A certain radioactive material can undergo three constant `lambda,2lambda` and `3 lambda`. Then, the effective decay constant `lambda_(eff)` is equal to `n lambda`. What is the value of n?

Options

A. ` 6 `
B. ` 4 `
C. `2 `
D. ` 3 `

Explanation

To solve the problem, we need to find the effective decay constant \( \lambda_{\text{eff}} \) when a radioactive material can undergo three different decay constants: \( \lambda \), \( 2\lambda \), and \( 3\lambda \).

### Step-by-Step Solution:

1. **Identify the decay constants**:
The decay constants given are:
- \( \lambda_1 = \lambda \)
- \( \lambda_2 = 2\lambda \)
- \( \lambda_3 = 3\lambda \)

2. **Calculate the effective decay constant**:
The effective decay constant \( \lambda_{\text{eff}} \) is the sum of the individual decay constants:
\[ \lambda_{\text{eff}} = \lambda_1 + \lambda_2 + \lambda_3 \]
Substituting the values:
\[ \lambda_{\text{eff}} = \lambda + 2\lambda + 3\lambda \]

3. **Simplify the expression**:
Combine the terms:
\[ \lambda_{\text{eff}} = (1 + 2 + 3)\lambda = 6\lambda \]

4. **Relate \( \lambda_{\text{eff}} \) to \( n\lambda \)**:
According to the problem, we have:
\[ \lambda_{\text{eff}} = n\lambda \]
From our previous calculation, we found that:
\[ \lambda_{\text{eff}} = 6\lambda \]

5. **Set the equations equal**:
Now we can set the two expressions for \( \lambda_{\text{eff}} \) equal to each other:
\[ n\lambda = 6\lambda \]

6. **Solve for \( n \)**:
To find \( n \), divide both sides by \( \lambda \) (assuming \( \lambda \neq 0 \)):
\[ n = 6 \]

### Final Answer:
The value of \( n \) is \( 6 \).

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