A radioactiev nuleus `X` deays to a stable nuleus `Y`. Then, time graph of rate of formation of `Y` against time `t` will be:

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining the time graph of the rate of formation of stable nucleus Y from radioactive nucleus X, we can follow these steps:

### Step 1: Understand the Decay Process
Radioactive nucleus X decays over time into stable nucleus Y. The decay process can be described by the exponential decay law, which states that the number of radioactive nuclei decreases over time.

### Step 2: Write the Decay Law
The number of undecayed nuclei \( N(t) \) at time \( t \) can be expressed as:
\[ N(t) = N_0 e^{-\alpha t} \]
where \( N_0 \) is the initial number of nuclei, and \( \alpha \) is the decay constant.

### Step 3: Determine the Rate of Formation of Y
As nucleus X decays, nucleus Y is formed. The rate of formation of Y, \( R_Y(t) \), is equal to the rate of decay of X, which can be expressed as:
\[ R_Y(t) = -\frac{dN(t)}{dt} \]
Calculating this derivative gives:
\[ R_Y(t) = \alpha N_0 e^{-\alpha t} \]
This shows that the rate of formation of Y is proportional to the number of undecayed nuclei of X.

### Step 4: Analyze the Graph of \( R_Y(t) \)
From the equation \( R_Y(t) = \alpha N_0 e^{-\alpha t} \), we can see that:
- At \( t = 0 \), \( R_Y(0) = \alpha N_0 \) (maximum rate of formation).
- As time increases, \( e^{-\alpha t} \) decreases, leading to a decrease in \( R_Y(t) \).

### Step 5: Sketch the Graph
The graph of \( R_Y(t) \) against time \( t \) will start at a maximum value when \( t = 0 \) and will decrease exponentially towards zero as time progresses. This graph will resemble a decaying exponential curve.

### Conclusion
Thus, the time graph of the rate of formation of Y against time \( t \) will show a decreasing exponential trend.

### Final Answer
The correct option for the graph of the rate of formation of Y against time is option C.

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