Chemical X, a powdered solid, is slowly and continuously fed for half an hour into a well-stirred vat of water. The solid quickly dissolves and hydrolyses to Y, which then slowly decomposes to Z as follows $\mathrm{Y} \rightarrow \mathrm{Z}, \quad-r_{\mathrm{Y}}=k C_{\mathrm{Y}}, \quad k=1.5 \mathrm{hr}^{-1}$ The volume of liquid in the vat stays close to $3 \mathrm{m}^{3}$ throughout this operation, and if no reaction of Y to Z occurred, the concentration of Y in the vat would be $100 mol/ m^{3}$ at the end of the half-hour addition of X. (a) What is the maximum concentration of Y in the vat and at what time is this maximum reached? (b) What is the concentration of product Z in the vat after 1 hour?

Solution

VerifiedSince Chemical $X$ fed to the vat of water is slowly and dissolving phenomenon is quickly its means this procedure follow zero order kinetics and in the vat, $Y$ to $Z$ is first order

Now, this series reaction is zero order followed by the first-order reaction. So here I am using results which are given in the textbook on page no. $180$

$\begin{align*} X \xrightarrow{n_{1}=0, \hskip 0.5em k_{1}}& Y \xrightarrow{n_{2}=1 \hskip 0.5em k_{2}}Z\\ -r_{X}=&k_{1} \hskip 0.5em \text{Because zero order}\\ -r_{Y}=&k_{2}C_{Y}\\ k_{2}=&1.5 \hskip 0.5em \mathrm{hr^{-1}}\\ C_{X_{0}}=&100 \hskip 0.5em \mathrm{\dfrac{mol}{m^{3}}}\\ k_{1}=&\dfrac{C_{X_{0}}}{\text{time required to pile up } }\\ k_{1}=&\dfrac{100}{0.5}=200 \hskip 0.5em \mathrm{\dfrac{mol}{m^{3}.hr}}\\ \end{align*}$