At time `t=0 N_(1)` nuclei of decay constant `lambda_(1)& N_(2)` nuclei of decay constant `lambda_(2)` are mixed. The decay rate of the mixture at time 't' is:

Options

A. `N_(1)N_(2)e^((lambda_(1)+lambda_(2))t)`
B. `+((N_(1))/(N_(2)))e^(-(lambda_(1)-lambda_(2))t)`
C. `+(N_(1)lambda_(1)e^(-lambda_(1)t) + N_(2)lambda_(2)e^(-lambda_(2)t))`
D. `+N_(1)lambda_(1)N_(2)lambda_(2)e^(-(lambda_(1)+lambda_(2))t)`

Explanation

To solve the problem, we need to determine the decay rate of a mixture of two different types of nuclei with given decay constants at a specific time \( t \).

### Step-by-Step Solution:

1. **Understanding Decay Rate**:
The decay rate of a radioactive substance is given by the formula:
\[ \frac{dN}{dt} = -\lambda N \]
where \( \lambda \) is the decay constant and \( N \) is the number of undecayed nuclei at time \( t \).

2. **Initial Conditions**:
At time \( t = 0 \), we have:
- \( N_1 \) nuclei with decay constant \( \lambda_1 \)
- \( N_2 \) nuclei with decay constant \( \lambda_2 \)

3. **Nuclei Remaining at Time \( t \)**:
The number of undecayed nuclei at time \( t \) for each type can be expressed as:
- For the first type:
\[ N_1(t) = N_1 e^{-\lambda_1 t} \]
- For the second type:
\[ N_2(t) = N_2 e^{-\lambda_2 t} \]

4. **Calculating Decay Rates**:
The decay rates for each type of nuclei at time \( t \) can be calculated as follows:
- For the first type:
\[ \frac{dN_1}{dt} = -\lambda_1 N_1(t) = -\lambda_1 (N_1 e^{-\lambda_1 t}) \]
- For the second type:
\[ \frac{dN_2}{dt} = -\lambda_2 N_2(t) = -\lambda_2 (N_2 e^{-\lambda_2 t}) \]

5. **Total Decay Rate of the Mixture**:
The total decay rate of the mixture at time \( t \) is the sum of the decay rates of both types:
\[ \text{Total Decay Rate} = \frac{dN_1}{dt} + \frac{dN_2}{dt} \]
Substituting the expressions from the previous step:
\[ \text{Total Decay Rate} = -\lambda_1 N_1 e^{-\lambda_1 t} - \lambda_2 N_2 e^{-\lambda_2 t} \]

6. **Final Expression**:
The decay rate can be expressed as:
\[ \text{Total Decay Rate} = N_1 \lambda_1 e^{-\lambda_1 t} + N_2 \lambda_2 e^{-\lambda_2 t} \]
This is the required decay rate of the mixture at time \( t \).

### Conclusion:
Among the options provided, the correct expression for the decay rate of the mixture at time \( t \) is:
\[ N_1 \lambda_1 e^{-\lambda_1 t} + N_2 \lambda_2 e^{-\lambda_2 t} \]