Past exam of the chemistry A course of the University of Cambridge / 2025 / Paper 2 / A8

A potential energy surface gives the molecular electronic energy, with nuclear repulsion, as a function of the nuclear coordinates under the Born-Oppenheimer approximation. Its minima describe stable configurations and its saddle regions help identify reaction barriers.

A reaction coordinate parametrizes progress along a chosen path on the potential energy surface. It usually combines several bond lengths or angles; it need not be one literal bond distance or elapsed time.

A transition state is the configuration or dividing-surface region associated with passage between reactant and product basins. In the elementary saddle-point picture it is a first-order saddle of the potential energy surface: stable against small displacements in transverse directions but unstable along the reaction direction. Transition state theory counts forward crossings and, in its simplest form, neglects recrossing.

The quantities $f_A$ and $f_B$ are molecular partition functions per unit volume for the two reactants, with their internal rotational, vibrational and electronic factors included. Thus their translational factors are $(2\pi m_{j}kT/h^2{)}^{3/2}$, rather than those factors multiplied by $V$. Their energy zeros must match the separately displayed barrier convention.

The primed $f_{\mathrm{TS}}^{\prime }$ is the corresponding transition state partition function per unit volume with the unstable reaction-coordinate degree of freedom omitted. Its replacement by the crossing-frequency factor $kT/h$ is already outside the partition-function ratio. For a diatomic atom-pair transition state, the omitted coordinate is the bond stretch, leaving translation and rotation.

The barrier $\Delta {\epsilon }_0^{‡}$ is the energy difference between the transition-state reference level and the separated reactant reference levels. Consistent zero-point energies of stable transverse modes belong in this reference-level difference. It is an energy per reacting pair; a molar activation energy uses $R$ instead of $k$ in the exponential.

Write $M=m_A+m_B$ and $\mu =m_A{m}_B/M$. An atom has no rotational or vibrational internal factor under the stated electronic assumption. The transition-state moment of inertia is $I=\mu R^2$, so the high-temperature rigid rotor result isThe translational factors therefore giveSubstituting into the transition state theory expression and simplifying,For distinct atoms, $\sigma =1$. This is a number-density rate coefficient, with units of volume per time. A rate law in molar concentrations uses $N_A{k}_{2nd}$. The derivation assumes a classical high-temperature rotor at the dividing surface and a transmission coefficient of one.

If the relevant ground manifolds have electronic degeneracies $g_A$, $g_B$ and $g_{\mathrm{TS}}$, and their components are treated as energetically degenerate, multiply the result byOnly reactive transition-state electronic channels should be counted. Resolved electronic splittings require electronic partition functions rather than simple integer degeneracies.

For identical atoms, use $\mu =m_A/2$ and the homonuclear rotational symmetry number $\sigma =2$. With a rate defined as reaction events per volume and time, this halves the geometric prefactor relative to distinguishable labelled pairs. Do not independently count both ordered pairs and then add another unrelated factor of two. If the rate is instead defined by disappearance of A, each event consumes two atoms, so $-d{n}_A/dt=2r$; that rate coefficient is twice the event-rate coefficient.

Identify the transition-state separation with contact, $R=r_A+r_B$, take distinct atoms so $\sigma =1$, and let $E_A=N_A\Delta {\epsilon }_0^{‡}$. Then $\pi R^2=\rho$ and $\Delta {\epsilon }_0^{‡}/kT=E_A/RT$, yieldingThe velocity factor is the mean relative speed from the Maxwell-Boltzmann distribution. Thus this simple transition state theory model reproduces the stated collision theory expression. Molecular orientation constraints, quantum transmission and reaction-surface recrossing can invalidate the equality outside these simplifying assumptions.