Question

$A$ and $B$ are two different chemical species undergoing 1st order decomposition with half lives equal to $5$ sec and $7.5$ sec respectively. If the initial concentration of $A$ and $B$ are in the ratio $3:2$. Calculate $\frac{C_{A_1}}{C_{B_1}}$ after three half lives of $A$. Report your answer after multiplying it with $100$.

Answer 1

Half life of a reaction is the time required for reaction to be half completed.

Amount of substance left unreacted after one half life= $\frac{[A{]}_0}{2}$

Amount of substance left unreacted after two half lives=$\frac{[A{]}_0}{2n}$

Half life of $A=5$ sec.

So, after $3$ half lives of $A,t=15$ sec

Half life of $B=7.5$ sec

No. of half lives $B$ undergoes=$\frac{15}{7.5}=2$

Now, Amount of $A$ left after $3$ half lives,

$A_1=\frac{[A{]}_0}{23}=\frac{[A{]}_0}{8}$

Amount of $B$ left after $2$ half lives,

$B_1=\frac{[B{]}_0}{2^2}=\frac{[B{]}_0}{4}$

So,$\frac{C_{A1}}{C_{B1}}=\frac{[A{]}_0}{8}\times \frac{4}{[B{]}_0}$

$=\frac{1}{2}\frac{[A{]}_0}{[B{]}_0}$

Now, $\frac{[A{]}_0}{[B{]}_0}=\frac{3}{2}$ (given)

$\Rightarrow \frac{C_{A1}}{C_{B1}}=\frac{3}{2}\times \frac{1}{2}=\frac{3}{4}$

As stated in the question,

$100\times \frac{C_{A1}}{C_{B1}}=\frac{3}{4}\times 100=75$ (after multiplying by 100)