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:

For reaction $A \rightarrow P$, rate constant $k = 1.5 \times 10^3 \text{ s}^{-1}$ at $27^\circ\text{C}$. If activation energy for the above reaction is $60 \text{ kJ mol}^{-1}$, then the temperature (in $^\circ\text{C}$) at which rate constant $k = 4.5 \times 10^3 \text{ s}^{-1}$ is . . . . . . .

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:

The correct reaction profile diagram for a positive catalyst reaction.

What is the value of the slope if $\log_{10} K$ ($y$-axis) is plotted versus $1/T$ ($x$-axis) for the Arrhenius equation?

The rate of a reaction doubles,when the temperature is changed from $300 \ K$ to $310 \ K$. Activation energy of the reaction is....... $(R=8.314 \ J \ K^{-1} \ mol^{-1}, \log 2=0.301)$