Question

Which of the following represents the fraction of molecules with energies equal to or greater than \(E_a\)?

• A. \(+\dfrac{E_a}{RT}\)
• B. \(e^{-E_a/RT}\)
• C. \(-\dfrac{E_a}{RT}\)
• D. \(e^{+E_a/RT}\)

Hint:

According to the Arrhenius equation, only molecules having energy equal to or more than the activation energy \(E_a\) can react. The fraction of such molecules is given by the exponential term in the Arrhenius equation.

Answer 1

The Correct Option is B

Step 1: Recall the Arrhenius equation.
The rate constant is given by \(k = A\, e^{-E_a/RT}\), where \(A\) is the frequency factor and \(E_a\) is the activation energy.

Step 2: Identify the meaning of each part.
The factor \(A\) relates to how often molecules collide with proper orientation. The exponential factor \(e^{-E_a/RT}\) tells us the fraction of molecules that actually have enough energy to react.

Step 3: Focus on the energy condition.
Only molecules whose energy is equal to or greater than \(E_a\) can cross the barrier. This fraction is exactly the term \(e^{-E_a/RT}\).

Step 4: Check the sign and form.
Since a larger \(E_a\) or a lower temperature should make this fraction smaller, the exponent must be negative, which matches \(e^{-E_a/RT}\). The other options have wrong signs or are not exponentials.

Step 5: Conclude.
The fraction of molecules with energy at least \(E_a\) is \(e^{-E_a/RT}\).

\[ \boxed{e^{-E_a/RT}} \]