The homogeneous decomposition of the free$-$radical polymerization initiator diethyl peroxydicarbonate $($DEPDC$)$ has been studied in supercritical carbon dioxide using and ideal CSTR$.$ The concentration of DEPDC in the feed to the CSTR was $0\mathrm{.}3mM.$ Because of this very low concentration, constant fluid density can be assumed. At $70C$ and a space time of $10$min, the fractional conversion of DEPDC was $0\mathrm{.}21.$ At $70C$ and a space time of $30$min, the fractional conversion of DEPDC was $0\mathrm{.}44.$

The rate equation for the DEPDC decomposition is believed to have the form:

$-r_{\text{D}\text{E}\text{P}\text{D}\text{C}}=k[DEPDC{]}^n$

What is the value of $n,$ and what is the value of $k$ at $70C?$

The hydrolysis of esters frequently is catalyzed by acids.

$R\frac{C}{O}OR+H_{2}OR\frac{C}{O}OH+R^{\text{'}}OH$

As the above hydrolysis reaction proceeds, more acid is produced and the concentration of catalyst increases. This phenomenon, known as "autocatalysis" is captured by the rate equation:

$-r_{R\frac{C}{O}O{R}^{\text{'}}}=(k_0+k_1[R\frac{C}{O}OH])[R\frac{C}{O}O{R}^{\text{'}}][H_{2}O]$

Here $k_0$ is the rate constant in the absence of the organic acid that is produced by the reaction. The initial$/$inlet concentrations are listed below:

$\{\begin{array}{cc}\text{:}[R\frac{C}{O}O& R^{\text{'}}{]}_0=1M\end{array}$

$\{\begin{array}{c}\text{:}[H_{2}O{]}_0=1M\end{array}$

$\{\begin{array}{cc}\text{:}[R\frac{C}{O}O& H{]}_0\\ [R^{\text{'}}OH{]}_0=0\end{array}$

The rate constants are $k_0=0\mathrm{.}01(Mh{)}^{-1},k_1=0\mathrm{.}20M^{-2}{h}^{-1}.$ The reaction takes place in the liquid phase.

$($a$)$ Calculate the values of $-r_{R\frac{C}{O}O{R}^{\text{'}}}$ at fractional conversions of ester of $x_{R\frac{C}{O}O{R}^{\text{'}}}=0,0\mathrm{.}1,0\mathrm{.}2,0\mathrm{.}3,0\mathrm{.}4,0\mathrm{.}5,0\mathrm{.}6,0\mathrm{.}7.$