can you help me just proposing the following problem P$3-17($Fogler $4$ta ed$)$ but following the steps of example $4-1($Fogler $4$ta ed$)$ i put the image below to calculate how long it takes to achieve the desired conversion. THANK YOU

P$3-17$ B Consider a cylindrical batch reactor that has one end fitted with a frictionless piston attached to a spring $($Figure P$3-17).$ The reaction: A $+$ B $=8$C

with the rate expression

$-r_A=k_1{C}_A^2{C}_B$

is taking place in this type of reactor.

$($a$)$ Write the rate law solely as a function of conversion, numerically evaluating all possible symbols. Ans.: $-r_A=5\mathrm{.}03{10}^{-9}[\frac{(1-x{)}^3}{(1+3x{)}^{\frac{3}{2}}}]$ lbmo$\frac{l}{f}{t}^3*s.)$

$($b$)$ What is the conversion and rate of reaction when $V=0\mathrm{.}2f{t}^3?($Ans$.$: $x=0\mathrm{.}259,-r_A=8\mathrm{.}63{10}^{-10}$lbmo$\frac{l}{f}{t}^3*s.)$

Additional information:

Equal moles of A and $B$ are present at $t=0$

Initial volume: $0\mathrm{.}15f{t}^3$

Value of $k_1$:$1\mathrm{.}0(f\frac{t^3}{l}$bmol${)}^2*s^{-1}$

The relationship between the volume of the reactor and pressure within the reactor is

$V=(0\mathrm{.}1)(P),(Vinf{t}^3,Pin$ atm $)$

Temperature of system $($considered constant$)$: $140F$

Gas constant: $0\mathrm{.}73f{t}^3*at\frac{m}{l}$bmol$*R$