The rate constant of a reaction is measured at different temperatures. A plot of the natural log of the rate constant as a function of the inverse of the temperature (in kelvins) yields a straight line with a slope of –8.55
×
103 K–1. What is the activation energy (Ea) for the reaction?

Question

The rate constant of a reaction is measured at different temperatures. A plot of the natural log of the rate constant as a function of the inverse of the temperature (in kelvins) yields a straight line with a slope of –8.55 
×
 103 K–1. What is the activation energy (Ea) for the reaction?

BlackTom AI · Accepted Answer

The question uses the Arrhenius relationship, where a plot of ln(k) versus 1/T has a slope equal to -Ea/R. Here, the given slope is -8.55 × 10^3 K^-1.

First, relate the slope to Ea: slope = -Ea/R, so Ea = -slope × R. Since the slope is negative, Ea will be positive as expected for an activation energy.

Plug in the universal gas constant R = 8.314 J mol^-1 K^-1: Ea = -(-8.55 × 10^3 K^-1) × 8.314 J mol^-1 K^-1 = (8.55 × 10^3) × 8.314 J mol^-1.

Carrying out the multiplication: 8.55 × 8.314 ≈ 71.0, and with the 10^3 factor, Ea ≈ 71.0 × 10^3 J mol^-1 = 7.1 × 10^4 J mol^-1 ≈ 71 kJ mol^-1.

Now evaluate each answer option:
- 1.0 kJ: This is far too small to be a typical activation energy for reactions and does not match the calculated magnitude. 
- –71 kJ: The activation energy must be positive; a negative Ea would imply a non-physical, increasing rate with decreasing temperature, which is not correct for standard Arrhenius behavior. 
- –1.0 kJ: Again, negative and also far from the calculated value. 
- 71 kJ: This matches the computed magnitude and has the correct positive sign, aligning with Ea ≈ 71 kJ mol^-1.

In summary, the key check is converting the slope to Ea via Ea = -slope × R and confirming the sign and magnitude against each option. The other choices fail either in magnitude, sign, or both, while the last option aligns with the Arrhenius-derived value.

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