a$.$ Show that a change of variables allows the material balance to be written as

null

$R\frac{dR}{dP}+\frac{1}{3}\frac{R^2}{P}=-\frac{}{P}$

where

$=\frac{k_c{R}_{g}T{C}^{**}}{{}_{L}g}<0$ since $<0$

b$.$ Solve the equation in part $($a$)$ subject to the initial condition $R(o)=R_O,P(o)=P_o=P_A+$

$g{z}_0$ and prove that

$\frac{P}{P_0}=(\frac{R_0^2+3}{R^2+3}{)}^{\frac{3}{2}}$

Now, find the expression for the time required to cause a soluble bubble to completely dis$-$

appear $(R0).$

Answer: $=\frac{[P_0{R}_0]}{k_c{C}^{**}{R}_{g}T}$;$\frac{t_f}{}=[1+\frac{2}{9}(\frac{R_0^2}{})]$

Only need b time expresssion