MHT CET · Chemistry · Chemical Kinetics

Half-life of first order reaction X \(\longrightarrow\) \(\mathrm{Y}+\mathrm{Z}\) is 3 minutes. What is the time required to reduce the concentration of \({ }^{\prime} \mathrm{X}^{\prime}\) by \(90 \%\) of it's initial concentration?

A \(4 \cdot 12\) minutes
B \(9.969\) minutes
C \(9 \cdot 105\) minutes
D \(12 \cdot 05\) minutes

Verified Solution

Answer & Solution

Correct Answer

(B) \(9.969\) minutes

Step-by-step Solution

Detailed explanation

(A)
\(\mathrm{t}_{1 / 2}=3 \mathrm{~min} \)
\( \therefore \mathrm{k}=\frac{0.693}{\mathrm{t}_{1 / 2}}=\frac{0.693}{3}=0.231 \mathrm{~min}^{-1}\)
\({[\mathrm{A}]_{0}=\text { Original amount of reactant }=100} \)
\([\mathrm{A}]_{t}=\text {Reactant remaining unreacted}=\) \(100-90=10\)
For first order reaction,
\(t=\frac{2.303}{k} \log _{10} \frac{[A]_{0}}{[A]_{t}}=\frac{2.303}{0.231 \mathrm{~min}^{-1}} \log _{10} \frac{100}{10}\)\(=9.969 \mathrm{~min}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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