Question

The relation between half-life $(T)$ and decay constant $(\lambda )$ is
$\begin{align}
  & \text{A}\text{. }\lambda T=1\text{ } \\
 & \text{B}\text{. }\lambda T=\dfrac{1}{2} \\
 & \text{C}\text{. }\lambda T={{\log }_{e}}2 \\
 & \text{D}\text{. }\lambda =\log 2T \\
\end{align}$

Answer 1

Hint: Radioactive decay has two important terms associated with it, Half-life and Decay constant. We will define both the terms and will find a relation between the two. The relation can be obtained by using the expression of radioactive decay law.

Formula used:
Radioactive decay law, $\dfrac{dN}{dt}=-\lambda N$

Complete step by step answer:
Radioactive decay is described as the process by which an unstable atomic nucleus loses energy by radiation. A sample material containing radioactive nuclei is considered as radioactive.
The decay of radioactive elements occurs at a fixed constant rate. The half-life of a radioisotope is the time required for one half of the concentration of unstable substance to degrade into a more stable material. We can say that half-life is the time required for a radioactive sample to reduce to half of its initial value. The half-life of a radioactive sample is represented by ${{T}_{\dfrac{1}{2}}}$.
Decay constant is proportionality between the size of a population of radioactive atoms and the rate at which the population decreases because of radioactive decay. The decay constant is represented by the symbol $\lambda $.
According to the radioactive decay law, the radioactive decays per unit time are directly proportional to the number of nuclei of radioactive compounds in the sample. Let the number of nuclei in a sample is $N$ and the number of radioactive decays per unit time $dt$ is $dN$, then,
$\dfrac{dN}{dt}=-\lambda N$
Integrating both sides,
$\int\limits_{{{N}_{o}}}^{N}{\dfrac{dN}{N}}=-\lambda \int\limits_{{{t}_{o}}}^{T}{dt}$
Taking initial time ${{t}_{o}}=0$, we get,
$\ln \dfrac{N}{{{N}_{o}}}=-\lambda T$
By definition of half-life, we have,
${{N}_{t}}=\dfrac{{{N}_{o}}}{2}$
Therefore,
$\ln 2=\lambda T={{\log }_{e}}2$
Relation between Decay constant and Half-life:
${{T}_{\dfrac{1}{2}}}=\dfrac{\ln 2}{\lambda }$
Hence, the correct option is C.

Note:
Half-life means how much time a radioactive sample takes to become half of its original concentration, while decay constant is a probability of decay per unit time. Half-life of a radioactive sample depends on the elimination rate of sample and the initial concentration of the sample, while decay constant is fixed for a particular radioactive nuclide.