Disintegration constant of a radioactive material is `lambda`:

Options

A. Its half life equal `(log_e^2)/lambda`
B. its means life equals `1/lambda`
C. At time equal to mean life, 63% of the initial radioactive material is left undecayed
D. After 3-half lives, `1/3`rd of the initial radioactive material is left undecayed

Answer

To solve the problem regarding the disintegration constant (λ) of a radioactive material, we need to analyze the given options step by step.

### Step 1: Understanding the Disintegration Constant
The disintegration constant (λ) is a measure of the probability per unit time that a nucleus will decay. It is related to the half-life (T₁/₂) and mean life (τ) of the radioactive material.

### Step 2: Finding the Half-Life
The half-life of a radioactive substance is the time required for half of the radioactive nuclei in a sample to decay. The relationship between the half-life and the disintegration constant is given by the formula:

\[ T_{1/2} = \frac{0.693}{\lambda} \]

Thus, the first option states that the half-life equals \( \log_e(2)/\lambda \). Since \( \log_e(2) \) is approximately 0.693, this option is correct.

### Step 3: Finding the Mean Life
The mean life (τ) of a radioactive material is the average time a nucleus exists before decaying. The relationship between mean life and disintegration constant is given by:

\[ \tau = \frac{1}{\lambda} \]

The second option states that the mean life equals \( 1/\lambda \), which is indeed correct.

### Step 4: Analyzing the Third Option
The third option states that at time equal to the mean life, 63% of the initial radioactive material is left undecayed.

To analyze this, we can use the exponential decay formula:

\[ N(t) = N_0 e^{-\lambda t} \]

At \( t = \tau \):

\[ N(\tau) = N_0 e^{-1} \approx N_0 \times 0.3679 \]

This means that approximately 36.79% of the material has decayed, and thus about 63.21% remains undecayed. Therefore, this option is correct.

### Step 5: Analyzing the Fourth Option
The fourth option states that after three half-lives, one-third of the initial radioactive material is left undecayed.

After three half-lives, the remaining quantity of the material can be calculated as follows:

1. After the first half-life: \( N_0/2 \)
2. After the second half-life: \( N_0/4 \)
3. After the third half-life: \( N_0/8 \)

Thus, after three half-lives, \( N_0/8 \) remains, not \( N_0/3 \). Therefore, this option is incorrect.

### Conclusion
The correct options are:
1. Half-life equals \( \frac{0.693}{\lambda} \) (Correct)
2. Mean life equals \( \frac{1}{\lambda} \) (Correct)
3. At time equal to the mean life, approximately 63% remains undecayed (Correct)
4. After three half-lives, one-third remains (Incorrect)

### Final Answer
The correct options are 1, 2, and 3.

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