DAT Organic Chemistry: Activation Energy and the Rate-Determining Step

DAT High-Yield Concept

Activation Energy Measurement: For Step 1, activation energy ( $E_{\mathrm{a}1}$ or $\Delta G^\ddagger_1$ ) is measured from initial reactants to Transition State 1 ( $\text{TS}_1$ ). For any subsequent step $n$ , $E_{\mathrm{a},n}$ is strictly measured from the local intermediate energy minimum ( $\text{I}_{n-1}$ ) immediately preceding that step to its transition state peak ( $\text{TS}_n$ ).
Rate-Determining Step (RDS): The RDS is the elementary step with the largest activation energy barrier (the "tallest climb"), governing the overall reaction rate and rate law.
Profile Topology: The number of peaks equals the number of elementary steps; the number of valleys between peaks equals the number of reactive intermediates.

1. Activation Energy of Each Step

Activation energy ( $E_{\mathrm{a}}$ or $\Delta G^\ddagger$ ) represents the minimum kinetic energy reacting species must possess to reach the high-energy transition state. On a reaction coordinate profile, the progress of reaction spans the horizontal axis while potential energy spans the vertical axis. Every elementary step in a reaction mechanism possesses its own distinct activation energy barrier.

Reaction Coordinate Diagram: Starting material or reactant A convert to product C via the transition state B. (Source: AimNature · CC BY-SA 3.0 · Wikimedia Commons)

In a multi-step reaction, measuring activation energy correctly is critical: Step 1 activation energy ( $E_{\mathrm{a}1}$ ) extends from the starting reactant baseline to $\text{TS}_1$ . However, Step 2 activation energy ( $E_{\mathrm{a}2}$ ) extends from the carbocation or reactive intermediate energy minimum up to $\text{TS}_2$ .

E1 Reaction Coordinate (Source: Kkukla · CC BY-SA 4.0 · Wikimedia Commons)

Worked Example & Exam Trap: Measuring Activation Energy

Problem: In the E1 elimination of 2-bromo-2-methylpropane, Step 1 heterolysis forms a carbocation: $\ce{(CH3)3C-Br -> (CH3)3C^+ + Br^-}$ ( $E_{\mathrm{a}1} = \pu{80 kJ/mol}$ ). The intermediate lies at $\pu{45 kJ/mol}$ relative to reactants, and $\text{TS}_2$ lies at $\pu{60 kJ/mol}$ relative to reactants during deprotonation: $\ce{(CH3)3C^+ + H2O -> (CH3)2C=CH2 + H3O^+}$ . Calculate $E_{\mathrm{a}2}$ .

Reasoning: $E_{\mathrm{a}2} = E(\text{TS}_2) - E(\text{Intermediate}) = \pu{60 kJ/mol} - \pu{45 kJ/mol} = \pu{15 kJ/mol}$ .

Common DAT Exam Trap: Subtracting the reactant baseline ( $\pu{0 kJ/mol}$ ) to get $\pu{60 kJ/mol}$ for Step 2. Always measure $E_{\mathrm{a},n}$ from its preceding local intermediate minimum!

2. Identifying the Rate-Determining Step (RDS)

A multi-step reaction mechanism cannot proceed faster than its slowest elementary step. The rate-determining step (RDS) acts as the bottleneck of the mechanism and corresponds to the elementary step requiring the largest activation energy barrier ( $E_{\mathrm{a}}$ ).

The rate-determining step is the tallest climb — The rate-determining step is the one with the tallest climb measured from its own starting well, not the one whose peak sits highest on the page. Here step one asks for far more energy than step two, so it is the bottleneck. Anything that happens after the bottleneck is fast by comparison and has no effect on the overall rate.

Worked Example & Exam Trap: Identifying the RDS

Problem: A reaction pathway has Reactants at $\pu{0 kJ/mol}$ , $\text{TS}_1$ at $\pu{70 kJ/mol}$ , Intermediate at $\pu{30 kJ/mol}$ , $\text{TS}_2$ at $\pu{80 kJ/mol}$ , and Products at $\pu{-20 kJ/mol}$ . Identify the RDS.

