Chemical and environmental engineers are increasingly interested in recovering resources from waste streams. Given the high costs of the Haber$-$Bosch process you decide to look into ammonia release and recovery from urea. Consider the enzymatic reaction,

NH$$CONH$2($Urea$)+$ Urease $+$ H$2$O $2$ NH$3+$ CO$2+$ Urease

The rate law for the reaction is

$-$r$\_\{$U$\}=\backslash$frac$\{$kC$\_\{$U$\}\}\{$K$\_\{$M$\}+$C$\_\{$U$\}\}$

The following data were collected at $25^\{\backslash$circ$\}$C:

$-$nU$($mol~g$^\{-1\}$min$^\{-1\})$

$5\mathrm{.}92\backslash$times$10^\{-6\}$

$\backslash$frac$\{5\mathrm{.}44\backslash$times$10^\{-5\}\}\{10^\{-5\}\}$

$5\mathrm{.}43\backslash$times$10^\{-4\}$

$3\mathrm{.}98\backslash$times$10^\{-3\}$

$\backslash$frac$\{\mathrm{.}36\backslash$times$10^\{-2\}\}\{10$c$^\{2\}\}$

$1\mathrm{.}66\backslash$times$10^\{-2\}$

$1\mathrm{.}86\backslash$times$10^\{-2\}$

Cu $\backslash$overline$\{($mol~L$^\{-1\})\}$

$\backslash$frac$\{5\mathrm{.}92\backslash$times$10^\{-6\}\}\{10^\{-6\}\}$

$10^\{-4\}$

$10^\{-3\}$

$10^\{-1\}$

$\backslash$overline$\{10^\{0\}\}$

$($a$)$ What are the values of k and K$\_\{$M$\}\backslash$overline$\{?\}$

$($b$)$ Generate a curve for $-$r$\_\{$v$\}$ as a function of C$\_\{$U$\}.$ Your x$-$axis should range from C$\_\{11\}=10^\{-6\}$ to $10^\{\backslash$circ$\}$ using a logarithmic scale. Comment on the goodness of fit..

$($c$)$ If the activation energy is $35\mathrm{.}3$ kJ mol, plot $-$r$\_\{$u$\}$ vs$.$ Cu at $37^\{\backslash$circ$\}$C Overlay the $-$r$\_\{$u$\}$ curve for $37^\{\backslash$circ$\}$C onto your plot from $($c$).$ Explain differences observed for the two temperatures.