Butadiene C4H6 reacts with itself to form a dimer with the formula C8H12. The reaction is second order in C4H6. Assume the rate constant at a particular temperature is 4.0 × 10^-2 M^-1 s^-1 and the initial concentration of C4H6 is 0.0200 M. (a) What is its molarity after a reaction time of 1.00 h?

Verified step by step guidance

1. Step 1: Identify the rate law for a second-order reaction. The rate law for a second-order reaction is given by \( \text{Rate} = k[A]^2 \), where \( k \) is the rate constant and \( [A] \) is the concentration of the reactant.
2. Step 2: Use the integrated rate law for a second-order reaction. The integrated rate law is \( \frac{1}{[A]_t} = \frac{1}{[A]_0} + kt \), where \( [A]_t \) is the concentration at time \( t \), \( [A]_0 \) is the initial concentration, and \( k \) is the rate constant.
3. Step 3: Substitute the given values into the integrated rate law. Here, \( [A]_0 = 0.0200 \text{ M} \), \( k = 4.0 \times 10^{-2} \text{ M}^{-1} \text{s}^{-1} \), and \( t = 1.00 \text{ h} \). Convert time from hours to seconds: \( 1.00 \text{ h} = 3600 \text{ s} \).
4. Step 4: Calculate \( \frac{1}{[A]_t} \) using the formula \( \frac{1}{[A]_t} = \frac{1}{0.0200} + (4.0 \times 10^{-2})(3600) \).
5. Step 5: Solve for \( [A]_t \) by taking the reciprocal of the result from Step 4 to find the concentration of C4H6 after 1.00 hour.