For a first-order reaction, `Aoverset(k)toB`, the degree of dissociation is equal to

Options

A. `e^(-kt)`
B. `1-e^(-kt)`
C. `e^(kt)`
D. `(1+e^(-kt))`

Explanation

To determine the degree of dissociation for the first-order reaction \( A \overset{k}{\rightarrow} B \), we can follow these steps:

### Step 1: Define the degree of dissociation
The degree of dissociation (\( \alpha \)) is defined as the fraction of the initial amount of reactant that has reacted. For a reaction \( A \rightarrow B \), if \( A \) dissociates to form \( B \), we can express the degree of dissociation as:
\[ \alpha = \frac{X}{A_0} \]
where \( X \) is the amount of \( A \) that has dissociated and \( A_0 \) is the initial concentration of \( A \).

### Step 2: Relate the concentration of \( A \) at any time \( t \)
For a first-order reaction, the concentration of \( A \) at time \( t \) can be expressed using the first-order rate equation:
\[ [A] = A_0 e^{-kt} \]
where \( k \) is the rate constant and \( [A] \) is the concentration of \( A \) at time \( t \).

### Step 3: Express \( X \) in terms of \( A_0 \) and \( [A] \)
At time \( t \), the amount of \( A \) that has reacted is:
\[ X = A_0 - [A] = A_0 - A_0 e^{-kt} = A_0(1 - e^{-kt}) \]

### Step 4: Substitute \( X \) into the degree of dissociation formula
Now, substituting \( X \) into the degree of dissociation formula:
\[ \alpha = \frac{X}{A_0} = \frac{A_0(1 - e^{-kt})}{A_0} = 1 - e^{-kt} \]

### Conclusion
Thus, the degree of dissociation for the first-order reaction \( A \overset{k}{\rightarrow} B \) is given by:
\[ \alpha = 1 - e^{-kt} \]