In a first order reaction the concentration of product ’x’ at time 't' is given by the expression (a=initial concentration, k=rate constant, n=order)

Options

A. `x=a(1-e^(-kt))`
B. `x=1/(a-x)`
C. `x=1/(2^(n-1))`
D. `x=a/(a-x)`

Explanation

To solve the question regarding the concentration of product 'x' in a first-order reaction, we can follow these steps:

### Step 1: Understand the Integrated Rate Law for First-Order Reactions
The integrated rate law for a first-order reaction is given by the equation:
\[ A = A_0 e^{-kt} \]
where:
- \( A \) is the concentration of the reactant at time \( t \),
- \( A_0 \) is the initial concentration,
- \( k \) is the rate constant,
- \( t \) is the time.

### Step 2: Relate the Concentration of Product 'x' to the Reactant
In this scenario, we are interested in the concentration of the product 'x' formed at time \( t \). If \( A \) is the concentration of the reactant at time \( t \), then the concentration of the product 'x' can be expressed as:
\[ A = A_0 - x \]
This means that the concentration of the reactant decreases as the product is formed.

### Step 3: Substitute into the Integrated Rate Law
Substituting \( A \) into the integrated rate law gives us:
\[ A_0 - x = A_0 e^{-kt} \]

### Step 4: Rearranging the Equation
Now, we can rearrange this equation to isolate 'x':
\[ x = A_0 - A_0 e^{-kt} \]

### Step 5: Factor Out \( A_0 \)
Factoring out \( A_0 \) from the right side:
\[ x = A_0 (1 - e^{-kt}) \]

### Conclusion
Thus, the final expression for the concentration of product 'x' at time \( t \) in a first-order reaction is:
\[ x = A_0 (1 - e^{-kt}) \]