The graph showing the variation of decay rate with number of nuclei is:

Options

A.
B.
C.
D.

Explanation

To solve the question regarding the graph that shows the variation of decay rate with the number of nuclei, we can follow these steps:

### Step 1: Understand the Decay Law
The decay of radioactive nuclei is described by the equation:
\[ n(t) = n_0 e^{-\lambda t} \]
where:
- \( n(t) \) is the number of undecayed nuclei at time \( t \),
- \( n_0 \) is the initial number of nuclei,
- \( \lambda \) is the decay constant.

### Step 2: Determine the Decay Rate
The decay rate (or activity) \( A \) is given by the derivative of the number of nuclei with respect to time:
\[ A = -\frac{dn}{dt} = \lambda n(t) \]
Substituting the expression for \( n(t) \):
\[ A = \lambda n_0 e^{-\lambda t} \]

### Step 3: Analyze the Relationship
From the equation \( A = \lambda n(t) \), we see that the decay rate \( A \) is directly proportional to the number of undecayed nuclei \( n(t) \). This means that as the number of nuclei decreases over time, the decay rate also decreases.

### Step 4: Graphical Representation
1. The graph of decay rate \( A \) versus the number of nuclei \( n \) will show a linear relationship since \( A \) is proportional to \( n \).
2. As \( n \) decreases, \( A \) will also decrease, indicating a negative slope.

### Step 5: Identify the Correct Graph
Among the options provided, we need to identify the graph that shows:
- A downward slope (negative slope) as the number of nuclei decreases.
- The graph should start from a positive value when \( n \) is at its maximum (initial number of nuclei).

### Conclusion
The correct graph that represents the decay rate as a function of the number of nuclei will be the one that shows a negative slope, indicating that as the number of nuclei decreases, the decay rate also decreases.

### Final Answer
The correct option is **Option A**.

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