Question

A and B are two different chemical species undergoing first order decomposition with half lives equal to 5 sec. and 7.5 sec, respectively. If the initial concentration A and B are in the ratio 3: 2, calculate $\dfrac{{{C}_{A}}}{{{C}_{B}}}$, after three half lives of A. Report your answer after multiplying it with 100.

Answer 1

Hint: The concept of half life is to be used in this question. Half life is the time which is taken by the reactant so that it gets reduced to $\dfrac{1}{2}$ of its initial concentration.

Complete Solution :
- In order to answer our question, we need to understand the concept of half life of a reaction. Now, we know that every substance in the universe undergoes radioactive decomposition. However, the decomposition of substances depends upon the amount the substance taken for decomposition, the time required for full decomposition as well as the nature of the material. Every radioactive decay that occurs is an exponential decay, which means that the concentration after infinite decays will not be exactly 0, but will tend to become zero. Let us consider a first order reaction. Now, a point will come during disintegration that the amount of reactant becomes exactly half of its initial concentration. The time which is taken for this to occur is called the half life${{t}_{1/2}}$. Considering the rate constant of the reaction to be ‘k’ we obtain the relation between k and ${{t}_{1/2}}$ as ${{t}_{1/2}}=\dfrac{0.693}{k}$.

Now, let us come to our question. Taking initial concentrations of A and B as 150 and 100 and using above formula, we get the half lives as:
\[\begin{align}
 & {{t}_{1/2}}(A)=\dfrac{0.693}{{{k}_{A}}}=5\Rightarrow {{k}_{A}}=0.1386\,{{\sec }^{-1}} \\
 & {{t}_{1/2}}(B)=\dfrac{0.693}{{{k}_{B}}}=7.5\Rightarrow {{k}_{B}}=0.0924\,{{\sec }^{-1}} \\
\end{align}\]

So, 3 half lives of A mean $5\times 3=15\,\sec $, which is equal to 2 half lives of B. So, the concentrations of A and B are:
$\begin{align}
  & {{C}_{A}}=\dfrac{150}{{{2}^{3}}}=18.75\,mol\,{{L}^{-1}} \\
 & {{C}_{B}}=\dfrac{100}{{{2}^{2}}}=25\,mol\,{{L}^{-1}} \\
\end{align}$
So, the final ratio is:
\[\dfrac{{{C}_{A}}}{{{C}_{B}}}=\dfrac{18.75}{25}=0.75\]
On multiplying by 100 we get the answer as $0.75\times 100=75$, which is the required answer for the question.

Note: The half life of a reaction can be used to determine the speed of the reaction. A reaction with more half life means the reaction is slow, and a reaction with less half life means time taken for reactant concentration to decrease is high, which means reaction is fast.