Reaction energy profiles

Result

Reaction energy profiles

T-064Home CU-206Threads kinetics · structure

Statement

Intermediates, transition states and the rate-determining step.

Why it matters

sn1-sn2, e1-e2-elimination, and nucleophilic-carbonyl-addition (together with electrophilic-addition-markovnikov and electrophilic-aromatic-substitution) each describe specific reaction mechanisms with their own sequence of elementary steps; reaction-energy-profiles supplies the common graphical and conceptual language — energy plotted against reaction coordinate — used to describe and compare all of them, distinguishing genuine intermediates from fleeting transition states and identifying which single step controls the overall rate. It is the essential vocabulary threading together every mechanism discussed elsewhere in this unit.

Hypotheses

The reaction coordinate is treated as a single collective coordinate summarising the progress of bond-breaking and bond-forming along the path.This is a simplification of the full, multidimensional potential-energy-surface picture down to its one-dimensional minimum-energy-path slice, adequate for comparing mechanisms without needing the full surface. Each elementary step proceeds through exactly one transition state, corresponding to a first-order saddle point on the full surface.This is what allows "one step, one transition state, one rate constant" to serve as the basic unit of mechanistic analysis used throughout this unit. The Hammond postulate — that a transition state's structure resembles whichever adjacent species (reactant, intermediate, or product) it is closer to in energy — is used to predict how substituent changes shift a transition state's height.This is an approximation useful for comparing closely related reactions, not a rigorously derived result, and is least reliable when applied as an absolute, standalone statement about a single, isolated reaction.

Proof

\text{Energy (vertical axis) vs reaction coordinate (horizontal axis)}

Reactants and products sit at energy minima at the two ends of the profile, by definition of a stable starting and ending species. A

\text{Intermediate: local energy minimum partway along the profile}

A genuine reaction intermediate is a real, however short-lived, chemical species that can in principle be detected, trapped, or, for a sufficiently stable case, isolated. A

\text{Transition state: local energy maximum partway along the profile}

A transition state is not a real, isolable species at all, but the fleeting, highest-energy configuration passed through during a single elementary step, matching potential-energy-surfaces' first-order-saddle-point definition applied along this one-dimensional slice. A

\text{Rate-determining step} = \arg\max_i\big(G_i^{\ddagger} - G_{\text{reactants}}\big)

For a multistep mechanism, the overall rate is controlled by the single highest-energy transition state relative to the starting materials, since it represents the highest activation-energy barrier that must be crossed en route to product. A

\text{Early (exergonic step): TS resembles reactant}; \qquad \text{Late (endergonic step): TS resembles product}

The Hammond postulate lets substituent effects on the relative stability of reactant, intermediate, or product be used to predict how they will, correspondingly, affect the nearby transition state's energy and hence the reaction's rate. B

Result

\text{Rate} \propto e^{-\Delta G^{\ddagger}_{\text{RDS}}/RT}, \qquad \Delta G^{\ddagger}_{\text{RDS}} = \max_i\big(G_i^{\ddagger} - G_{\text{reactants}}\big)

Reading. Only the single highest-energy transition state along the whole multistep pathway, measured relative to the starting materials, controls the overall observed rate.

Scope. Assumes each step genuinely proceeds through one transition state; when two steps have comparably high barriers, both can contribute measurably to the overall rate, a less common but genuine complication.

Corollaries & converses

sn1-sn2's contrasting mechanisms are most cleanly distinguished by their energy profiles: SN1 shows a genuine carbocation intermediate (a real minimum, two flanking transition states) while SN2 shows a single concerted transition state with no intermediate at all.
The Hammond postulate directly explains why more stable carbocation intermediates (tertiary \(>\) secondary \(>\) primary) correspond to lower-energy, earlier transition states for their formation, rationalising SN1's characteristic reactivity order.
Kinetic versus thermodynamic control is precisely the distinction between which product forms fastest (lowest transition-state energy) versus which product is most stable (lowest final energy) — two generally different points read directly off the same profile.

Fails without

Assume the first step of a multistep mechanism is always rate-determining (violating Step 4's actual criterion): misidentifies the true rate-determining step whenever a later step's transition state is actually the highest in energy, giving a wrong prediction of which factors control the overall rate.
Apply the Hammond postulate (Hypotheses, third assumption) to make an absolute statement about a single, isolated reaction rather than a comparison between closely related reactions: the predicted transition-state structure or energy trend can be qualitatively unreliable, since the postulate is an approximation, not a rigorous derivation.

