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 $\frac{1}{\left[A\right]_{t}}=\frac{1}{\left[A\right]_{0}}+kt \ldots\left(i\right)$
For half-life time $t=t_{1 /2}, \left[A\right]_{t}=\left[A\right]_{0}/ 2 $
On putting the values in Eq. $\left(i\right)$
$\frac{1}{\left[A\right]_{t/ 2}}=\frac{1}{\left[A\right]_{0}}+kt_{1 /2}$
$kt_{1/ 2}=\frac{2}{\left[A\right]_{0}}-\frac{1}{\left[A\right]_{0}}$
$t_{1/ 2}=\frac{1}{k} \left[\frac{1}{\left[A\right]_{0}}\right] \ldots\left(ii\right)$
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. $\left(i\right)$
$\frac{1}{\left[A\right]_{0} /4}=\frac{1}{\left[A\right]_{0}}+kt_{3 /4}$
$t_{3/ 4}=\frac{1}{k}\left[\frac{3}{\left[A\right]_{0}}\right] \ldots\left(iii\right)$
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