MHT CET · Chemistry · Chemical Kinetics

What time is required for 100 g of reactant to reduce to 25 g in a first order reaction having half life 5760 year?

A 4760 year
B 8640.26 year
C 2880.15 year
D 11526.48 year

Verified Solution

Answer & Solution

Correct Answer

(D) 11526.48 year

Step-by-step Solution

Detailed explanation

For a first order reaction,
\({\left[\mathrm{A}_0\right]=100 \mathrm{~g} ;\left[\mathrm{A}_{\mathrm{t}}\right]=25 \mathrm{~g} ; \mathrm{t}_{1 / 2}=5760 \text { years } } \)
\( \mathrm{k}=\frac{0.693}{\mathrm{t}_{1 / 2}}=\frac{0.693}{5760 \mathrm{yr}}=1.203 \times 10^{-4} \text { year }^{-1} \)
\( \mathrm{t}=\frac{2.303}{\mathrm{k}} \log _{10} \frac{\left[\mathrm{~A}_0\right]}{\left[\mathrm{A}_{\mathrm{t}}\right]} \)
\( \therefore \mathrm{t}=\frac{2.303}{1.203 \times 10^{-4} \text { year }^{-1}} \log _{10} \frac{100}{25} \)
\( =19141.8 \times 0.6021=11526.5 \text { years}\)

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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