The production process to produce Y is carried out in a batch reactor using raw material X$,$ the equilibrium reaction follows the first order of reaction as follows:

$x{k}_2^{k_1}Y$

If the initial concentrations of X and Y

C x$0=100$ gmol$/$L and

C Y$0=10$ gmol$/$L

and the coefficient of reaction rate

k $1=5$ hours$^-1$ and

k $2=2$ hours$^-1$

Calculate the optimal residence time to yield a maximum value of C Y