Two radiactive material `A_(1)` and `A_(2)` have decay constants of `10 lambda_(0)` and `lambda_(0)`. If initially they have same number of nyclei, the ratio of number of their undecayed nuclei will be `(1//e)` after a time

Options

A. `1/(lambda_(0))`
B. `1/(9lambda_(0))`
C. `(1)/(10lambda_(0))`
D. `1`

Explanation

To solve the problem step by step, we will use the formula for radioactive decay and the given decay constants for the two materials.

### Step 1: Understand the decay formula
The number of undecayed nuclei \( N \) at any time \( t \) can be expressed using the formula:
\[ N(t) = N_0 e^{-\lambda t} \]
where:
- \( N_0 \) is the initial number of nuclei,
- \( \lambda \) is the decay constant,
- \( t \) is the time.

### Step 2: Write the equations for both materials
For material \( A_1 \) with decay constant \( 10 \lambda_0 \):
\[ N_1(t) = N_0 e^{-10 \lambda_0 t} \]
For material \( A_2 \) with decay constant \( \lambda_0 \):
\[ N_2(t) = N_0 e^{-\lambda_0 t} \]

### Step 3: Find the ratio of undecayed nuclei
To find the ratio of undecayed nuclei \( \frac{N_1(t)}{N_2(t)} \):
\[ \frac{N_1(t)}{N_2(t)} = \frac{N_0 e^{-10 \lambda_0 t}}{N_0 e^{-\lambda_0 t}} = \frac{e^{-10 \lambda_0 t}}{e^{-\lambda_0 t}} = e^{-10 \lambda_0 t + \lambda_0 t} = e^{-9 \lambda_0 t} \]

### Step 4: Set the ratio equal to \( \frac{1}{e} \)
We need to find the time \( t \) when this ratio equals \( \frac{1}{e} \):
\[ e^{-9 \lambda_0 t} = \frac{1}{e} \]
Taking the natural logarithm of both sides:
\[ -9 \lambda_0 t = -1 \]
This simplifies to:
\[ 9 \lambda_0 t = 1 \]

### Step 5: Solve for \( t \)
Now, we can solve for \( t \):
\[ t = \frac{1}{9 \lambda_0} \]

Thus, the time at which the ratio of undecayed nuclei of the two materials equals \( \frac{1}{e} \) is:
\[ t = \frac{1}{9 \lambda_0} \]

### Conclusion
The answer to the question is \( t = \frac{1}{9 \lambda_0} \).

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