Question

The ratio of the half-life time $(t_{1/2})$ , to the three quarter life-time, $(t_{3/4})$ , for a reaction that is second order

• A. depends directly on concentration of reactants
• B. is independent of concentration of reactant
• C. depends inversely on the concentration of reactants
• D. depends directly to the square of concentration of reactants

Answer 1

Answer: (B)

Integrated rate law for the second order
reaction is $\frac{1}{{\left[A\right]}_t}=\frac{1}{{\left[A\right]}_0}+kt\dots \left(i\right)$
For half-life time $t=t_{1/2},{\left[A\right]}_t={\left[A\right]}_0/2$
On putting the values in Eq. $\left(i\right)$
$\frac{1}{{\left[A\right]}_{t/2}}=\frac{1}{{\left[A\right]}_0}+k{t}_{1/2}$
$k{t}_{1/2}=\frac{2}{{\left[A\right]}_0}-\frac{1}{{\left[A\right]}_0}$
$t_{1/2}=\frac{1}{k}\left[\frac{1}{{\left[A\right]}_0}\right]\dots \left(ii\right)$
For three quarter half-life time $t=t_{3/4},{\left[A\right]}_t={\left[A\right]}_0/4$
On putting the values in Eq. $\left(i\right)$
$\frac{1}{{\left[A\right]}_0/4}=\frac{1}{{\left[A\right]}_0}+k{t}_{3/4}$
$t_{3/4}=\frac{1}{k}\left[\frac{3}{{\left[A\right]}_0}\right]\dots \left(iii\right)$
Now, ratio of $t_{1/2}$ to $t_{3/4}$ is given by
$\frac{t_{1/2}}{t_{3/4}}=\frac{\frac{1}{k}\left[\frac{1}{{\left[A\right]}_0}\right]}{\frac{1}{k}\left[\frac{3}{{\left[A\right]}_0}\right]}$
$t_{1/2}:t_{3/4}=1:3$
Hence, $t_{1/2}:t_{3/4}$ is independent of the concentration of reactant