Many unstable nuclie can decay spontaneously to a nucleus of lower mass but differnet combination of nucleons. The process of spontaneous emission of radiation is called radioactivity substance.
Radioactive decay is a statistical process. Radiaactivity is independent of all external conditions. The number of decays per unit time or decay rate is called activity. Activity exponentially decrease with time. Mean lifetime is always greater than half-life time.
If `T_(H)` is the half-life and `T_(M)` is the mean life. Which of the following statement is correct.

Options

A. `T_(M) gtT_(H)`
B. `T_(M)ltT_(H)`
C. Both ar directly proportional to square of the decay constant.
D. `T_(M) prop lambda_(0)`

Explanation

To solve the problem regarding the relationship between half-life (T_H) and mean life (T_M) of a radioactive substance, we will follow these steps:

### Step 1: Understand the Definitions
- **Half-life (T_H)**: The time taken for half of the radioactive nuclei in a sample to decay.
- **Mean life (T_M)**: The average time that a nucleus remains undecayed before it decays.

### Step 2: Write the Formulas
- The mean life (T_M) is related to the decay constant (λ) by the formula:
\[ T_M = \frac{1}{\lambda} \]
- The half-life (T_H) is also related to the decay constant (λ) by the formula:
\[ T_H = \frac{\ln 2}{\lambda} \]

### Step 3: Compare T_H and T_M
To compare T_H and T_M, we can express T_H in terms of T_M:
- From the formulas, we can substitute for λ:
\[ T_H = \frac{\ln 2}{\lambda} = \ln 2 \times T_M \]

### Step 4: Evaluate the Relationship
- Since \(\ln 2\) is approximately 0.693, we can see that:
\[ T_H = 0.693 \times T_M \]
- This shows that the half-life (T_H) is less than the mean life (T_M).

### Conclusion
Thus, the correct statement is that the half-life is less than the mean life:
\[ T_H < T_M \]