The reason for almost doubling the rate of reaction on increasing the temperature of the reaction system by $10 \, ^\circ C$ is

The velocity of the chemical reaction doubles every $10^\circ C$ rise of temperature. If the temperature is raised by $50^\circ C$,the velocity of the reaction increases to about .......... times.

For reaction $A \to B$,rate constant $K_1 = A_1 e^{-E_{a_1}/RT}$ and for the reaction $X \to Y$,rate constant $K_2 = A_2 e^{-E_{a_2}/RT}$. If $A_1 = 10^8, A_2 = 10^{10}$ and $E_{a_1} = 600 \ cal \ mol^{-1}$,$E_{a_2} = 1800 \ cal \ mol^{-1}$,then the temperature at which $K_1 = K_2$ is (given: $R = 2 \ cal \ K^{-1} \ mol^{-1}$):

Consider the following statements related to the temperature dependence of rate constants. Identify the correct statements:
$A.$ The Arrhenius equation holds true only for an elementary homogeneous reaction.
$B.$ The unit of $A$ is the same as that of $k$ in the Arrhenius equation.
$C.$ At a given temperature,a low activation energy means a fast reaction.
$D.$ $A$ and $E_a$ as used in the Arrhenius equation depend on temperature.
$E.$ When $E_a > RT$,$A$ and $E_a$ become interdependent.
Choose the correct answer from the options given below:

Two reactions of the same order have equal Pre-exponential factors but their activation energies differ by $24.9 \ kJ/mol$. Calculate the ratio between the rate constants $\left( \frac{K_2}{K_1} \right)$ of these reactions at $27 \ ^\circ C$.

The temperature coefficient of a reaction is $2$. When the temperature is increased from $30^{\circ}C$ to $90^{\circ}C$, the rate of reaction is increased by (in $times$)