(Solved): The Peng-Robinson equation of state is given by \[ p=\frac{R T}{v-b}-\frac{a \alpha}{v^{2}+2 b v-b ...

The Peng-Robinson equation of state is given by \[ p=\frac{R T}{v-b}-\frac{a \alpha}{v^{2}+2 b v-b^{2}} \] where \( R= \) the universal gas constant \( [=0.518 \mathrm{k}] /(\mathrm{kg} \mathrm{K})], \mathrm{T}= \) absolute temperature \( (\mathrm{K}), p= \) absolute pressure \( (\mathrm{kPa}) \), and \( v= \) the volume of a \( \mathrm{mol} \) of \( g a s\left(\mathrm{~m}^{3} / \mathrm{kg}\right) \). The parameters \( a, b \) and \( \alpha \) are calculated by \[ \begin{array}{c} a=0.45724 \frac{\alpha\left(R T_{C}\right)^{2}}{p_{C}} \\ b=0.07780 R \frac{T_{C}}{p_{C}} \\ \times \quad \text { (2) } \\ \alpha=\left(1+\left(0.37464+1.54226 \omega-0.26992 \omega^{2}\right)\left(1-T_{r}^{0}\right.\right. \end{array} \] where \[ T_{r}=\frac{T}{T_{C}} \] As a chemical engineer, you are asked to determine the amount of methane fuel that can be held in a \( 3-\mathrm{m}^{3} \) tank at a temperature of \( 233 \mathrm{~K} \) with a pressure of \( 65,000 \mathrm{kPa} \). Methane's critical temperature, critical pressure and acentric factor are \( T_{c}=191.15 \mathrm{~K}, p_{c}=4641 \mathrm{kPa} \) and \( \omega=0.0115 \). Use Newton-Raphson method to calculate \( v \) with stopping error of \( 0.1 \% \) and then determine the mass of methane contained in the tank in \( \mathrm{kg} \). For all calculation, use at least 5 significant figures for better accuracy. Use numerical differentiation methods to differentiate the function.