Consider the first order reaction: `A rarr 2B`. Which of the following figure correctly describes the rate of disappearance of `A(r_(1))` and rate of appearance of `B(r_(2))` with time ?

Options

A.
B.
C.
D.

Answer

To solve the problem regarding the first-order reaction \( A \rightarrow 2B \), we need to analyze the rates of disappearance of \( A \) and the appearance of \( B \) over time. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Reaction
The reaction given is a first-order reaction where one mole of \( A \) produces two moles of \( B \). The stoichiometry indicates that for every mole of \( A \) that disappears, two moles of \( B \) appear.

### Step 2: Define the Rates
- The rate of disappearance of \( A \) can be expressed as:
\[ R_1 = -\frac{d[A]}{dt} \]
- The rate of appearance of \( B \) can be expressed as:
\[ R_2 = \frac{d[B]}{dt} \]

From the stoichiometry of the reaction, we can relate these rates:
\[ R_1 = -\frac{1}{2} R_2 \]
This means that the rate at which \( A \) disappears is half the rate at which \( B \) appears.

### Step 3: Express Concentrations Over Time
For a first-order reaction, the concentration of \( A \) at time \( t \) is given by:
\[ [A] = [A]_0 e^{-kt} \]
Where \( [A]_0 \) is the initial concentration of \( A \) and \( k \) is the rate constant.

### Step 4: Calculate the Rate Expressions
Using the expression for \( [A] \):
- The rate of disappearance of \( A \) becomes:
\[ R_1 = k[A] = k[A]_0 e^{-kt} \]
- The rate of appearance of \( B \) becomes:
\[ R_2 = -\frac{d[A]}{dt} = k[A]_0 e^{-kt} \]
However, since \( R_2 \) is related to the stoichiometry, we have:
\[ R_2 = -\frac{d[A]}{dt} = 2k[A]_0 e^{-kt} \]

### Step 5: Analyze the Graphs
- The graph of \( R_1 \) (rate of disappearance of \( A \)) will show an exponential decay since it is proportional to \( e^{-kt} \).
- The graph of \( R_2 \) (rate of appearance of \( B \)) will also show an exponential decay, but it will be steeper because it is multiplied by 2.

### Step 6: Conclusion
The correct graph will show:
- \( R_1 \) decreasing gently over time.
- \( R_2 \) decreasing more steeply over time, reflecting that \( B \) is produced at a rate that is double that of the disappearance of \( A \).

### Final Answer
The figure that correctly describes the rates of disappearance of \( A \) and appearance of \( B \) with time is the one where \( R_1 \) decreases gently and \( R_2 \) decreases more steeply.

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