Trajectories and Energy Disposal

Introduction

A potential-energy surface tells us which molecular geometries are energetically accessible, but molecules also have momentum. Two collisions with the same total energy can behave differently if one carries energy as relative translation and the other as vibration of a reactant bond. Trajectory studies follow motion across the surface. Polanyi's rules offer a useful qualitative link between where a barrier lies and which form of energy most effectively promotes reaction, while product energy distributions reveal what happened after the barrier.

Core explanation

For a simplified atom–diatom reaction A + BC → AB + C, relative translation describes how A approaches BC. Internal vibration describes stretching and contracting of the BC bond before impact. A barrier located early along the route, when the geometry still resembles separated reactants, is often promoted efficiently by adding translational collision energy: it helps the partners reach and cross the entrance barrier. A late barrier, near a product-like geometry with a bond substantially rearranged, is often promoted more efficiently by vibrational excitation of the relevant reactant mode. These qualitative patterns are commonly called Polanyi's rules. A primary research discussion states the trends and explicitly notes their limits.

The rules compare energy put into different modes at similar total energy; they do not claim that vibration never helps an early-barrier reaction or translation never helps a late-barrier one. The excited vibration must project onto the coordinate that matters for reaching the barrier. Exciting an unrelated motion may do little. Molecular rotation, orientation, tunneling and multiple electronic surfaces can change the outcome. For polyatomic molecules, identifying one barrier as simply “early” or “late” may be ambiguous because many bonds and angles change. A state-resolved dynamics study tests mode-specific trends and illustrates why total-energy comparisons matter.

A trajectory is a time-dependent path of nuclear positions and momenta on a chosen potential-energy surface. In a classical calculation, initial positions, velocities and internal states are sampled, then equations of motion are integrated. Some trajectories return to reactants; some form products; some temporarily occupy a complex. The minimum-energy path of page 3113 is only one reference path. A real trajectory may cross a saddle region away from its valley floor, recross a proposed dividing surface or branch toward another product. Quantum dynamics can add interference and tunneling that a purely classical trajectory omits.

After products form, conservation of energy still applies. Define ΔE = E products,reference − E reactants,reference for internal reference energies. If reactants bring collision energy E coll and internal excitation E int, then a simplified available energy is E avail = E coll + E int − ΔE. This can appear as relative product translation, product rotation, product vibration or other allowed channels. For an exothermic reaction ΔE < 0, released chemical energy increases E avail. The equation does not specify the distribution by itself; dynamics and product-state measurements do.

For example, let E coll = 0.20 eV, initial internal excitation be zero and ΔE = −0.50 eV. Available product energy is 0.20−(−0.50) = 0.70 eV above the selected product reference state. One possible measured distribution could be 0.30 eV translation, 0.20 eV rotation and 0.20 eV vibration. The sum is 0.70 eV. A different reaction path could allocate the same total differently, so product-state measurements contain mechanistic information beyond the net exothermicity.

Angular distributions add another clue. Products concentrated in certain directions relative to the incoming beam can suggest a direct collision geometry, while more isotropic scattering may be consistent with a longer-lived complex, subject to detailed analysis. Translational energy and internal-state distributions must be interpreted together with the apparatus and potential-energy surface. No single pattern automatically proves one mechanism.

The energetic form of a thermal rate constant averages all this mode-specific behaviour over a broad distribution of reactant states. A molecular-beam experiment can select one collision energy or initial vibrational state to reveal a dependence that is invisible in a single bulk k. This is why reaction dynamics complements empirical rate laws rather than replacing them.

Step-by-step reasoning

1. Identify the reaction and the location of its barrier along a proposed path. 2. Distinguish relative translational energy from specific vibrational or rotational excitation. 3. Apply Polanyi's rule as a qualitative prediction, with the relevant mode named. 4. Compare collisions at equal total energy when testing mode efficacy. 5. Calculate total energy available to products from collision, excitation and ΔE. 6. Use product-state and angular measurements to test trajectory-based explanations.

