Use the data below

$=0\mathrm{.}099$

Molecular weight: $30\mathrm{.}07$

Critical temperature: $305\mathrm{.}33K$

Melting point: $89\mathrm{.}88K$

Critical pressure: $4\mathrm{.}872$MPa

and apply the newton method

$x_{i+1}=x_i-\frac{f(x_i)}{f^{\text{'}}(x_i)}$

to find the molar volume of the ethane at $40*{10}^5$ pascal and $370K.$ Using the Ping Robinson equation.

$p=\frac{RT}{V-b}-\frac{a}{(b)+b(V-b)}$

$a=0\mathrm{.}45724\frac{R^2{T}_c^2}{p_c}$

$b=0\mathrm{.}07780\frac{R{T}_c}{p_c}$

$=[1+m(1-\sqrt[\phantom{2}]{T_r}){]}^2$

$m=0\mathrm{.}379642+1\mathrm{.}48503-0\mathrm{.}1644{}^2+0\mathrm{.}016667{}^3$