$5\mathrm{.}84.$ Using the SRK equation of state $($or any other cubic equation of state$)$ to determine a specific volume from a specified temperature and pressure requires a trial$-$and$-$error calculation. Three computer$-$based approaches to solving this problem may be used: $(1)$ spreadsheet programs such as Excel's Solver; $(2)$ mathematical packages such as Mathcad, Mathematica, Matlab, and Polymath; and $(3)$ programming languages such as Fortran and $C^{++}.$ The goal of this problem is to use the first two approaches to determine $)$ for $C{O}_2$ at $($i$)200K$ and $6\mathrm{.}8$atm; $($ii$)250K$ and $12\mathrm{.}3$atm; $($iii$)300K$ and $6\mathrm{.}8$atm; $($iv$)300K$ and $21\mathrm{.}5$atm; and $($v$)300K$ and $50\mathrm{.}0$atm.

$($a$)$ Starting with Equation $5\mathrm{.}3-8,$ derive the following equivalent expression for the SRK equation of state:

$f(hat(V))=$Phat$(V{)}^3-$RThat$(V{)}^2+(a-b^{2}P-bRT)hat(V)-ab=0$

$($b$)$ Prepare a spreadsheet to take as inputs a species identifier $($such as $C{O}_2),$ the critical temperature, critical pressure, Pitzer acentric factor, and the temperatures and pressures for which hat$(V)$ is to be calculated, and to calculate hat$(V)$ using Equations $5\mathrm{.}3-8$ to $5\mathrm{.}3-13$ for each of the specified conditions. The spreadsheet should have the following structure:

$\backslash$table$[[$PROBLEM $5\mathrm{.}84-$SRK EQUATION SPREADSHEET$],[$Species$,CO2,,,,],[Tc(K),304\mathrm{.}2,,,,],[Pc(atm),72\mathrm{.}9,,,,],[$w$,0\mathrm{.}225,,,,],[a,***,,,,],[$b$,***,,,,],[m,***,,,,],[T(K),P(atm),$alpha,V$($ideal$),V(SRK),f(V)$