Question

If $T_{H}$ is the half-life and $T_{M}$ is the mean life. Which of the following statements is correct?
A. $T_{M} > T_{H}$
B. $T_{M} < T_{H}$
C. both are directly proportional to square of the decay constants
D. $T_{M}\propto \lambda_{0}$

Answer 1

Hint: Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration or nuclear disintegration) is the process by which an unstable atomic nucleus emits radiation.$t_{1/2}=\dfrac{0.693}{\lambda}$ where $t_{1/2}$ is the half-life and $\dfrac{1}{\lambda}$ is the mean life.
Formula used: $t_{1/2}=\dfrac{0.693}{\lambda}$ where ${{t}_{1/2}}$is the half-life$T_{H}$ and $\dfrac{1}{\lambda}$ is the mean life. i.e$T_{M}$

Complete step-by-step answer:
By the definition of half-life of radioactive elements, we know that half-life is the time taken by one half of the elements to have decayed or disintegrated. Whereas, mean life of an element is the average time of the radioactive element ie. the sum of the lifetimes of all the individual unstable nuclei in a sample, divided by the total number of unstable nuclei present.
It observed that the mean life of the atoms, is longer than their half-life.ie) The mean life of a particular species of unstable nucleus is always 1.443 times longer than its half-life or The mean life of an element equals the half-life of the substance divided by the natural logarithm of 2 which is about 0.693.
$t_{1/2}=\dfrac{0.693}{\lambda}$ where ${{t}_{1/2}}$ is the half-life $T_{H}$ and $\dfrac{1}{\lambda}$ is the mean life. i.e. $T_{M}$
Hence the answer is A.$T_{M} > T_{H}$

Additional Information:
Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material is said to be radioactive if the element contains unstable nuclei. Three of the most common types of decay are alpha decay, beta decay, and gamma decay, all of which involve emitting one or more particles or photons. An older unit of radioactivity is the curie, Ci, which was originally named after Marie Curie is defined as "the quantity or mass of radium emanation in equilibrium with one gram of radium (element)"The International System of Units (SI) unit of radioactive activity is the becquerel (Bq), named after the scientist Henri Becquerel. One Bq is defined as one transformation (or decay or disintegration) per second.

Note: Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation .SI unit of radioactive activity is the becquerel (Bq), earlier it was curie, Ci.