For a first-order reaction, the rate constant \( k \) is given by \[ k = \frac{2.303}{t} \log \frac{[A]_0}{[A]} \] When a reaction is 25% complete, 75% of the reactant remains. Therefore, the ratio of initial to remaining reactant concentration is \[ \frac{[A]_0}{[A]} = \frac{100}{75} = \frac{4}{3} \] and the time elapsed is \( t = 30 \text{ min} \). Substituting these values, the rate constant is \[ k = \frac{2.303}{30} \log \left( \frac{4}{3} \right) = \frac{2.303}{30} \times 0.1249 \approx 0.00958 \text{ min}^{-1} \] The half-life for a first-order reaction is calculated using the formula \[ t_{1/2} = \frac{0.693}{k} \] Using the determined rate constant, the half-life is \[ t_{1/2} = \frac{0.693}{0.00958} \approx 72.33 \ \text{min} \]