Two redioactive nuclei A and B have disintegration constants `lamda_(A)andlamda_(B)` and initially `N_(A)andN_(B)` number of nuclei of them are taken, then the time after which their undisintegrated nuclei are same is

Options

A. `(lambda_Alambda_B)/((lambda_A-lambda_B))"In"(N_B/N_A)`
B. `(1)/((lambda_A+lambda_B))"In"(N_B/N_A)`
C. `(1)/((lambda_B-lambda_A))"In"(N_B/N_A)`
D. `(1)/((lambda_A-lambda_B))"In"(N_B/N_A)`

Explanation

To solve the problem of finding the time after which the number of undecayed nuclei of two radioactive substances A and B becomes the same, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Decay Law**:
The number of undecayed nuclei at time \( t \) can be expressed using the radioactive decay formula:
\[ N(t) = N_0 e^{-\lambda t} \]
where \( N_0 \) is the initial number of nuclei, \( \lambda \) is the disintegration constant, and \( t \) is time.

2. **Applying the Decay Law to Both Nuclei**:
For nucleus A, the number of undecayed nuclei at time \( t \) is:
\[ N_A(t) = N_A e^{-\lambda_A t} \]
For nucleus B, the number of undecayed nuclei at the same time \( t \) is:
\[ N_B(t) = N_B e^{-\lambda_B t} \]

3. **Setting the Undecayed Nuclei Equal**:
We want to find the time \( t \) when the number of undecayed nuclei of A equals that of B:
\[ N_A e^{-\lambda_A t} = N_B e^{-\lambda_B t} \]

4. **Rearranging the Equation**:
Dividing both sides by \( N_A e^{-\lambda_B t} \):
\[ \frac{N_A}{N_B} = e^{(-\lambda_A + \lambda_B)t} \]

5. **Taking the Natural Logarithm**:
Taking the natural logarithm of both sides gives:
\[ \ln\left(\frac{N_A}{N_B}\right) = (-\lambda_A + \lambda_B)t \]

6. **Solving for Time \( t \)**:
Rearranging the equation to solve for \( t \):
\[ t = \frac{\ln\left(\frac{N_B}{N_A}\right)}{\lambda_B - \lambda_A} \]

### Final Result:
Thus, the time after which the number of undecayed nuclei of A and B becomes the same is:
\[ t = \frac{1}{\lambda_B - \lambda_A} \ln\left(\frac{N_B}{N_A}\right) \]