\(k=\frac{2.303}{t} \log \frac{a}{a-x}\)

Where, \(k=\) Rate constant

\(a=\) initial concentration

\(a-x=\) concentration after time \({ }^{\prime} t^{\prime}\)

For half-life, \(t=t_{1 / 2}, x=\frac{a}{2}\)

On substituting the values, we get

\(k=\frac{2.303}{t_{1 / 2}} \log \frac{a}{a-\frac{a}{2}}=\frac{2.303}{t_{1 / 2}} \log 2\)

\(=\frac{0.693}{t_{1 / 2}}\)

\(k=\frac{0.693}{t_{1 / 2}}\)

Thus, \(t_{1/2}\) of a first order reaction does not depend upon the concentration.