What is the half-life of a radioactive substance if its decay constant is $\lambda$?

Solution

1. The radioactive decay law is given by $N(t)=N_0{e}^{-\lambda t}$, where $N(t)$ is the amount remaining at time $t$.
2. At the half-life $T$, the remaining amount is half of the initial amount, so $N(T)=N_0/2$.
3. Substituting this into the decay equation gives $1/2=e^{-\lambda T}$, which simplifies to $\ln (2)=\lambda T$.
4. Solving for $T$ yields $T=\ln (2)/\lambda$, and since $\ln (2)\approx 0.693$, we have $T=0.693/\lambda$.
5. Hence the answer is (A).