Q.Consider the Arrhenius equation given below and mark the correct option.
$k = A\, e^{-E_a/RT}$

The key idea is that the Arrhenius equation $k = A e^{-E_a/RT}$ shows how $k$ depends on $E_a$ and $T$ through the exponential factor.

1. The exponent is $-\frac{E_a}{RT}$. A larger $E_a$ makes the exponent more negative, so $k$ decreases. A smaller $E_a$ makes the exponent less negative, so $k$ increases.
2. A higher $T$ makes $\frac{E_a}{RT}$ smaller, so the exponent is less negative and $k$ increases. A lower $T$ does the opposite — $k$ decreases. …

The Arrhenius equation is one of the most elegant relationships in chemical kinetics. It tells us how the rate constant $k$ — the speedometer of a reaction — changes with two key factors: the activation energy barrier $E_a$ and the temperature $T$. The equation is:

Here, $A$ is the pre-exponential factor (a constant for a given reaction), $R$ is the gas constant, and the exponential term $e^{-E_a/RT}$ is the fraction of molecules that have enough energy to overcome the barrier.

The core idea: the exponent is negative, so anything that makes $-\frac{E_a}{RT}$ less negative (i.e., closer to zero) will make $e^{\text{that}}$ larger, and hence $k$ larger. Let’s see how $E_a$ and $T$ affect this.

1. Effect of activation energy $E_a$

   The exponent is $-\frac{E_a}{RT}$. If $E_a$ is large, the negative number is more negative, so $e^{-E_a/RT}$ is very small — $k$ is small. If $E_a$ is small, the negative number is less negative, so $e^{-E_a/RT}$ is larger — $k$ is larger.

   So: decreasing $E_a$ increases $k$ (exponentially, because $E_a$ sits in the exponent).

2. Effect of temperature $T$

   $T$ appears in the denominator of the exponent. If $T$ increases, the fraction $\frac{E_a}{RT}$ becomes smaller, so $-\frac{E_a}{RT}$ becomes less negative, and $e^{-E_a/RT}$ increases — $k$ increases.

   So: increasing $T$ increases $k$ (again, exponentially).

3. Putting it together

   The rate constant $k$ increases exponentially when:

   • Activation energy $E_a$ decreases
   • Temperature $T$ increases …

Method: Arrhenius Equation Analysis — Direct Interpretation of the Exponential Form

Step 1: Write the Arrhenius equation in its standard form

Where:

• $k$ = rate constant
• $A$ = pre-exponential factor (frequency factor)
• $E_a$ = activation energy (J/mol)
• $R$ = gas constant (8.314 J/mol·K)
• $T$ = absolute temperature (K)

Step 2: Identify the key relationship from the exponent

The exponent is $-\frac{E_a}{RT}$.

• Effect of $E_a$: If $E_a$ increases, the magnitude of the negative exponent becomes larger (more negative), so $e^{-E_a/RT}$ becomes smaller → $k$ decreases.
• Effect of $T$: If $T$ increases, the denominator $RT$ increases, so $-\frac{E_a}{RT}$ becomes less negative, making $e^{-E_a/RT}$ larger → $k$ increases.

Step 3: Combine the effects …

Common Mistakes on the Arrhenius Equation Plot Question

Mistake 1: Confusing the Exponential Relationship

The error: Students think that a negative exponent means the function decreases when the quantity inside increases — but they forget the sign of the exponent.

Why it happens: The equation is $k = A e^{-E_a/RT}$. The exponent is $-E_a/RT$. Many students see "$-E_a$" and assume that increasing $E_a$ makes $k$ larger.

How to avoid:

• Remember: A negative exponent means the function decreases as the exponent's magnitude increases.
• Think of $e^{-x}$: as $x$ gets bigger, $e^{-x}$ gets smaller.
• Here $x = E_a/RT$, so:
  • Higher $E_a$ → larger $x$ → smaller $k$
  • Lower $T$ → larger $x$ (since $T$ is in denominator) → smaller $k$

Mistake 2: Misinterpreting "Exponentially"

The error: Students think "exponentially" always means "increases rapidly" — but it can also mean "decreases rapidly."

Why it happens: In everyday language, "exponential growth" is common, but "exponential decay" is equally valid in science.

How to avoid:

• In the Arrhenius equation, $k$ follows an exponential decay with respect to $E_a/RT$.
• The phrase "increases exponentially" in options refers to the direction of change in $k$, not the mathematical form.
• Always check: does the quantity inside the exponent increase or decrease?

Mistake 3: Forgetting Temperature is in the Denominator

The error: Students think decreasing temperature always decreases the exponent's magnitude.

Why it happens: They see $T$ in the denominator but treat it like a numerator.

How to avoid:

• Write it clearly: $-\frac{E_a}{R} \cdot \frac{1}{T}$
• Lower $T$ → larger $1/T$ → larger $E_a/RT$ → more negative exponent → smaller $k$
• Higher $T$ → smaller $1/T$ → smaller $E_a/RT$ → less negative exponent → larger $k$

Mistake 4: Mixing Up Activation Energy and Temperature Effects

The error: Students think both $E_a$ and $T$ affect $k$ in the same direction.

Why it happens: They memorize "higher temperature = faster reaction" but don't connect it to the equation.

How to avoid:

• Make a quick table: …