The effect of $pH$ on the rate constant for the hydrolysis of an ester is given by:

At $100\,^\circ C,$ the gaseous reaction $A_{(g)} \rightarrow 2B_{(g)} + C_{(g)}$ is found to be first order. Starting with pure $A,$ if at the end of $10\,\min$,the total pressure of the system is $160\,mm$ and after a long time,when dissociation of $A$ was complete,it was $300\,mm.$ The partial pressure of $A$ at the end of $10\,\min$ is .......... $mm$

Fill in the blanks:
$(1)$ If $t_{1/2} = \frac{0.693}{k}$,then the order of reaction is .......... .
$(2)$ For a zero order reaction,$t_{1/2}$ is proportional to .......... of the initial concentration.
$(3)$ In a first order reaction,the rate of reaction is proportional to the .......... exponent of the concentration of the reactant.

Which of the following reaction mechanisms will have a fractional order with respect to $A_2$ or $B_2$?

Match the rate expressions in List-$I$ for the decomposition of $X$ with the corresponding profiles provided in List-$II$. $X_s$ and $k$ are constants having appropriate units.

The following graph shows the variation of rate constant $(k)$ with temperature $(T)$. Which graph follows the Arrhenius equation?