Find the correct equation among the following.

The rate constant of a reaction is $2 \times 10^{-3} \ min^{-1}$ at $300 \ K$. When the temperature is increased by $20 \ K$,the rate constant becomes three times its initial value. Calculate the activation energy $(E_a)$ of the reaction. Also,determine the rate constant at $310 \ K$.

The variation of the rate constant with temperature is given by the Arrhenius equation $k = A e^{-E_a / (RT)}$. If $T \to \infty$,the rate constant $k$ will be equal to:

For a complex reaction $A \xrightarrow{K} \text{products}$,where $Ea_1 = 180 \ kJ/mol$,$Ea_2 = 80 \ kJ/mol$,and $Ea_3 = 50 \ kJ/mol$,the overall rate constant $K$ is related to individual rate constants by the equation $K = (\frac{K_1 \cdot K_2}{K_3})^{2/3}$. The activation energy $(kJ/mol)$ for the overall reaction is:

The rate of a reaction at $273 \ K$ is $R_0$. What will be the rate of the reaction at $313 \ K$? (Take the temperature coefficient as $2$)

The rate constant for the decomposition of $N_{2}O_{5}$ at various temperatures is given below:

$T / ^{\circ}C$	$0$	$20$	$40$	$60$	$80$
$10^{5} \times k / s^{-1}$	$0.0787$	$1.70$	$25.7$	$178$	$2140$

Draw a graph between $\ln k$ and $1 / T$ and calculate the values of $A$ and $E_{a}.$ Predict the rate constant at $30^{\circ}C$ and $50^{\circ}C$.