Question

Consider the first order reaction R → P. The fraction of molecules decomposed in the given first order reaction can be expressed as

• A. \(1 - e^{-k_1 t}\)
• B. \(1 + e^{-k_1 t}\)
• C. \(1 + e^{k_1 t}\)
• D. \(1 - e^{k_1 t}\)

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
For a first-order reaction, the rate of reaction is proportional to the concentration of the reactant. The integrated rate law provides the concentration of the reactant remaining at any time $t$.

Step 2: Key Formula or Approach:
1. Integrated rate law: $[R]_t = [R]_0 e^{-k_1 t}$ 2. Amount decomposed: $[R]_0 - [R]_t$ 3. Fraction decomposed: $\frac{[R]_0 - [R]_t}{[R]_0}$

Step 3: Detailed Explanation:
1. Let the initial concentration be $[R]_0$. 2. The concentration at time $t$ is $[R]_t = [R]_0 e^{-k_1 t}$. 3. The amount that has decomposed is $[R]_0 - [R]_0 e^{-k_1 t} = [R]_0 (1 - e^{-k_1 t})$. 4. The fraction decomposed is: \[ \text{Fraction} = \frac{[R]_0 (1 - e^{-k_1 t})}{[R]_0} = 1 - e^{-k_1 t} \]

Step 4: Final Answer:
The fraction of molecules decomposed is \(1 - e^{-k_1 t}\).