For a first order reaction,
\(k=\frac{2.303}{t}log\frac{[R]_0}{[R]}\)
At \(t_\frac{1}{2}\), [R] = \(\frac{[R]_0}{2}\)
So, the above equation becomes,
\(k=\frac{2.303}{t}log\frac{[R]_0}{[R]/2}\)
or, \(t_\frac{1}{2}\) = \(\frac{2.303}{k}log2\)
\(t_\frac{1}{2}\) = \(\frac{2.303}{k}\times0.3010\)
\(t_\frac{1}{2}\) = \(\frac{0.693}{k}\)
Thus, for a first order reaction, half-life period is constant, i.e., it is independent of initial concentration of the reacting species.