$8\mathrm{.}7$ The Redlich$-$Kwong equation of state is given by

$p=\frac{RT}{v-b}-\frac{a}{v(v+b)\sqrt[\phantom{2}]{T}}$

where $R=$ the universal gas constant $=0\mathrm{.}518k\frac{J}{kgK},T=$ absolute temperature $(K),p=$ absolute pressure $(kPa),$ and $v=$ the volume of a $kg$ of gas $(\frac{m^3}{k}g).$ The parameters a and $b$ are calculated by

$a=0\mathrm{.}427\frac{R^2{T}_c^{2\mathrm{.}5}}{p_c},b=0\mathrm{.}0866R\frac{T_c}{p_c}$

where $p_c=$ critical pressure $(kPa)$ and $T_c=$ critical temperature $(K).$ As a chemical engineer, you are asked to determine the amount of methane fuel $(p_c=4600$kPa and $T_c=191K)$ that can be held in a $3-m^3$ tank at a temperature of $-40C$ with a pressure of $65,000$kPa. Use a root$-$locating method of your choice to calculate $v$ and then determine the mass of methane contained in the tank.