Question

What is the integrated rate equation for a first-order reaction?

• A. \( [A] = [A_0] e^{-kt} \)
• B. \( [A] = \frac{[A_0]}{e^{-kt}} \)
• C. \( [A] = [A_0] e^{-t} \)
• D. \( [A] = [A_0] e^{-k} \)

Hint:

For first-order reactions, the concentration of reactants decreases exponentially with time, and the rate constant \( k \) is essential in the rate equation.

Answer 1

The Correct Option is A

Step 1: First-Order Reaction Integrated Rate Equation 
For a first-order reaction, the rate of reaction is directly proportional to the concentration of the reactant.
The integrated rate equation for a first-order reaction is: \[ [A] = [A_0] e^{-kt} \] 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: Explanation of Other Options 
Option (b) is incorrect because it suggests an inverse relationship with time and the rate constant, which is not the case for first-order reactions.
Option (c) is incorrect because it lacks the rate constant \( k \) and is not the correct form of the first-order integrated rate equation.
Option (d) is incorrect because it shows an equation where the concentration is exponentially decreasing with respect to \( k \), which does not match the first-order rate equation.

Step 3: Conclusion 
Thus, the correct integrated rate equation for a first-order reaction is \( [A] = [A_0] e^{-kt} \).