In a first order reaction A arrow B if k is rate constant and...

For a first order reaction. $k=\frac{2.303}{t} \log _{10} \frac{a}{a-x}$ when $t=t_{1/2}$ $k=\frac{2.303}{t_{1/2}} \log _{10} \frac{a}{a-\frac{a}{2}}$ or $t_{1/2}=\frac{2.303}{k} \log _{10} 2=\frac{\ln 2}{k}$ A Caution The half-life offirst order reaction is constant because the initial concentration of the reactant has no influence on the half-life of the reaction, i.e. the half-life is constant regardless of the concentration of the reactant.