Question

The rate of first order reaction is $0.04\text{ }molL/s$ at 10 minutes and $0.03\text{ }molL/s$ at 20 minutes after initiation. Find the half-life of the reaction.
A. 2.406 min
B. 24.06 min
C. 240.6 min
D. 0.204 min

Answer 1

Hint: Try to recall that the half-life of a reaction is time duration during which the concentration of a reactant is reduced to half of its initial concentration. Also, the half period of a first order reaction is independent of initial concentration. Now by using this and the formula given below you can easily find the half-life of reaction.

Complete step by step answer:
 Recall the equation $k=\dfrac{2.303}{t}{{\log }_{10}}\dfrac{a}{A}$, where $k$ is the rate constant for first order reaction, $a$ is the initial concentration and $A$ is the final concentration.
\[{{t}_{1/2}}=\dfrac{0.693}{k}\], where $k$ is the rate constant and ${{t}_{1/2}}$ is the half-life for the first order reaction.
- The rate law expression for a first order reaction rate is given, $Rate=k[A]$ where $[A]$ is the concentration of reactant at time $t$.

Calculations:
Let ${{r}_{1}}$ the rate of reaction after 10 min. and ${{r}_{2}}$ be the rate of reaction after 20 min.
So, ${{r}_{1}}=k\left[ A \right]$ and ${{r}_{2}}=k\left[ B \right]$.
Given that,
\[\begin{align}
 & {{r}_{1}}=0.04\text{ }molL/s \\
 & {{r}_{2}}=0.03\text{ }molL/s \\
\end{align}\]
\[{{r}_{1}}=k\left[ A \right]\]
or, Eq.1: $0.04=k\left[ A \right]$
Similarly, ${{r}_{2}}=k\left[ B \right]$
or, Eq.2: $0.03=k\left[ B \right]$
Dividing eq. 1 by eq. 2, we get
\[\dfrac{\left[ A \right]}{\left[ B \right]}=\dfrac{4}{3}\]

Let $[A]$ be the initial concentration of reactant and $[B]$ be the final concentration after a time gap of $t=10\min $
So, using above formula
\[\begin{align}
 & k=\dfrac{2.303}{t}{{\log }_{10}}\dfrac{\left[ A \right]}{\left[ B \right]} \\
 & k=\dfrac{2.303}{10}{{\log }_{10}}\dfrac{4}{3} \\
 & \Rightarrow k=0.0285{{s}^{-1}} \\
\end{align}\]
We know that,
\[\begin{align}
 & {{t}_{1/2}}=\dfrac{0.693}{k} \\
 & \Rightarrow {{t}_{1/2}}=\dfrac{0.693}{0.0285} \\
 & or,\text{ }{{t}_{1/2}}=24.06\min \\
\end{align}\]
So, the correct answer is “Option B”.

Note: It should be remembered to you that radioactive disintegration reactions are first order reactions. In these reactions, the whole of substance never disintegrates. The formula for the half-life of any species changes as per the order of the reaction.