For a chemical reaction, the rate constant at 42.0 °C is 0.00395 s-1, while the rate constant at 67.4 °C is 0.0133 s-1. Calculate the value of the energy of activation, in kilojoules.

Question

BlackTom AI · Accepted Answer

The question asks us to determine the activation energy Ea from two rate constants at different temperatures, which points us to the Arrhenius relationship k = A e^{-Ea/(RT)}.

First, identify the temperatures in Kelvin: T1 = 42.0 °C = 315.15 K, T2 = 67.4 °C = 340.55 K. The corresponding rate constants are k1 = 0.00395 s^-1 and k2 = 0.0133 s^-1.

Use the logarithmic form of the Arrhenius equation for two temperatures: ln(k2/k1) = Ea/R * (1/T1 - 1/T2).

Compute the ratio k2/k1: 0.0133 / 0.00395 ≈ 3.367. Then ln(k2/k1) ≈ ln(3.367) ≈ 1.213.

Compute the difference in reciprocals: 1/T1 - 1/T2 = 1/315.15 − 1/340.55 ≈ 0.003174 − 0.002937 ≈ 0.000237 K^-1.

Now solve for Ea: Ea = (R * ln(k2/k1)) / (1/T1 - 1/T2).

Using R = 8.314 J/mol·K, Ea ≈ (8.314 * 1.213) / 0.000237 ≈ 10.089 / 0.000237 ≈ 42,500 J/mol ≈ 42.5 kJ/mol.

Rounding and slight differences in intermediate rounding typically yield a value around 42–43 kJ/mol, which aligns with the option provided.

Option-by-option analysis:
- Option a: 42.65 kJ — This value is consistent with the calculation above and falls within typical rounding ranges for Ea from the given data, so it is plausible as the activation energy.
- Option b: 1.13 kJ — This is far too small for a chemical activation energy; typical Ea values are on the order of tens of kJ/mol to hundreds, not around 1 kJ. This likely results from misapplying logarithms or ignoring temperature dependence.
- Option c: 0.421 kJ — Also far too small; it underestimates by over two orders of magnitude and would imply negligible temperature sensitivity, inconsistent with the given rate change.
- Option d: 18.5 kJ — While closer to typical values than the previous two, it does not match the calculated Ea from the provided k1, k2, T1, and T2; thus it is inconsistent with the data.
- Option e: 0.617 kJ — Similar to options b and c, this is far too small to represent a reasonable activation energy for a chemical reaction based on the provided rate constants.

In summary, the calculation using the Arrhenius relationship with the two provided temperatures and rate constants yields an Ea near 42–43 kJ/mol, making option a the options that best fits the data.

CHM1051 MUM S1 2025 Week 12: Weekly quiz (1%)

For a chemical reaction, the rate constant at 42.0 °C is 0.00395 s-1, while the rate constant at 67.4 °C is 0.0133 s-1. Calculate the value of the energy of activation, in kilojoules.

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