$($a$)$ The dependency of Gibbs energy on temperature is related in the following equations:

$\frac{d(\frac{G^o}{R}T)}{dT}=\frac{-H^o}{R{T}^2}$

with, $K=$exp$(\frac{-G^o}{RT})$

Derive an expression that relates the equilibrium constant, $K$ with the temperature from an initial state to a final state. Sate any assumption in deriving the expression. Based on the expression, describe the effect of $K$ value for both endothermic and exothermic reactions if the temperature increases.

$($b$)$ A study was performed on the following reaction an a dilute solution:

$NA{D}^{+}(aq)+$ ethanol$(I)=$NADH$(aq)+acldehyde(aq)+H^{+}(aq)$

The reaction is catalyzed by alcohol dehydrogenase enzyme, and the following data are the equilibrium concentration for the reaction

$\backslash$table$[[T(C),pH,$NADH$(M),$Acetaldehyde $(M),NA{D}^{+}(M),$Ethanol $(mM)$