Question 27 of 37intermediate🔧 ApplyNumerical3 marks

A first order reaction takes 40 min for 30% decomposition. Calculate t1/2t_{1/2}.

Correct Answer

The half-life of the reaction is 77.7 minutes.

Exercise: EXERCISES | Q: 3.19 | (Page 27)
For More Understanding

Explanation

This question tests the understanding of first order reaction kinetics. The student needs to first calculate the rate constant using the integrated rate law for first order reactions, then use the relationship between rate constant and half-life. The key insight is that 30% decomposition means 70% of the reactant remains, which gives the ratio [R]₀/[R] = 100/70.

Solution Steps

  1. Step 1: Identify the given data. Time (t) = 40 min, 30% decomposition means 70% reactant remains. So [R] = 0.70[R]₀.

  2. Step 2: Calculate the rate constant (k) using the first order integrated rate law: k = (2.303/t) × log([R]₀/[R]) = (2.303/40) × log(100/70) = (2.303/40) × log(1.4286) = (2.303/40) × 0.1549 = 0.00892 min⁻¹.

  3. Step 3: Calculate half-life using the formula: t₁/₂ = 0.693/k = 0.693/0.00892 = 77.7 min.

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