$A$ first-order reaction decomposes such that the time taken for $1/8$ and $1/10$ of the initial concentration to decompose are $t_{1/8}$ and $t_{1/10}$ respectively. Find the value of $\frac{t_{1/8}}{t_{1/10}} \times 10$. (Given: $\log_{10} 2 = 0.3$)

For a reaction,the rate constant is $2.34 \ s^{-1}$. The half-life period for the reaction is ....... $s.$

For a first-order reaction, $20 \text{ mmol}$ of reactant is reduced to $10 \text{ mmol}$ in $1.151 \text{ min}$. Find the rate constant $k$. (in $\text{ min}^{-1}$)

For a first-order reaction,the rate constant is $4 \times 10^{-3} \, s^{-1}$. If the concentration of the reactant is $0.02 \, M$,what will be the rate of the reaction?

The slope of a graph $\log [A]_t$ versus '$t$' for a first-order reaction is $-2.5 \times 10^{-3} \,s^{-1}$. Find the rate constant of the reaction.

For the decomposition of $N_2O_5$ according to the equation $2N_2O_5(g) \to 4NO_2(g) + O_2(g)$,the reaction is of first order. After $30 \, \min$ from the start of the reaction in a closed vessel,the total pressure is $305.5 \, mm \, Hg$,and at the end of complete decomposition,the total pressure is $587.5 \, mm \, Hg$. Calculate the rate constant of the reaction.