Chemistry · Lesson

Reaction coordinates, transition states and molecular collisions

Connect collision conditions and reaction-coordinate geometry to a rate model without treating a potential curve as time. Keep activation free energy, standard-state conventions and dynamical corrections explicit.

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By the end, you can

Distinguish a path barrier, a saddle and a stable intermediate.
Calculate encounters using a relative-speed and pair-counting convention.
Retain the standard concentration in a bimolecular transition-state coefficient.

A coordinate describes configuration rather than elapsed time

A reaction coordinate maps configurations toward reactant or product regions. It need not be one physical bond length, and its horizontal axis is not automatically time. A minimum-energy path describes a selected potential landscape; a molecular trajectory additionally needs equations of motion, initial conditions and the potential used.

Potential energy, zero-point-corrected energy and activation free energy are different quantities. A free-energy barrier includes the ensemble and thermal reference. Neither a bare path maximum nor one imaginary saddle frequency alone supplies a complete rate, transmission probability or product distribution.

A path maximum can be stable in a transverse direction

Consider the original two-coordinate model U(q,s) = 20(1−q²)² + 5s² kJ mol⁻¹, with q and s dimensionless. Along s=0, q=±1 gives U=0 and q=0 gives U=20 kJ mol⁻¹. At (0,0), the curvature is negative along q and positive along s, so the point is a saddle, not a local minimum.

The two Hessian eigenvalues there are −80 and +10 kJ mol⁻¹ per squared dimensionless coordinate. At (±1,0), they are +160 and +10. This model illustrates geometry; it is not an actual molecular potential or an activation free energy at a specified temperature. A stable intermediate instead occupies an appropriate local minimum in the relevant landscape.

A path maximum and a two-coordinate saddle describe geometry

Potential U / kJ mol⁻¹

Dimensionless coordinate q

Selected s = 0 configurations
— Potential along s = 0

Underlying data

Dimensionless coordinate q	Potential U / kJ mol⁻¹
-1	0 · Selected s = 0 configurations
-0.5	11.25 · Selected s = 0 configurations
0	20 · Selected s = 0 configurations
0.5	11.25 · Selected s = 0 configurations
1	0 · Selected s = 0 configurations

Original polynomial U(q,s)=20(1−q²)²+5s² kJ mol⁻¹, with dimensionless coordinates. The plot fixes s=0. The dashed curve and selected markers show potential energy along q, not elapsed time or activation free energy. Positive transverse curvature makes the central path maximum a saddle of this stipulated surface. No molecular masses or trajectories are supplied.

Use relative motion and a declared pair count

For dilute independent Maxwell velocities, the mean relative speed of distinct particles A and B is ⟨v_rel⟩ = √[8k_B T/(πm_red)], with m_red=m_A m_B/(m_A+m_B), particle masses m_A and m_B, absolute temperature T, and Boltzmann constant k_B in J K⁻¹. The RMS relative speed is √(3k_B T/m_red), a different average. A fixed hard-sphere collision diameter d gives cross-section σ=πd² and number-density encounter coefficient k_n=σ⟨v_rel⟩ in m³ s⁻¹.

Distinct-species pair events per volume and time are k_nρ_Aρ_B, where ρ denotes number density. Identical pair events instead have a one-half counting factor in the dilute macroscopic limit; counting each particle’s encounters would count each pair twice. A coefficient multiplying molar concentrations in mol m⁻³ is N_A k_n, where N_A is the Avogadro constant in mol⁻¹. Encounters do not guarantee reactions: impact parameter, orientation, internal energy and quantum effects can matter.

An activated-complex model also needs its reference and dynamics

In a stipulated ideal concentration transition-state model, a dimensionless activation constant is K‡=exp[−ΔG°‡/(RT)]. R is the molar gas constant. For a unimolecular first-order model, k₁=κ(k_B T/h)K‡. Here h is the Planck constant and κ is the specified transmission factor. Classical recrossing can reduce the no-recrossing estimate; quantum corrections require their own definitions.

For a separate bimolecular model using activities c/c°, the activated-complex concentration is K‡[A][B]/c°, giving k₂=κ(k_B T/h)K‡/c°. Its units depend on the concentration basis: c°=1 mol L⁻¹ equals 1000 mol m⁻³. Using a first-order coefficient without that reference factor would mix units. These are model estimates under stated equilibrium and dynamics assumptions, not rates determined by the earlier potential curve.

Worked examples

Calculate an encounter coefficient before a reaction yield

Use stipulated T=300 K, m_A=2.00 × 10⁻²⁶ kg, m_B=3.00 × 10⁻²⁶ kg and hard-sphere collision diameter d=4.00 × 10⁻¹⁰ m. Find the distinct-species encounter coefficient.

The reduced mass is m_red=1.20 × 10⁻²⁶ kg.
With k_B=1.380649 × 10⁻²³ J K⁻¹, mean relative speed is approximately 937.523 m s⁻¹.
σ=πd²≈ 5.02655 × 10⁻¹⁹ m².
k_n=σ⟨v_rel⟩≈ 4.71250 × 10⁻¹⁶ m³ s⁻¹.

k_n≈ 4.71 × 10⁻¹⁶ m³ s⁻¹ for the stipulated dilute hard-sphere model. A reactive fraction and any needed other physics remain additional inputs.

Check a potential barrier without inventing a rate

For the stipulated U(q,s), compare (−1,0), (0,0) and (1,0).

The two endpoint minima both have U=0.
The central point has U=20 kJ mol⁻¹.
Its q curvature is negative and its s curvature positive.
The forward and reverse path potential barriers are both 20 kJ mol⁻¹, with endpoint potential change 0.

The model has a central saddle and symmetric potential barriers. No elapsed time or thermal activation free energy follows from those values alone.

Retain the bimolecular standard-concentration factor

For a separate stipulated ideal bimolecular model, T=300 K, ΔG°‡=60.0 kJ mol⁻¹, κ=0.400 and c°=1 mol L⁻¹. At current [A]=0.0200 and [B]=0.0300 mol L⁻¹, find its initial formation rate.

K‡=exp[−60000/(RT)]≈ 3.57500 × 10⁻¹¹.
k_B T/h≈ 6.25099 × 10¹² s⁻¹.
k₂=κ(k_B T/h)K‡/c°≈ 89.3891 L mol⁻¹ s⁻¹, equivalently 0.0893891 m³ mol⁻¹ s⁻¹.
Multiply by the supplied current concentrations: v=k₂[A][B]≈ 0.0536334 mol L⁻¹ s⁻¹.

Initial v≈ 0.0536 mol L⁻¹ s⁻¹ under this separate model. It is an instantaneous rate; depletion would change a later rate.

Try it yourself

Does the horizontal reaction-coordinate axis automatically measure elapsed time?
Show the answer and reasoning
No.
It describes configuration; dynamics requires additional information.
If d doubles in the same fixed hard-sphere model, how does σ change?
Show the answer and reasoning
It increases by a factor of 4.
The cross-section isπd².
Why do identical-particle pair events have a one-half macroscopic counting factor?
Show the answer and reasoning
To count each unordered pair once.
Summing every particle’s encounters counts the same pair twice.
Which factor supplies the concentration basis in the stated bimolecular Eyring coefficient?
Show the answer and reasoning
1/c°.
The activation constant is dimensionless; molecularity changes the concentration units.

What to carry forward

Separate configuration, potential, free energy and time.
A saddle is not a stable intermediate.
Name the speed average and event-counting basis.
Retain standard-state and transmission assumptions in a rate estimate.

References

Teaching resources and reference material supporting this lesson’s explanations and examples.