$($a$)$ Explain how a temperature jump experiment can be used to determine the rate constants for the forward and reverse first$-$order steps of a chemical equilibrium.

$($b$)$ In a temperature jump experiment the temperature of $1,2$difluoroethylene, $C_2{H}_2{F}_2,$ was suddenly increased from $1000K$ to $1300K.$ The equilibrium constant for the cis$/$trans isomerisation of $C_2{H}_2{F}_2$ was observed to relax to a new value of $0\mathrm{.}54$ with a half$-$life of $0\mathrm{.}6ms.$ Determine the values of the forward and reverse rate constants for the cis$/$trans isomerisation of $C_2{H}_2{F}_2.$

$($c$)$ Given that the activation energy for the cis$-$trans process is $263kJmo{l}^{-1}$ and that for the reverse trans$-$cis process is $251kJmo{l}^{-1}$ determine the equilibrium constant at room temperature.