For a first-order reaction, the half-life period ($t_{1/2}$) is given by the expression $t_{1/2} = \frac{0.693}{k}$. This equation clearly shows that the half-life of a first-order reaction is constant and independent of the initial concentration of the reacting species. In general, for an $n^{\text{th}}$ order reaction, the half-life is inversely proportional to the initial concentration raised to the power of $(n-1)$, i.e., $t_{1/2} \propto \frac{1}{[R]_0^{n-1}}$. For $n=1$, $t_{1/2} \propto [R]_0^0$, meaning it is independent of the initial concentration.