If half life of a radioactive substnace is 1 month, then which of these are true ?

Options

A. 7/8 part of substance disintegrate in 3 months
B. 1/8 part of substance disintegrate in 4 months
C. Substance disintegrates completely in 4 months
D. the substence disintegrates completely in 2 months

Answer

To solve the problem, we need to analyze the half-life of the radioactive substance and how it affects the quantity of the substance over time. Given that the half-life (t_half) is 1 month, we can determine the amount of substance remaining after specific time intervals.

### Step-by-Step Solution:

1. **Understanding Half-Life**:
- The half-life of a radioactive substance is the time required for half of the radioactive nuclei in a sample to decay. Here, t_half = 1 month.

2. **Calculating Remaining Substance After 1 Month**:
- If we start with N0 (initial quantity), after 1 month, half of the substance will remain:
\[ N(1 \text{ month}) = \frac{N_0}{2} \]

3. **Calculating Remaining Substance After 2 Months**:
- After another month (2 months total), half of the remaining substance will decay again:
\[ N(2 \text{ months}) = \frac{N(1 \text{ month})}{2} = \frac{N_0}{2^2} = \frac{N_0}{4} \]

4. **Calculating Remaining Substance After 3 Months**:
- Continuing this process, after 3 months:
\[ N(3 \text{ months}) = \frac{N(2 \text{ months})}{2} = \frac{N_0}{2^3} = \frac{N_0}{8} \]
- Therefore, the amount that has disintegrated in 3 months is:
\[ \text{Disintegrated} = N_0 - N(3 \text{ months}) = N_0 - \frac{N_0}{8} = \frac{7N_0}{8} \]
- This confirms that **7/8 of the substance disintegrates in 3 months**, making the first option true.

5. **Calculating Remaining Substance After 4 Months**:
- After 4 months, we can calculate:
\[ N(4 \text{ months}) = \frac{N(3 \text{ months})}{2} = \frac{N_0}{2^4} = \frac{N_0}{16} \]
- The amount that has disintegrated in 4 months is:
\[ \text{Disintegrated} = N_0 - N(4 \text{ months}) = N_0 - \frac{N_0}{16} = \frac{15N_0}{16} \]
- Therefore, only **1/16 of the substance remains**, which means **1/16 of the substance has disintegrated**, not 1/8. Thus, the second option is false.

6. **Complete Disintegration**:
- The third option states that the substance disintegrates completely in 4 months. However, radioactive decay is an exponential process, and it never completely reaches zero in a finite time. Thus, this option is also false.

### Conclusion:
- **True Statements**:
- 7/8 of the substance disintegrates in 3 months (True).
- **False Statements**:
- 1/8 of the substance disintegrates in 4 months (False).
- The substance disintegrates completely in 4 months (False).