Visual explanation

Draw two energy profiles for A + BC → AB + C: one with an entrance-side peak near reactants, one with an exit-side peak near products. Under the first, draw a long incoming translation arrow; under the second, draw a vibrating BC bond. Then show a product energy bar divided into translation, rotation and vibration sections whose lengths add to the calculated E avail. The bar is a possible distribution, not a prediction from enthalpy alone.

Real-world analogy

To cross a doorway threshold, forward motion may help most; to reshape an object just before it exits, internal flexing may help more. That loosely evokes early versus late barrier energy effectiveness. Molecular pathways are not literal doorways, and different vibrations couple differently, so the analogy is only a guide to the position of the difficult step.

Real-world example

Researchers prepare a vibrationally excited reactant in a molecular beam and measure its reaction probability with another species at a controlled collision energy. They compare it with an unexcited beam whose translational energy is adjusted so total energy matches. If vibration gives greater enhancement, they consider a late barrier or strong vibrational coupling. A calculated surface and product-state measurements test that interpretation before generalising it.

Why?

Why does equal total energy not guarantee equal reaction probability? Energy is distributed among motions that couple differently to the barrier-crossing coordinate. Translation can drive approach; a specific vibration can stretch a bond that must break; another vibration may scarcely affect the needed geometry. Molecular dynamics therefore depends on where the energy is placed, not just its sum.

Common misconception

“Polanyi's rules are exact laws for every molecule.” They are qualitative trends developed for relatively simple activated reactions; polyatomic pathways, surface chemistry and quantum effects can produce exceptions. Another mistake is to assume all released reaction energy becomes product speed. Rotation, vibration and electronic excitation can also carry energy, and measured distributions must determine the shares.

Worked example

An idealised exothermic channel has ΔE = −0.50 eV and begins with 0.20 eV relative translational energy and negligible initial internal excitation. Conservation gives E avail = 0.20−(−0.50) = 0.70 eV. If product-state analysis assigns 0.30 eV to relative translation and 0.20 eV to rotation, the remaining energy available for vibration or other allowed internal modes is 0.70−0.30−0.20 = 0.20 eV. This arithmetic checks energy disposal; it does not identify the trajectory or barrier location by itself.

Quick check

1. Under the simple Polanyi trend, which energy input is often more effective for a late, product-like barrier? Answer: Excitation of a reactant vibration that couples to the bond rearrangement is often more effective than the same added relative translation energy.

Exam focus

Define “early” and “late” relative to reactant- and product-like geometry, not clock time. State the Polanyi trends as qualitative and mode-specific. Write a consistent energy-balance sign convention before calculating product energy. Distinguish one minimum-energy path from an ensemble of dynamical trajectories, and do not infer product-state distribution solely from ΔE.

Advanced insight

Mode specificity can be examined with state-selected molecular beams or laser-prepared reactants. The coupling of an excited mode to the transition-state direction can be estimated from overlap of vibrational displacement vectors and reaction-coordinate motion. In complex systems, post-transition-state bifurcations can determine product branching after the nominal barrier has been crossed. These effects show why transition-state location is informative but does not always dictate every product channel.

Summary

Reaction trajectories connect molecular motion with outcomes on a potential-energy surface. Early barriers are often promoted by relative translation, while late barriers can respond strongly to relevant vibration; Polanyi's rules describe these trends with important limits. Energy conservation sets the total available to products, but dynamics partitions it among translation, rotation, vibration and other channels. State-resolved experiments reveal that partition and test proposed mechanisms.

Practice questions

1. What is meant by an early barrier? Answer: It lies near reactant-like geometries along a proposed reaction path. 2. Why must mode-efficacy comparisons use equal total energy when possible? Answer: Otherwise greater reaction probability could merely reflect more total energy rather than a special advantage of translation or vibration. 3. A reaction has ΔE = +0.10 eV and E coll = 0.30 eV with no initial excitation. What is E avail? Answer: 0.30−0.10 = 0.20 eV above the selected product reference state. 4. Name one measured quantity that reveals energy disposal. Answer: Product translational energy, rotational state, vibrational state or angular scattering distribution can provide evidence.