For a first-order reaction, the half-life period ($t_{1/2}$) is given by the integrated rate equation : $t_{1/2} = \frac{\ln 2}{k} = \frac{2.303 \log 2}{k} = \frac{0.693}{k}$ As seen from the formula, the half-life of a first-order reaction is constant and independent of the initial concentration of the reactant . Therefore, the given initial concentration of $0.5 \text{ M}$ does not change the formula, and the half-life remains $\frac{\ln 2}{k}$.