Consider a nuclear reaction :
`Aoverset(lambda_(1))rarrB+C`
and `Boverset(lambda_(2))rarrC`
A converts into B and to C with decay with decay constant `lambda_(1)` B is alos stable nucleus which futher decays into stable nucleus C with decay constant `lambda_(2).` Mark the correct statement(s)

Options

A. `(dN_(C))/(dt)=-(lambda_(1)N_(A)+lambda_(2)N_(B))`
B. `(dN_(A))/(dt)=-lambda_(1)N_(A)`
C. `(dN_(A))/(dt)=(lambda_(1)N_(A)+lambda_(2)N_(B))`
D. `(dN_(B))/(dt)=lambda_(1)N_(A)+lambda_(2)N_(B)`

Answer

To solve the problem, we need to analyze the nuclear reactions and their decay constants. The reactions can be summarized as follows:

1. \( A \overset{\lambda_1}{\rightarrow} B + C \)
2. \( B \overset{\lambda_2}{\rightarrow} C \)

Where:
- \( \lambda_1 \) is the decay constant for the reaction of A to B and C.
- \( \lambda_2 \) is the decay constant for the reaction of B to C.

### Step-by-Step Solution:

1. **Understanding the Decay of A**:
- The decay of A can be described by the equation:
\[ \frac{dN_A}{dt} = -\lambda_1 N_A \]
- This equation states that the rate of change of the number of atoms of A (\( N_A \)) is proportional to the number of atoms present, with a proportionality constant of \( -\lambda_1 \).

2. **Understanding the Decay of B**:
- The decay of B can be described by the equation:
\[ \frac{dN_B}{dt} = -\lambda_2 N_B \]
- This indicates that the rate of change of the number of atoms of B (\( N_B \)) is also proportional to the number of atoms present, with a proportionality constant of \( -\lambda_2 \).

3. **Relating the Decay of A and B**:
- As A decays to form B and C, the change in the number of B atoms can be expressed as:
\[ \frac{dN_B}{dt} = \lambda_1 N_A - \lambda_2 N_B \]
- Here, \( \lambda_1 N_A \) represents the formation of B from A, and \( -\lambda_2 N_B \) represents the decay of B into C.

4. **Total Decay Rate**:
- The total decay rate of A can be considered as:
\[ \frac{dN_A}{dt} = -(\lambda_1 + \lambda_2) N_A \]
- This indicates that A is effectively decaying into C directly with a combined decay constant of \( \lambda_1 + \lambda_2 \).

5. **Conclusion**:
- The correct statement is that the decay of A can be represented by the equation:
\[ \frac{dN_A}{dt} = -\lambda_1 N_A \]
- This means that the decay of A is solely dependent on \( \lambda_1 \) and not affected by the decay of B.