The conversion of cyclobutane (C₄H₈) to ethylene (C₂H₄) is a first-order reaction. This means that the concentration of cyclobutane at time t is given by ln C = ln C₀ − kt, where C is the concentration at time t, C₀ is the initial concentration, t is the time since the reaction started, and k is the rate constant. Assume that C₀ = 0.2 mol/L with negligible uncertainty. After 300 seconds at a constant temperature, the concentration is measured to be C = 0.174 ± 0.005 mol/L. Assume that time can be measured with negligible uncertainty.

a. Estimate the rate constant k, and find the uncertainty in the estimate. The units of k will be s⁻¹.

b. Find the relative uncertainty in k.

c. The half-life t_1/2 of the reaction is the time it takes for the concentration to be reduced to one-half its initial value. The half-life is related to the rate constant by t_1/2 = (ln 2)/k. Using the result found in part (a), find the uncertainty in the half-life.

d. Find the relative uncertainty in the half-life.