1 markMCQSection A

Which of the following represents the fraction of molecules with energies equal to or greater than Eₐ?

Q: Which of the following represents the fraction of molecules with energies equal to or greater than Eₐ?

**Correct option: (B)** — $e^{-E_a/RT}$. **Key idea:** The Boltzmann/Arrhenius factor gives the fraction of molecules whose energy is at least the activation energy Eₐ: - fraction = $e^{-E_a/RT}$ This is the same exponential factor that appears in the Arrhenius equation k = A·$e^{-E_a/RT}$.

Q: What does the factor $e^{-E_a/RT}$ physically represent?

It is the fraction of colliding molecules that possess kinetic energy equal to or greater than the activation energy Eₐ at temperature T. Only these energetic molecules can cross the energy barrier and react, which is why this Boltzmann factor controls the reaction rate.

Q: Why must the exponent carry a negative sign?

A fraction must lie between 0 and 1. With a negative exponent, $e^{-E_a/RT}$ is always less than 1 and correctly increases when temperature rises (barrier easier to cross) and decreases when Eₐ rises. A positive exponent would give a value above 1, which is physically impossible for a fraction.

Chemical Kinetics

Arrhenius equation and activation energy

Options

(A)

+E_a/RT

(B)

e^{-E_a/RT}

(C)

-E_a/RT

(D)

e^{+E_a/RT}

Official Answer

Correct option: (B) — $e^{-E_a/RT}$ .

Key idea: The Boltzmann/Arrhenius factor gives the fraction of molecules whose energy is at least the activation energy Eₐ:

fraction = $e^{-E_a/RT}$

This is the same exponential factor that appears in the Arrhenius equation k = A· $e^{-E_a/RT}$ .

Arrhenius equationactivation energy Eae^(-Ea/RT)Boltzmann factorfraction of moleculestemperature dependence of rateMaxwell-Boltzmann distribution

Marking Scheme

11 mark: correctly selecting option (B) $e^{-E_a/RT}$ .
2Accepted reason: it is the Boltzmann fraction of molecules with energy $\ge E_a$ .

Hint

It is the exponential factor of the Arrhenius equation $k = A \cdot e^{-E_a/RT}$ ; a fraction must lie between 0 and 1.

Quick Oral Answer

The fraction of molecules with energy equal to or greater than the activation energy is e to the power minus Eₐ over RT — the same negative exponential factor that appears in the Arrhenius equation.

Analysis & Explanation

Concept:

At temperature T, from the Maxwell–Boltzmann distribution of molecular energies, the fraction of molecules possessing energy $\ge E_a$ is the exponential factor $e^{-E_a/RT}$ . Multiplying by the collision/frequency factor A gives the rate constant, k = A· $e^{-E_a/RT}$ .

Why the distractors are wrong:

(A) $+E_a/RT$ and (C) $-E_6/RT$ are just the exponents (dimensionless numbers), not a fraction of molecules — a fraction must be a value between 0 and 1, which only an exponential term gives.
(D) $e^{+E_a/RT}$ is greater than 1 for positive Eₐ, so it cannot be a fraction of molecules; the exponent must be negative.
Only (B) $e^{-E_a/RT}$ is a proper fraction (0 < value < 1) and matches the Arrhenius factor.

Exam trap: The minus sign in the exponent is essential — it makes the fraction decrease as Eₐ rises and increase as T rises, exactly as chemical intuition demands.

Common Mistakes

1Choosing $e^{+E_a/RT}$ ; a positive exponent gives a value greater than 1, which cannot be a fraction.
2Selecting the bare exponent $E_a/RT$ instead of the exponential expression.
3Forgetting the negative sign, which is what makes the fraction rise with temperature and fall with higher activation energy.

Interesting Facts

Svante Arrhenius proposed this equation in 1889; he later won the 1903 Nobel Prize in Chemistry for his electrolytic dissociation theory.

The strong exponential dependence explains the rough 'rule of thumb' that reaction rates roughly double for every 10 °C rise in temperature near room temperature.

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Frequently Asked Questions

What does the factor $e^{-E_a/RT}$ physically represent?