Catalyst Reduces Activation Energy to One-Third – Rate Constant Ratio Using Arrhenius Equation

Understanding how a catalyst reduces activation energy and increases the reaction rate
is an important concept in Chemical Kinetics. Using the Arrhenius equation, we can
calculate the ratio of rate constants for catalyzed and uncatalyzed reactions easily.

Step 1: Arrhenius Equation

The Arrhenius equation is:

k = A exp( −E_a / RT )

• k = rate constant
• A = frequency factor
• E_a = activation energy
• R = gas constant
• T = temperature

Step 2: Write Rate Constants

Uncatalyzed Reaction

k_u = A exp( −E_a / RT )

Catalyzed Reaction

Given:
E’_a = E_a / 3

k_c = A exp( −E_a / 3RT )

Step 3: Take Ratio

k_c / k_u
=
[A exp( −E_a / 3RT )] /
[A exp( −E_a / RT )]

Cancel A:

= exp( −E_a/3RT + E_a/RT )

Step 4: Simplify

= exp( (3E_a − E_a) / 3RT )

= exp( 2E_a / 3RT )

Final Answer

k_catalyzed / k_uncatalyzed = exp( 2E_a / 3RT )

Correct Option: (C)

Explanation of Each Option

(A) 1

If ratio = 1, catalyst has no effect. Catalysts always increase rate. ❌ Incorrect

(B) 1/3

Rate constant does not change linearly with activation energy. It follows exponential relation. ❌ Incorrect

(C) exp(2E_a/3RT)

Correctly derived using Arrhenius equation. Shows exponential increase in rate. ✅ Correct

(D) exp(E_a/3RT)

Mathematical simplification is wrong. ❌ Incorrect

Concept Summary

• Catalyst lowers activation energy
• More molecules cross energy barrier
• Rate increases exponentially
• Small change in E_a → large change in rate

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