= Catalytic barriers and reaction free energy
A directional <Gibbs energy of activation> measures the climb from a starting state to a <transition state>. The net standard reaction change in <Gibbs free energy> measures the difference between endpoints. Changing the climb can strongly change a <rate constant> while leaving that endpoint difference intact. For two illustrative single-barrier routes with common endpoint levels $G_{\mathrm A}^\circ$ and $G_{\mathrm B}^\circ$,
$$
\Delta G_f^{\ddagger\circ}=G_{\mathrm{TS}}^\circ-G_{\mathrm A}^\circ,
\qquad
\Delta G_r^{\ddagger\circ}=G_{\mathrm{TS}}^\circ-G_{\mathrm B}^\circ,
\qquad
\Delta G_f^{\ddagger\circ}-\Delta G_r^{\ddagger\circ}=\Delta_rG^\circ.
$$
Lowering the common <transition state> level lowers both directional <Gibbs energies of activation>; it does not lower just the forward climb. The usual statement that a <catalyst> lowers an <activation energy> describes a kinetic route, not a change in the net <Gibbs free energy>. An <activation energy> inferred from an <Arrhenius plot> is also distinct from a <Gibbs energy of activation> at one <temperature>.
\Image[/catalysis-barriers-and-net-free-energy.png]
{title=Illustrative <catalysis> schematic: different barriers, identical endpoint <Gibbs free energy> difference}
{id=image-catalysis-barriers-and-net-free-energy}
{description=Chosen levels in $\mathrm{kJ\,mol^{-1}}$: A plus regenerated C at $0$, B plus regenerated C at $-3$, and schematic maxima at $12$ and $7$. Forward barriers are $12$ and $7$; reverse barriers are $15$ and $10$. Neither curve is a measured molecular path. The <reaction coordinate> is not time. These values illustrate the barrier bookkeeping and do not parametrize the separate <reversible catalytic relaxation> example.}
With a common prefactor in <transition state theory>, subtracting the directional barriers yields
$$
\frac{k_f}{k_r}=\exp\!\left(-\frac{\Delta_rG^\circ}{RT}\right)=K,
$$
where $R$ is the <molar gas constant> and $T$ is the absolute <temperature>. This particular <rate constant> ratio requires the simple compatible directional <rate laws> and consistent <standard states>; a general multistep <reaction mechanism> does not supply two arbitrary measured <rate constants> whose ratio must equal $K$. At fixed <temperature> a <catalyst> preserves $K$ for the same net conversion. The actual reaction change in <Gibbs free energy>, $\Delta_rG=\Delta_rG^\circ+RT\ln Q$, is likewise unchanged at the same <reaction quotient> $Q$. Composition changes during the faster approach to <chemical equilibrium>, so comparing mixtures at the same elapsed time need not compare the same $Q$.<Chemistry 2e by Paul Flowers and others/13.3 Shifting Equilibria: Le Châtelier’s Principle>[§13.3, Effect of a Catalyst: both directional rates accelerate while equilibrium composition is unchanged]{ref}
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