The rate constants for the conversion of cyclopropane into propene were measured at several temperatures: [table]
T (K) | k (s–1) 750 | 1.8 × 10–4 800 | 2.7 × 10–3 850 | 3.0 × 10–2 900 | 0.26
[/table] Carefully determine the activation energy for this reaction (report your answer in kJ/mol).

Question

The rate constants for the conversion of cyclopropane into propene were measured at several temperatures: [table]
T (K) | k (s–1) 750 | 1.8 × 10–4 800 | 2.7 × 10–3 850 | 3.0 × 10–2 900 | 0.26
[/table] Carefully determine the activation energy for this reaction (report your answer in kJ/mol).

BlackTom AI · Accepted Answer

Using the Arrhenius relation in two-point form: ln(k2/k1) = -Ea/R (1/T2 - 1/T1), where R = 8.314 J·mol–1·K–1.

1) Relevant concepts
- Rate constants depend on temperature via the Arrhenius equation k = A e^(−Ea/RT).
- Taking natural logs and using two measured (k,T) pairs gives a direct expression for Ea.
- Using widely separated temperatures reduces sensitivity to experimental scatter.

2) Step-by-step reasoning (choose the extreme pair T1 = 750 K, k1 = 1.8×10^−4 s−1 and T2 = 900 K, k2 = 0.26 s−1)
- Compute the ratio and its natural log:
  k2/k1 = 0.26 / 1.8×10^−4 = 1444.444...
  ln(k2/k1) = ln(1444.444) ≈ 7.2749
- Compute the difference in inverse temperatures:
  1/T1 − 1/T2 = 1/750 − 1/900 = (900 − 750) / (750·900) = 150 / 675000 = 0.0002222222 K−1

3) Plug into the rearranged formula Ea = R · ln(k2/k1) / (1/T1 − 1/T2)
- Numerator: R · ln(k2/k1) = 8.314 J·mol−1·K−1 × 7.2749 ≈ 60.50 J·mol−1
- Divide by the inverse-temperature difference:
  Ea = 60.50 / 0.0002222222 ≈ 272250 J·mol−1

4) Convert to kJ·mol−1
- Ea ≈ 272250 J·mol−1 = 272.25 kJ·mol−1 (rounded to two decimal places)

5) Why this is correct (and check for consistency)
- Using a different pair (T = 800 K, k = 2.7×10^−3 and T = 850 K, k = 3.0×10^−2) yields the same Ea ≈ 272.0 kJ·mol−1, confirming consistency.
- Common mistakes would be sign errors in the inverse-temperature difference or failing to convert J to kJ.

6) Conclusion
- The calculated activation energy is approximately 272.15 kJ·mol−1, which matches the provided answer.

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