Gaseous $A$ is to be absorbed by liquid $B$ in an experimental packed column. The reaction is elementary second order, $A+BR+S,$ and the system conditions are estimated as follows

$k_{Ag}={10}^{-3}mo\frac{l}{m^2}*h*Pa$;$k={10}^5\frac{m^3}{h}*mol$;$H_A={10}^{4}Pa.\frac{m^3}{m}ol$;

$k_{Al}=1\frac{m}{h}$;$D_A=D_B={10}^{-6}\frac{m^2}{h}$;$f=0\mathrm{.}1$;$a=100m^{-1}$;$C_B={10}^{3}mo\frac{l}{m^3}$;

We intend to run our experiments where the liquid film is in the pseudo first$-$order regime. To insure this, the pressure of A must lie in some range of values. Find this range and the corresponding absorption rate $r_A^{\text{'}\text{'}\text{'}}.$