Question

The reaction follows 1st order reaction \( \text{R} \rightarrow \text{P} \) Find the fraction of molecules dissociated in time \( t \). \([k_1 = \text{Rate constant}]\)

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

Answer 1

The Correct Option is A

textbf{Step 1: First-order reaction rate law.}
For a first-order reaction, the rate law is given by: \[ \ln \left( \frac{[R]}{[R_0]} \right) = -k_1 t \] where \( [R] \) is the concentration of reactant at time \( t \), and \( [R_0] \) is the initial concentration of reactant. Step 2: Finding the fraction dissociated.
The fraction dissociated is the ratio of the change in concentration to the initial concentration: \[ \text{Fraction dissociated} = \frac{[R_0] - [R]}{[R_0]} \] Now, using the equation for first-order reaction: \[ \frac{[R]}{[R_0]} = e^{-k_1 t} \] Substitute this in the equation for fraction dissociated: \[ \text{Fraction dissociated} = 1 - e^{-k_1 t} \] \[ \boxed{1 - e^{-k_1 t}} \]