Reasoning: Calculate individual step barriers:
Step 1 barrier: $E_{\mathrm{a}1} = 70 - 0 = \pu{70 kJ/mol}$ .
Step 2 barrier: $E_{\mathrm{a}2} = 80 - 30 = \pu{50 kJ/mol}$ .
Since $E_{\mathrm{a}1} > E_{\mathrm{a}2}$ , Step 1 is the slowest step and serves as the rate-determining step.

Common DAT Exam Trap: Incorrectly selecting Step 2 as the RDS simply because $\text{TS}_2$ reaches a higher absolute peak ( $\pu{80 kJ/mol}$ ) than $\text{TS}_1$ ( $\pu{70 kJ/mol}$ ). The RDS depends on the step barrier height relative to its starting valley, not absolute height from zero.

DAT Mnemonic: "RDS = Tallest Climb" — Identify the RDS by finding the step with the tallest vertical climb from its preceding energy well to its peak.

3. Relating Profile Shape to Mechanism

The overall topology of a reaction energy profile reveals key mechanistic details:

Peaks (Transition States): Each peak represents a high-energy transition state ( $\text{TS}^\ddagger$ ) where bonds are partially formed and broken.
Valleys (Intermediates): Each valley between peaks represents a distinct reactive intermediate species (e.g., carbocation, carbanion, or radical).
Concerted vs. Stepwise: Concerted mechanisms (e.g., $\mathrm{S_N2}$ , $\mathrm{E2}$ ) display a single peak and zero intermediate wells. Stepwise mechanisms (e.g., $\mathrm{S_N1}$ , $\mathrm{E1}$ ) exhibit multiple peaks separated by intermediate wells.

Worked Example: Concerted $\mathrm{S_N2}$ Profile

Reaction: Nucleophilic substitution of bromomethane by hydroxide ion: $\ce{CH3Br + OH^- -> CH3OH + Br^-}$ .

Mechanism Analysis: Hydroxide attacks back-side while bromide departs simultaneously. The profile contains 1 single peak for the pentacoordinate transition state $[\ce{HO \cdots CH3 \cdots Br}]^{\ddagger}$ and zero intermediate wells.

Comparison Table: Profile Features of Major Mechanisms

Feature	Concerted Mechanisms ( $\mathrm{S_N2}$ , $\mathrm{E2}$ )	Stepwise Mechanisms ( $\mathrm{S_N1}$ , $\mathrm{E1}$ )
Number of Elementary Steps	1 step	2 or more steps
Transition States (Peaks)	1 peak	Multiple peaks ( $n$ steps = $n$ peaks)
Intermediates (Valleys)	0 intermediate wells	$n - 1$ local energy wells
Rate-Determining Step	The single concerted step	Step with highest $E_{\mathrm{a},n}$ (typically Step 1 carbocation formation)

Critical Thinking Questions

How does adding a catalyst alter a multi-step reaction profile without changing the energy levels of initial reactants or final products?
In an $\mathrm{S_N1}$ reaction, why is carbocation formation (Step 1) almost always the rate-determining step compared to nucleophilic capture (Step 2)?
A reaction has $\text{TS}_1$ at $\pu{50 kJ/mol}$ , Intermediate at $\pu{10 kJ/mol}$ , and $\text{TS}_2$ at $\pu{70 kJ/mol}$ relative to starting reactants ( $\pu{0 kJ/mol}$ ). Which step is the RDS?

References

Sources returned by Google Search grounding during lesson verification.

Describing a Reaction - Energy Diagrams and Transition States (LibreTexts) — chem.libretexts.org
Reaction Mechanisms, Catalysts, and Reaction Coordinate Diagrams (Chad's Prep) — chadsprep.com

Activation Energy and the Rate-Determining Step

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