A sample of a radioactive nucleus A disintegrates to another radioactive nucleus B, which in turn disintegrates to some other stable nucleus C. Plot of a graph showing the variation of number of atoms of nucleus B vesus time is : (Assume that at t = 0, there are no B atoms in the sample)

Options

A.
B.
C.
D.

Answer

To solve the problem of plotting the variation of the number of atoms of nucleus B versus time, we need to understand the decay process involving the radioactive nuclei A, B, and C.

### Step-by-Step Solution:

1. **Understanding the Decay Process**:
- Nucleus A decays to nucleus B.
- Nucleus B decays to a stable nucleus C.
- At time \( t = 0 \), there are no B atoms in the sample.

2. **Defining the Decay Constants**:
- Let \( \lambda_A \) be the decay constant for nucleus A.
- Let \( \lambda_B \) be the decay constant for nucleus B.
- The decay of A to B can be described by the equation:
\[ \frac{dN_A}{dt} = -\lambda_A N_A \]
- The number of A nuclei at time \( t \) can be expressed as:
\[ N_A(t) = N_{A0} e^{-\lambda_A t} \]
where \( N_{A0} \) is the initial number of A nuclei.

3. **Rate of Formation of B**:
- The rate of formation of B from A is given by:
\[ \frac{dN_B}{dt} = \lambda_A N_A(t) - \lambda_B N_B(t) \]
- Substituting \( N_A(t) \):
\[ \frac{dN_B}{dt} = \lambda_A N_{A0} e^{-\lambda_A t} - \lambda_B N_B(t) \]

4. **Solving the Differential Equation**:
- This is a first-order linear differential equation in \( N_B \). We can solve it using an integrating factor or by recognizing it as a standard form.
- The integrating factor \( e^{\lambda_B t} \) can be used:
\[ e^{\lambda_B t} \frac{dN_B}{dt} + \lambda_B e^{\lambda_B t} N_B = \lambda_A N_{A0} e^{(\lambda_B - \lambda_A)t} \]
- Integrating both sides will yield the function for \( N_B(t) \).

5. **Behavior of Nucleus B Over Time**:
- Initially, \( N_B(0) = 0 \).
- As time progresses, \( N_B \) will increase as A decays into B.
- Eventually, as B decays into C, the number of B nuclei will reach a maximum (steady state) and then start to decrease.

6. **Plotting the Graph**:
- The graph of \( N_B \) versus time will show an initial increase, reaching a peak, and then a decline as B decays into C.
- The shape of the graph will resemble a curve that rises to a maximum and then falls off.

### Final Graph Description:
- The graph starts at \( N_B = 0 \) at \( t = 0 \).
- It increases rapidly initially due to the decay of A.
- It reaches a peak when the rate of formation of B equals the rate of decay of B.
- After reaching the peak, it starts to decline as more B decays into C.