SCIENCE · INVESTIGATE AND EXPLAIN

Construct transition states from local energy barriers

Repair a two-step energy diagram, compare directional barriers and separate a common datum shift from a changed pathway.

Learning outcome: Construct a transition-state level from its adjacent minimum and barrier, reconcile forward/reverse differences and avoid inferring overall rates from peak height alone.

Starting recap

Energy differences subtract two levels. A reaction coordinate orders configurations; it is not elapsed time. A transition state lies above neighboring minima in this schematic.

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Try

An intermediate lies 40 kJ/mol below reactants. A 70 kJ/mol rise from that intermediate places the next transition state at +30, not +70.

Open the experiment

Predict

Predict whether the highest plotted peak must also have the largest local forward barrier. Predict what happens to every barrier after adding 100 kJ/mol to all levels.

Your prediction and reason

Experiment

Repair the second peak to match the supplied local barrier. Test an invalid below-minimum proposal, shift the common datum, compare an alternative path with unchanged endpoints and transfer to an endothermic product. Capture both valid and invalid proposals with explanations.

Construct local barriers on a multistep energy profile

Worked teaching model · synthetic values · Original schematic potential-energy profiles, not measured free-energy surfaces. Each proposed transition state must lie strictly above its adjacent minima. The horizontal reaction coordinate is neither time nor distance. A common energy-datum shift changes no differences. Local barriers alone do not determine an overall rate, dominant pathway or equilibrium constant without further kinetic and thermodynamic assumptions.

Original energy-profile task: Deep intermediate: reconstruct the second peak packet Second transition-state level before the common shift: 0 kJ/mol

Reset keeps your prediction, captures and practice history.

Place a transition state from a local barrier

Supplied second forward barrier: 70 kJ/mol, measured upward from the intermediate. The common plotted offset is 0 kJ/mol. Your unshifted second-peak proposal is 0; its plotted level is 0 kJ/mol. These are original schematic potential energies, not measured thermochemical or free-energy data.

Valid shape, wrong constraint. Recalculate the second peak from the intermediate, not from zero.

Step	Forward: TS − left minimum	Reverse: TS − right minimum	Step change: right − left	Forward − reverse
1	50	90	-40	-40
2	40	20	20	20

All table energies are kJ/mol. Step changes sum to -40 + (20) = -20, equal to product minus reactant energy. Reversing the overall path changes this difference to 20. The forward and reverse barriers differ by each step's energy change; their absolute values are not generally equal.

Highest plotted peak: step(s) 1. Largest local forward barrier: step(s) 1 at 50 kJ/mol. These are different questions. Neither label alone establishes the rate-controlling step: populations, rate prefactors, reversibility and temperature also matter.

Keep the same proposal and shift the whole baseline path by +100. All directional barriers and endpoint differences stay unchanged. Compare the alternative pathway: its reactant/product energies remain the same, while its intermediate and peaks differ. This illustrates a possible changed pathway; it does not establish actual catalytic acceleration, a new equilibrium composition or a numerical equilibrium constant.

Sources and interpretation limits

OpenStax Chemistry 2e §12.7 explains alternative energy pathways with unchanged endpoints. The constructed piecewise diagram is a teaching schematic, not a computed molecular surface. Potential-energy differences are not automatically Gibbs free-energy differences; no rate constant, Arrhenius calculation or equilibrium constant is inferred here.

The profile is geometrically valid, but the second local barrier does not match the supplied constraint.

Compute each barrier from its own adjacent minimum, not from the global energy zero. Negative differences remain visible as invalid proposals, not physical negative activation barriers.

Observation to attach to a capture

Captures retain the original energy task and proposed transition-state level, including invalid placements. Replay reconstructs directional differences, not measured barriers, rates or catalyst performance.

Understand

Try: place the second peak

The original two-step diagram puts reactants at 0, the intermediate at −40 and products at −20 kJ/mol. The first transition state is at +50. The supplied second forward barrier is 70 kJ/mol. Predict the second transition-state level before changing the control: start from the intermediate and rise by 70, giving −40+70=+30.

Understand the neighboring reference

At that proposal, the forward barriers are 50 and 70 kJ/mol. The second peak is lower on the page, yet its local forward barrier is larger. Reverse barriers are 50−(−40)=90 and 30−(−20)=50. A barrier is a difference from the adjacent starting minimum, not a peak’s height above the arbitrary global zero.

The two step energy changes are −40 and +20 kJ/mol, summing to −20. For each step, forward barrier minus reverse barrier equals its endpoint change. Changing direction reverses the endpoint difference but does not make both activation barriers equal.

Experiment with invalid and shifted proposals

Place the second peak below or equal to a neighboring minimum. The signed ledger retains the error, but the interface refuses to call that location a valid transition-state maximum. A negative difference here diagnoses an invalid proposed profile, not a physical negative activation barrier.

Return to +30 and shift every plotted level by +100. Every difference stays unchanged: subtraction removes the shared offset. The common vertical scale makes the plotted shift visible without suggesting a change in the barriers. Then change the product endpoint to +20 and calculate the new reverse second-step barrier: 30−20=10 kJ/mol, while the forward second barrier remains 70.

Compare an alternative path honestly

The alternative task retains reactants at 0 and products at −20, changes the intermediate to −10 and the first peak to 30, and asks for a second forward barrier 20. Its second peak belongs at 10. The path and local barriers change while the overall endpoint difference remains −20. This is a schematic illustration of an alternative pathway, not measured catalyst performance.

Do not label the overall slowest step solely by its highest absolute peak or largest local barrier. Rate prefactors, concentrations, intermediate populations, reversibility and temperature also enter kinetic behavior. Nor does a potential-energy diagram alone give Gibbs free energies or an equilibrium constant. The horizontal coordinate orders configurations and is not a clock.

Apply and continue

Capture a repaired baseline, the same proposal after the datum shift and an invalid peak. Explain which changes affect barriers and which only change their plotted reference. Apply the unfamiliar neighboring-minimum calculation in practice. Revisit mechanism construction for species accounting and HS single-barrier reasoning for the simpler case.

OpenStax Chemistry 2e §12.7 supplies alternative-pathway concepts. All energy levels here are original teaching assumptions. Other authored mechanism comparisons appear in acid/base catalytic cycles and enzyme inventory. This one activity does not certify full-unit learner understanding; empirical surfaces and independent practical review still require actual evidence. Quantitative Arrhenius calculations are not an AP Exam prerequisite.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Distinguish intermediate minima from transition-state maxima, local barriers from overall energy change, and a schematic energy comparison from evidence for a rate-controlling step or actual catalyst.

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Distinguish energetic collisions from suitably oriented ones Next: Construct a catalytic surface without double-counting its sites

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