Consider the reaction `A to B`. The concentration of both the reactants and the products varies exponentially with time. Which of the following figure correctly describes the change in concentration of reactants and products with time ?

Options

A.
B.
C.
D.

Explanation

To solve the problem regarding the reaction \( A \to B \) and the exponential variation of concentrations with time, we can follow these steps:

### Step 1: Understand the Reaction
The reaction involves a reactant \( A \) that transforms into a product \( B \). As the reaction proceeds, the concentration of \( A \) will decrease while the concentration of \( B \) will increase.

### Step 2: Analyze Concentration Changes
Since it is stated that both concentrations vary exponentially with time, we can express this mathematically:
- The concentration of \( A \) decreases exponentially with time, which can be represented as:
\[ [A] = [A_0] e^{-kt} \]
where \( [A_0] \) is the initial concentration of \( A \), \( k \) is the rate constant, and \( t \) is time.

- The concentration of \( B \) increases exponentially with time, which can be represented as:
\[ [B] = [B_0] (1 - e^{-kt}) \]
where \( [B_0] \) is the final concentration of \( B \) that can be achieved as \( t \) approaches infinity.

### Step 3: Sketch the Graphs
1. **Graph for \( A \)**: The graph of \( [A] \) versus time will show an exponential decay. It starts at \( [A_0] \) and approaches zero as time increases.
2. **Graph for \( B \)**: The graph of \( [B] \) versus time will show an exponential increase. It starts at zero and approaches \( [B_0] \) as time increases.

### Step 4: Compare with Given Options
Now, we need to compare the sketches of \( [A] \) and \( [B] \) with the provided options to identify which one correctly represents the exponential changes in concentration.

- **Option A**: Shows \( [A] \) decreasing exponentially but \( [B] \) increasing incorrectly.
- **Option B**: Correctly shows \( [A] \) decreasing exponentially and \( [B] \) increasing exponentially.
- **Option C**: Shows \( [A] \) decreasing correctly but \( [B] \) increasing incorrectly.
- **Option D**: Incorrectly shows both concentrations.

### Conclusion
The correct figure that describes the change in concentration of reactants and products with time is **Option B**, where the concentration of \( A \) decreases exponentially and the concentration of \( B \) increases exponentially.

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