The rate constant for a certain chemical reaction is 0.00250 L mol-1 s-1 at 25.0 °C and 0.0125 L mol-1 s-1 at 50.0 °C. What is the activation energy for the reaction, expressed in kilojoules?

Question

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We’re given the rate constants at two temperatures and asked to find the activation energy Ea using the Arrhenius relationship.

To compare two temperatures, the natural log form is ln(k2/k1) = -Ea/R (1/T2 - 1/T1). It’s often convenient to rearrange as Ea = R * [ln(k2) - ln(k1)] / [ (1/T1) - (1/T2) ]. Here, T1 = 25.0 °C = 298 K and T2 = 50.0 °C = 323 K. The given rate constants are k1 = 0.00250 L mol⁻¹ s⁻¹ and k2 = 0.0125 L mol⁻¹ s⁻¹.

Step 1: Compute natural logs of the rate constants.
- ln(k1) = ln(0.00250) ≈ -5.9915
- ln(k2) = ln(0.0125) ≈ -4.3821
Step 2: Compute the difference in ln(k): ln(k2) - ln(k1) ≈ (-4.3821) - (-5.9915) = 1.6094. 
Step 3: Compute the difference in reciprocal temperatures: (1/T1) - (1/T2) = (1/298) - (1/323) ≈ 0.003356 - 0.003096 = 0.000260 s/K. (Using more precise values: 1/298 ≈ 0.003356, 1/323 ≈ 0.003096; difference ≈ 0.000260 s/K.) 
Step 4: Assemble Ea = R * [ln(k2) - ln(k1)] / [(1/T1) - (1/T2)]. With R = 8.314 J/mol·K, Ea ≈ 8.314 * 1.6094 / 0.000260 ≈ 8.314 * 6196 ≈ 5.15 × 10⁴ J/mol. 
Step 5: Convert to kilojoules per mole: Ea ≈ 51.5 kJ/mol (rounded to about 51.6 kJ/mol).

Option-by-option assessment:
- a. 25.1 kJ: This would require a much smaller slope or a larger denominator in the ratio; the calculated value is clearly higher, so this underestimates the activation energy.
- b. 51.6 kJ: This closely matches the computed value using the two-temperature Arrhenius relation, making it a plausible result.
- c. 37.6 kJ: This lies between a and b but does not align with the slope derived from the given k1, k2, T1, and T2; the difference in ln(k) versus the temperature change would not support this.
- d. 45.3 kJ: This is somewhat closer but still does not fit the precise ratio of ln(k2)−ln(k1) to (1/T1)−(1/T2) as calculated above.
- e. 60.3 kJ: This would imply a steeper temperature dependence than observed; the computed ratio yields a smaller Ea than this value.

In summary, the activation energy is derived from the temperature dependence of the rate constants, and the numbers align with Ea ≈ 51.6 kJ/mol when you plug the values into the Arrhenius form. The analysis shows why the other options deviate from the calculated slope and ratio.

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

The rate constant for a certain chemical reaction is 0.00250 L mol-1 s-1 at 25.0 °C and 0.0125 L mol-1 s-1 at 50.0 °C. What is the activation energy for the reaction, expressed in kilojoules?

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