Which of the following options correctly represents the relationship between $t_{7/8}$ and $t_{1/2}$ where $t_{7/8}$ represents the time required for the concentration to become $1/8$ of the original for a reaction of order $n$?

When the temperature increases from $300 \, K$ to $310 \, K$,the rate of reaction increases by $2.3$ times. If the rate constant at $300 \, K$ is $x$,then the rate constant at $310 \, K$ will be equal to .......

For the first order decomposition reaction of $N_2O_5$,it is found that -
$(a)$ $2N_2O_5 \rightarrow 4NO_{2(g)} + O_{2(g)}$ ; $-\frac{d[N_2O_5]}{dt} = k[N_2O_5]$
$(b)$ $N_2O_5 \rightarrow 2NO_{2(g)} + 1/2 O_{2(g)}$ ; $-\frac{d[N_2O_5]}{dt} = k'[N_2O_5]$
Which of the following is true?

For a particular reaction,the variation of the rate constant with temperature is given by $\ln k_t = \ln k_0 + \left( \frac{\ln (2.5)}{10} \right) \times t$ $(t \geqslant 0 \, ^{\circ}C)$. Here,$k_0$ is the rate constant at $0 \, ^{\circ}C$ and $k_t$ is the rate constant at $t \, ^{\circ}C$. The temperature coefficient of the reaction,assuming that the rate constant increases by the same factor for every $10 \, ^{\circ}C$ rise in temperature,is:

For the reaction $Cl_{2(aq)} + H_2S_{(aq)} \to S_{(s)} + 2H^{+}_{(aq)} + 2Cl^{-}_{(aq)}$,the rate law is $r = K[Cl_2][H_2S]$. Which of the following mechanisms is/are consistent with this rate law?
$A. \ H_2S \rightleftharpoons H^{+} + HS^{-}$ (fast)
$Cl_2 + HS^{-} \to 2Cl^{-} + H^{+} + S$ (slow)
$B. \ Cl_2 + H_2S \to H^{+} + Cl^{-} + Cl^{+} + HS^{-}$ (slow)
$Cl^{+} + HS^{-} \to H^{+} + Cl^{-} + S$ (fast)

The correct difference between $1^{st}$-order and $2^{nd}$-order reaction is that: