For a reaction whose rate constant increases by a factor of 100 when temperature rises from 400 K to 500 K, estimate E/R (using the Arrhenius/transition-state form).

Introduction / Context:
Temperature sensitivity of reactions is quantified by Arrhenius or transition state relationships. Estimating E/R from two temperature-rate points is a common design and analysis task.

Given Data / Assumptions:

• k2/k1 = 100 between T1 = 400 K and T2 = 500 K.
• Arrhenius form: ln(k2/k1) = (E/R) * (1/T1 - 1/T2).

Concept / Approach:
Rearrange the Arrhenius equation to isolate E/R. Use natural logarithms and the reciprocal temperatures.

Step-by-Step Solution:
1) Compute ln(k2/k1) = ln(100) = 4.60517.2) Compute (1/T1 - 1/T2) = (1/400 - 1/500) = 0.0025 - 0.0020 = 0.0005 K^-1.3) Solve for E/R: E/R = 4.60517 / 0.0005 ≈ 9210 K.

Verification / Alternative check:
Check units: since the RHS uses inverse Kelvin, E/R has units of Kelvin, which is consistent. A 100-fold increase over a 100 K rise is plausible for such E/R values.

Why Other Options Are Wrong:
Values like 8621–8987 K result from arithmetic slips or rounding errors. 9210 K is obtained from the exact logarithmic and reciprocal-temperature difference.

Common Pitfalls:
Using log base 10 instead of natural log without compensating; mixing T2 and T1 order; not converting to Kelvin.

Final Answer:
9210 K.