At $100C$ pure gaseous A reacts with stoichiometry $2AR$ in a constant volume batch reactor, and the data were obtained as follows

$\backslash$table$[[t,s,P_A,$ atm,,,,,$],[0,1,,,,,],[40,0\mathrm{.}6,,,,,],[80,0\mathrm{.}3,,,,,],[120,0\mathrm{.}1,,,,,],[160,0\mathrm{.}01,,,,,],[,,,,,,],[,,,,,,]]$

a$.$ Find the specific rate constant and order of reaction if $-(r_A)=kC{A}_A^{}$ using differentialfinite difference method. $(7$ pts$).$

b$.$ If this reaction is done in a PFR$,$ at molar flow rate of $2mo\frac{l}{sec},$ calculate the volume of PFR required to achieve $80\%$ conversion. $(3$ pts$)$

$S_x=\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{}}}{x}_i,S_{}=\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{}}}{x}_i^2,y=ax+b,b=\frac{S_{}{S}_y-S_{xy}{S}_x}{S_{}-S_x^2}$

$S_y=\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{}}}{y}_i,S_{xy}=\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{}}}{x}_i{y}_i,a=\frac{S_{xy}-S_x{S}_y}{S_{}-S_x^2}$