)*exp(x)/(x*exp(3*x) + x*exp(x)
    - exp(3*x) + exp(x)) + 1/3 $
vl(x)  := -1/6*(4*x*exp(2*x) - exp(4*x) + 1)*exp(-x)/(x*exp(3*x) + x*exp(x)
    - exp(3*x) + exp(x)) + 1/3$

T(d) :=
            -27/4*((4*d*exp(2*d) - exp(4*d) + 1)*((4*d*exp(2*d) - exp(4*d) + 1)*
                    exp(-d)/(d*exp(3*d) + d*exp(d) - exp(3*d) + exp(d)) - 2)^2*exp(d)
                /(d*exp(3*d) + d*exp(d) - exp(3*d) + exp(d)) + (((4*d*exp(2*d) - exp(4*d) + 1)
                        *exp(d)/(d*exp(3*d) + d*exp(d) - exp(3*d) + exp(d)) - 2)^2 +
                    4*(4*d*exp(2*d) - exp(4*d) + 1)*exp(d)/(d*exp(3*d)
                        + d*exp(d) - exp(3*d) + exp(d)) - 4)*((4*d*exp(2*d) - exp(4*d) + 1)
                    *exp(-d)/(d*exp(3*d) + d*exp(d) - exp(3*d) + exp(d)) - 2) +
                2*((4*d*exp(2*d) - exp(4*d) + 1)*exp(d)/(d*exp(3*d) + d*exp(d) - exp(3*d)
                        + exp(d)) - 2)^2 + 4*(4*d*exp(2*d) - exp(4*d) + 1)*exp(d)/(d*exp(3*d)
                    + d*exp(d) - exp(3*d) + exp(d)) - 8)/(((4*d*exp(2*d) - exp(4*d) + 1)
                    *exp(-d)/(d*exp(3*d) + d*exp(d) - exp(3*d) + exp(d)) - 2)^2*((4*d*exp(2*d)
                        - exp(4*d) + 1)*exp(d)/(d*exp(3*d) + d*exp(d)
                        - exp(3*d) + exp(d)) - 2)^2)$
pg(d) := 8*T(d)/(3*vg(d)-1)-3/vg(d)^2$       
pl(d) := 8*T(d)/(3*vl(d)-1)-3/vl(d)^2$

wxplot2d([vg(x),vl(x)],[x,0.1,1]);
wxplot2d(T(x),[x,0.1,1]);
wxplot2d(pg(x),[x,0.1,1]);
s1:parametric(vg(d),pg(d),d,0.01,0.9)$
s2:parametric(vl(d),pl(d),d,0.01,0.9)$
wxdraw2d(nticks=21,line_width=2,color=red,key="gas",s1,line_width=2,color=blue,key="liquid",s2,
xlabel="p_r",ylabel="v_r",terminal   = 'png,file_name = "vdw8_out");
    /*wxdraw3 draws here; else draw3 draws in rotatable gnuplot window and
       draw_file(terminal =  'png, file_name = "~/Desktop/vdw8_out");*/

To put the first plot (above) in context, below is the pseudo-3D plot of the same coexistence locus, this time showing the explicit temperature (reduced) dependence. The above plot can be thought of as placing one's eyes along the reduced temperatue axis at a high value of T, and looking backwards at the p-V projection.

The pseudo three dimensional representation of the vapor-liquid coexistence locus for the van der Waals fluid.