The half-life period and the mean life period of a radioactive element are denoted respectively by Th and Tm. ThenTh=TmTh>TmTh<TmTh≥Tm

To address the question regarding the relationship between the half-life period ($T_h$) and the mean life period ($T_m$) of a radioactive element, let’s analyze the concepts and mathematical relationships involved.

Explanation of Terms

1. Half-Life ($T_h$): This is the time required for half of the radioactive atoms in a sample to decay.
2. Mean Life ($T_m$): This is the average lifetime of a particle before it decays; it's generally calculated as the expected value of the lifetime of a particle.

Relevant Concepts

The relationship between half-life and mean life is mathematically defined. For a radioactive decay process, the mean life ($T_m$) is given by the formula:

$T_m = \frac{T_h}{\ln(2)}$

where $\ln(2)$ is the natural logarithm of 2, which approximately equals 0.693.

Analysis of The Relationship

1. Substituting the Relationship: From the equation $T_m = \frac{T_h}{\ln(2)}$, we can infer that:

   $T_h = T_m \cdot \ln(2)$

2. Comparison: Since $\ln(2) \approx 0.693$, it follows that:

   $T_h < T_m$

This means that the half-life ($T_h$) of a radioactive substance is always less than its mean life ($T_m$).

Verification and Summary

From the derived relationship, we conclude that half-life is always shorter than mean life for any radioactive substance. Thus, we can summarize the relationship as:

$T_h < T_m$

Final Answer

Based on these calculations and analyses, the correct relationship is:

$T_h < T_m$

This implies that the half-life period is less than the mean life period.