(Solved): Question 3 Peng-Robinson equation given in equation (3) is one of the Equations of States (EOS) tha ...

Question 3 Peng-Robinson equation given in equation (3) is one of the Equations of States (EOS) that relates specific volume 1 (V) of gases to temperature $(\mathrm{T})$ and pressure $(\mathrm{P})$ $P=\frac{RT}{V?b}?\frac{a}{V(V+b)+b(V?b)}$ The constants a, b are determined as follows: $\begin{array}{rl}a& =0.45724\left(\frac{R^2{T}_c^2}{p_c}\right)\left[1+m\left(1?T_r^{\frac{1}{2}}\right)\right]\\ b& =0.07780\left(\frac{R^2{T}_c}{p_c}\right)\\ m& =0.37464+1.5422??0.26992{?}^2\\ T_r& =\frac{T}{T_c}\end{array}$ where $T$ is the temperature in $K,p$ is the pressure in atm, $\mathrm{V}$ is the specific molar volume in L/kgmol, $R$ is the molar gas constant in $\frac{\mathrm{atm}.\mathrm{L}}{\mathrm{kgmol}.K},?$ is the acentric factor, $T_c$ and $p_c$ are critical temperature and critical pressure, in $K$, and $atm$, respectively, and $T_r$ is reduced temperature. (a) For n-hexane at $1\mathrm{atm}$ and $25^{?}\mathrm{C}$, perform the first three iterations to estimate the specific molar volume using bisection method, taking the boundary of the solution to be in the range 50 to 200 . Take $p_c=29.92$ atm, $T_c=507.85\mathrm{K},?=0.3007$, and $R=83.14\frac{\mathrm{atm},\mathrm{L}}{\mathrm{kgmol}.\mathrm{K}}$ (b) Perform the first three iterations to estimate the specific volume for the same case in (a) using regula-falsi method, taking the boundary of the solution to be in the range 50 to 200 . (c) Perform the first three iterations to estimate the specific volume for the same case in (a) using Newton method, taking the guess for the solution to be 200 .