JEE Main · 2023 · 25 Jan 2023 · Shift 2 · Chemical Kinetics

For a first-order reaction the time is $t=\frac{1}{k}\ln \frac{a}{a-x}$ and the rate constant alone carries units of ${\mathrm{s}}^{-1}$. Such a reaction approaches completion only asymptotically, so it never truly completes in a finite time, and its half-life is $\frac{0.693}{k}=\frac{0.693}{4.6\times 10^{-3}}\approx 150\mathrm{s}$, not $500\mathrm{s}$, so both of those statements are wrong. The time for ninety per cent completion is $\frac{\log (100/10)}{\log (100/90)}=\frac{1}{1-2\log 3}=\frac{1}{0.04}=25$ times the time for ten per cent completion, so stating it the other way round makes that statement wrong as well. The degree of dissociation does equal $1-e^{-kt}$, since $\frac{a-x}{a}=e^{-kt}$, and the rate, $k[A]$, carries different units from $k$, so of the five only the statement on the degree of dissociation is correct. The number of correct statements is one.