$9-24.$ Use the vapor pressure data given in Problem $9-7$ and the density data given in Problem $913$ to calculate $\frac{{}_{\text{v}\text{a}\text{p}}}{b}ar(H)$ for methanol from the triple point $(175\mathrm{.}6K)$ to the critical point $(512\mathrm{.}6$ K$).$ Plot your result.

Data from $9-7$:

$9-7.$ The vapor pressure of methanol along the entire liquid$-$vapor coexistence curve can be expressed very accurately by the empirical equation

$ln(\frac{P}{b}ar)=-\frac{10\mathrm{.}752849}{x}+16\mathrm{.}758207-3\mathrm{.}603425x$

$+4\mathrm{.}373232x^2-2\mathrm{.}381377x^3+4\mathrm{.}572199(1-x{)}^{1\mathrm{.}70}$

where $x=\frac{T}{T_c},$ and $T_c=512\mathrm{.}60K.$ Use this formula to show that the normal boiling point of methanol is $337\mathrm{.}67K.$

Data from $9-13$:

$9-13.$ The densities of the coexisting liquid and vapor phases of methanol from the triple point to the critical point are accurately given by the empirical expressions

$\frac{{}^{\text{I}}}{{}_c}-1=2\mathrm{.}51709(1-x{)}^{0\mathrm{.}350}+2\mathrm{.}466694(1-x)$

$-3\mathrm{.}066818(1-x^2)+1\mathrm{.}325077(1-x^3)$

and

$ln(\frac{{}^g}{{}_c})=-10\mathrm{.}619689\frac{1-x}{x}-2\mathrm{.}556682(1-x{)}^{0\mathrm{.}350}$

$+3\mathrm{.}881454(1-x)+4\mathrm{.}795568(1-x{)}^2$

where ${}_c=8\mathrm{.}40$mol$*L^{-1}$ and $x=\frac{T}{T_c},$ where $T_c=512\mathrm{.}60K.$ Use these expressions to plot ${}^1$ and ${}^g$ against temperature, as in Figure $9\mathrm{.}7.$ Now plot $\frac{{}^l+{}^g}{2}$ against $T.$ Show that this line intersects the ${}^l$ and ${}^g$ curves at $T=T_c.$