A molecule undergoes two independent first order reactions whose respective half lives are 12 min and 3 min. If both the reactions are occurring then the time taken for the 50% consumption of the reactant is ___________ min. (Nearest integer)

Solution

Chemistry ma’am · 3 steps

Parallel first order paths add their rate constants, so add the reciprocals of the half lives.

1. Rate constants add. Both reactions use up the same reactant at the same time, so its total rate is (k₁ + k₂)[A]. The overall reaction is still first order, with k = k₁ + k₂.
2. Switch to half lives. For a first order reaction k = 0.693/t_(1/2). Put this into k = k₁ + k₂ and the 0.693 cancels from every term.

   1/t_(1/2) = 1/12 + 1/3 = (1 + 4)/12 = 5/12 min⁻¹

3. So. t_(1/2) = 12/5 = 2.4 min, which rounds to 2. It is shorter than either half life alone, as it should be with two paths working together.

The common slip: 15. Adding the two half lives directly gives 15 min, not 2. It is the rate constants that add, so the reciprocals of the half lives add.

The idea to keep. For parallel first order reactions, the overall k is the sum of the individual k values. Half lives combine like resistors in parallel, and the overall one is always shorter than the fastest path's.

Answer from the NTA final answer key. Solution by PrepWiser.