USA Biology Olympiad (USABO) Practice Exam

If the rate constant for a first order reaction is 0.693 min^-1, what is the half-life of this reaction?

1. 0.500 min

2. 1.000 min

3. 2.000 min

4. 3.464 min

The correct answer is: 1.000 min

In a first-order reaction, the relationship between the rate constant and the half-life is particularly straightforward. The half-life (t₁/₂) of a first-order reaction is given by the formula: \[ t_{1/2} = \frac{0.693}{k} \] where k is the rate constant. Given that the rate constant (k) is 0.693 min⁻¹, you can substitute this value into the equation to find the half-life: \[ t_{1/2} = \frac{0.693}{0.693 \, \text{min}^{-1}} \] When you perform the calculation, the 0.693 in the numerator and denominator cancel each other out, leading to: \[ t_{1/2} = 1.000 \, \text{min} \] This result indicates that the half-life of the reaction is 1.000 min, which embodies the characteristic of first-order kinetics where the half-life remains constant regardless of the initial concentration of the reactants.