Common errors

Confusing an intermediate (a real local minimum, a genuine albeit short-lived species) with a transition state (a local maximum, never an isolable species).
Assuming the rate-determining step is always the first step of a mechanism, rather than correctly identifying it as whichever step has the highest-energy transition state relative to the starting materials.
Applying the Hammond postulate uncritically to every transition state without recognising it specifically links structure to whether a step is strongly exergonic or strongly endergonic.
Comparing the absolute energies of two different reaction profiles rather than the local activation energy relevant to a given elementary step within a specific profile.

Discussion

The energy-profile/reaction-coordinate diagram, alongside transition state theory itself, was formalised substantially through the work of Henry Eyring and, independently, Michael Polanyi and Meredith Evans in the 1930s, connecting the graphical picture directly to a quantitative rate expression.

George Hammond's 1955 postulate, while an approximation rather than an exact derivation, has proven remarkably durable and predictive across organic mechanism; it is most safely applied to compare closely related reactions, such as the same reaction with different substituents, rather than to make absolute, standalone statements about a single reaction's transition-state structure in isolation.

Common misconception: that a "high-energy" species drawn partway along a mechanism's arrow-pushing scheme is automatically an intermediate. Whether it is a genuine intermediate or merely a transition state depends entirely on whether it corresponds to a local minimum or local maximum on the actual energy profile, not on how it happens to be drawn in a mechanism.

Worked examples

\text{SN1: R-LG} \to [\text{R}^+ \cdots \text{LG}^-]^{\ddagger} \to \text{R}^+ \to [\text{R}\cdots\text{Nu}]^{\ddagger} \to \text{R-Nu}

The carbocation intermediate sits as a local minimum flanked by two transition states; ionisation (formation of the higher-energy carbocation) is generally the rate-determining step, since it is typically the higher of the two barriers, well above the subsequent, generally faster nucleophilic capture step. A

\text{Tertiary carbocation: more stable} \Rightarrow \text{earlier, lower TS for its formation}

Applying the Hammond postulate (Step 5) to the ionisation step: a more stable carbocation lies lower in energy, and since ionisation is endergonic (uphill) from starting material to carbocation, a lower-energy carbocation product pulls its preceding transition state earlier and lower as well — directly rationalising the SN1 reactivity order (tertiary \(>\) secondary \(>\) primary) from the energy profile itself. B

\text{Rate}_{\text{SN1}} \propto e^{-\Delta G^{\ddagger}_{\text{ionisation}}/RT}, \text{ lower for more stable carbocations}

Reading. The Hammond postulate connects a purely thermodynamic fact (carbocation stability) to a kinetic outcome (reaction rate) via the shape of the energy profile.

Scope. The same reasoning — identify the rate-determining step, then apply the Hammond postulate to its transition state — extends to any multistep mechanism with a clearly higher-energy step.

Problems

A three-step mechanism has transition-state energies (relative to reactants) of \(+45\), \(+62\), and \(+38\,\text{kJ/mol}\) for steps 1, 2, and 3. Identify the rate-determining step.
Solution
Step 2, since its transition state (\(+62\,\text{kJ/mol}\)) is the highest of the three relative to the reactants, matching the Result's \(\arg\max\) criterion.
Using the Hammond postulate, state whether the transition state for a highly exergonic step is expected to resemble reactant or product more closely.
Solution
Reactant. A highly exergonic (strongly downhill) step has its transition state occurring early along the reaction coordinate, energetically and structurally closer to the reactant than to the much lower-energy product (Step 5's early-transition-state case).
Explain why only the height of the highest transition state relative to the reactants, not the depth of any intermediate valley, determines the overall rate of a multistep reaction.
Solution
The overall rate is governed by the Boltzmann factor associated with crossing the highest activation-energy barrier along the path (Result); once that highest barrier is crossed, subsequent, lower barriers are traversed comparatively quickly and do not further limit the rate. The depth of any intermediate's energy well affects only how long-lived that intermediate is, not how fast the overall multistep process proceeds, since it is not itself a barrier that must be surmounted.