Domain: Chemical kinetics

At 25 °C, the geometric isomerization between cis-2-butene and trans-2-butene proceeds reversibly and is first order in each direction. The experimentally measured equilibrium constant (defined as Kc = [trans]eq / [cis]eq) is 0.406. The forward rate constant for conversion of the cis isomer to the trans isomer is 4.21 × 10^-4 s^-1.

A sealed reaction vessel initially contains only cis-2-butene at a concentration of 0.115 mol dm^-3.

Determine the time required for the trans isomer to reach one-half of its equilibrium concentration under these conditions.

Grading rubric:
criterion: The solution must explicitly derive the time expression starting from the cis(t) form rather than from the trans(t) form.
weight: 10
criterion: The solution must explicitly write the integrated equation as cis(t) = cis_eq + (cis_0 − cis_eq) exp[−(k1 + k−1)t] before any substitution.
weight: 20
criterion: The solution must explicitly construct the logarithmic ratio as (cis_0 − cis_eq)/(cis_t − cis_eq)
weight: 10
criterion: The solution must explicitly evaluate ln(2.0625) = 0.724 before multiplying by the reciprocal rate constant sum.
weight: 10
criterion: The solution must show that the final time arises from the logarithmic expression and not from a half-life identity.
weight: 10
criterion: The solution correctly algebraically rearranges the exponential form into the logarithmic time expression.
weight: 20
criterion: The final numerical time must fall between 490 s and 500 s.
weight: 20

Answer:
approximately 497 s