Questions: Arrhenius Equation and Temperature Dependence

Question 1

Reaction A has Eₐ = 50 kJ/mol and Reaction B has Eₐ = 100 kJ/mol. Both start at 300 K and are heated to 310 K. Which reaction's rate constant increases by a larger factor?

• A. Reaction A, because the lower barrier is easier to overcome with added energy
• B. Reaction B, because higher activation energy makes the rate more sensitive to temperature
• C. Both increase by the same factor because the temperature change is identical
• D. Neither changes significantly because 10 K is too small a change

Answer: (B)

The Arrhenius equation shows that temperature sensitivity scales with Eₐ: in the exponential term −Eₐ/RT, a larger Eₐ means a larger fractional change in the exponent for a given ΔT. Mathematically, d(ln k)/dT = Eₐ/RT², so reactions with higher activation energies have a steeper ln(k) vs T slope. Reaction B will show a greater rate increase per degree. The common misconception (option A) imagines a lower barrier as 'easier to push over' — but it's the opposite: the high-Eₐ reaction is more starved of reactive collisions, so additional thermal energy proportionally helps it more.

Question 2

In an Arrhenius plot of ln(k) versus 1/T, what does the slope of the best-fit line equal?

• A. ln(A), the natural log of the pre-exponential factor
• B. −Eₐ/R, which allows the activation energy to be calculated from the slope
• C. Eₐ × R, giving the activation energy in units of J²·mol⁻²
• D. The reaction rate order with respect to temperature

Answer: (B)

Taking the natural log of the Arrhenius equation gives ln(k) = ln(A) − (Eₐ/R)(1/T). Plotting ln(k) on the y-axis versus 1/T on the x-axis gives a line with slope = −Eₐ/R and y-intercept = ln(A). The slope is negative (faster reactions at higher T means higher k at lower 1/T). Multiplying the slope by −R gives Eₐ in J/mol. Option A is the y-intercept, not the slope.

Question 3

A reaction with a low activation energy is more sensitive to temperature changes than a reaction with a high activation energy.

• T. True
• F. False

Answer: False

This is a persistent misconception. The Arrhenius equation's exponential form means that high-Eₐ reactions are *more* sensitive to temperature. The derivative d(ln k)/dT = Eₐ/RT² shows sensitivity is proportional to Eₐ. Intuitively, a high-Eₐ reaction has few reactive collisions at low temperatures — a small increase in T shifts many more molecules above the threshold. A low-Eₐ reaction already has many reactive collisions, so adding more energy has a smaller proportional effect.

Question 4

The pre-exponential factor A in the Arrhenius equation captures geometric and frequency factors independent of energy, and is approximately constant over moderate temperature ranges.

• T. True
• F. False

Answer: True

A incorporates the collision frequency and the fraction of collisions with correct orientation for reaction — factors that depend weakly on temperature compared to the exponential term. Over a modest temperature range (say, 50–100 K), A is approximately constant, which is why plotting ln(k) vs 1/T gives a straight line. The dramatic temperature dependence of k comes from the e^(−Eₐ/RT) term, not from A.

Question 5

Why does a 10°C temperature increase often double or triple a reaction rate, and why does the magnitude of this effect depend on the activation energy?

Model answer: The Arrhenius equation's exponential term e^(−Eₐ/RT) represents the Boltzmann fraction of molecules with enough energy to react. Because the exponent contains temperature in the denominator, even a modest increase in T significantly enlarges this fraction. The effect is larger for high-Eₐ reactions because the exponent −Eₐ/RT changes more when Eₐ is large, meaning the fraction of reactive collisions grows faster per degree of heating.

A 10°C rule of thumb (rate doubling) applies to reactions with Eₐ around 50–60 kJ/mol at room temperature — coincidentally typical of many biochemical reactions. For Eₐ = 100 kJ/mol, the same 10°C increase would roughly quadruple the rate. Understanding this relationship is essential for designing experiments (choosing useful temperature ranges) and interpreting kinetic data (knowing what temperature control precision is needed).