= Catalyst
A <catalyst> makes a <chemical reaction> proceed faster by participating in an alternative <reaction mechanism> and being regenerated. This makes repeated conversion possible: the same <catalyst> can help many successive <reactants> become <reaction products>. An <enzyme> is a biological <catalyst>. The overall standard change in <Gibbs free energy> is unchanged; see the https://goldbook.iupac.org/terms/view/C00876[IUPAC Gold Book: catalyst].
Regeneration does not mean that the <catalyst> sits unchanged throughout every step. For example, a schematic <reaction mechanism> can temporarily bind A to a <catalyst> C in a <reaction intermediate> AC:
$$
\begin{aligned}
\mathrm A+\mathrm C&\rightleftharpoons\mathrm{AC},\\
\mathrm{AC}&\rightleftharpoons\mathrm B+\mathrm C,\\
\text{net:}\qquad\mathrm A&\rightleftharpoons\mathrm B.
\end{aligned}
$$
Adding the steps cancels both the <reaction intermediate> AC and the <catalyst> C from the net <chemical equation>. AC is produced and later consumed; C is used and later recovered. These letters describe possible roles, not a measured <reaction mechanism>. The alternative route is useful only if its <chemical kinetics> permits faster conversion. See https://openstax.org/books/chemistry-2e/pages/12-7-catalysis[Flowers et al., Chemistry 2e, §12.7: catalysts and alternative mechanisms].
For a worked barrier example, consider an idealized reversible A-to-B conversion at fixed <temperature> $T=300\,\mathrm K$. Stipulate standard molar <Gibbs free energy> levels $G_{\mathrm A}^\circ=0$ and $G_{\mathrm B}^\circ=-20\,\mathrm{kJ\,mol^{-1}}$, with effective <transition state> levels $80$ without a <catalyst> and $50\,\mathrm{kJ\,mol^{-1}}$ with a <catalyst>. Use the same <standard states> and the same zero for both routes; the catalytic route includes the regenerated C at both endpoints. Unlike the separate <potential-energy profile> example, this illustration uses <Gibbs free energy> at one specified <temperature>.
\Image[/catalyst-alternative-reaction-path.png]
{title=A <catalyst> lowers the stipulated <Gibbs energy of activation> in both directions while retaining the same endpoints}
{description=Original smooth schematic <Gibbs free energy> profiles at $300\,\mathrm K$, with levels in $\mathrm{kJ\,mol^{-1}}$. The horizontal <reaction coordinate> describes progress along each schematic route, not time. These are chosen effective barriers, not measured <potential energy surfaces>, actual molecular paths, or the two-step <reaction mechanism> above. A real catalytic route can have several <transition states> and <reaction intermediates>.}
The <Gibbs energy of activation> is measured from the starting endpoint in the chosen direction. Thus the four barriers are
$$
\begin{aligned}
\Delta G_{f,\mathrm{uncat}}^{\ddagger\circ}&=80-0=80,&
\Delta G_{f,\mathrm{cat}}^{\ddagger\circ}&=50-0=50,\\
\Delta G_{r,\mathrm{uncat}}^{\ddagger\circ}&=80-(-20)=100,&
\Delta G_{r,\mathrm{cat}}^{\ddagger\circ}&=50-(-20)=70,
\end{aligned}
\qquad\text{all in }\mathrm{kJ\,mol^{-1}}.
$$
Both directional barriers fall by $30\,\mathrm{kJ\,mol^{-1}}$. The endpoint difference stays $\Delta_rG^\circ=-20\,\mathrm{kJ\,mol^{-1}}$.
To turn those barriers into <rate constants>, use the Eyring form of <transition state theory> for this effective <first-order reaction> pair, with a common prefactor and unit probability of successful passage:
$$
k=\nu\exp\left(-\frac{\Delta G^{\ddagger\circ}}{RT}\right),
\qquad \nu=\frac{k_{\mathrm B}T}{h}.
$$
Here $k_{\mathrm B}$ is the <Boltzmann constant>, $h$ is the <Planck constant>, and $R$ is the <molar gas constant>. The common prefactor cancels from the comparisons. These <Gibbs energies of activation> are not the empirical <activation energy> $E_a$ of the <Arrhenius equation>. With the stipulated value $R=8.314462618\,\mathrm{J\,mol^{-1}\,K^{-1}}$, the speedup is
$$
\frac{k_{f,\mathrm{cat}}}{k_{f,\mathrm{uncat}}}
=\frac{k_{r,\mathrm{cat}}}{k_{r,\mathrm{uncat}}}
=\exp\left(\frac{30000}{8.314462618\times300}\right)
\approx1.67248\times10^5.
$$
The exponent is a <dimensionless quantity>: convert $30\,\mathrm{kJ\,mol^{-1}}$ to $30000\,\mathrm{J\,mol^{-1}}$ before dividing by $RT$. The <exponential function> explains why a modest barrier reduction can have a dramatic kinetic effect. This number follows from the chosen free-energy barriers and common prefactor; it is not a universal catalytic multiplier or a prediction across <temperatures>.
\Image[/catalyst-forward-reverse-rate-ratios.png]
{title=Both directional <rate constants> increase by about $1.67\times10^5$, while their forward-to-reverse ratio stays about $3036$}
{description=Original bar plots at $300\,\mathrm K$. Left: base-ten <logarithms> of the <dimensionless> normalized <rate constants> $k/\nu$ for the four stipulated barriers. A less negative value means faster conversion. Right: the <logarithm> of $k_f/k_r$ is identical for the two routes. Absolute units factor out through the common $\nu=k_{\mathrm B}T/h$; no temperature dependence outside this one model point is asserted.}
Assume ideal <thermodynamic activities> with a common concentration <standard state>, and directional <rate laws> $J_f=k_f[\mathrm A]$ and $J_r=k_r[\mathrm B]$. At <chemical equilibrium>, equal <reaction rates> give $K=[\mathrm B]/[\mathrm A]=k_f/k_r$ for this model. The barriers give
$$
K_{\mathrm{uncat}}=\frac{e^{-80000/(RT)}}{e^{-100000/(RT)}}
=e^{20000/(RT)}
=\frac{e^{-50000/(RT)}}{e^{-70000/(RT)}}=K_{\mathrm{cat}}
\approx3035.58.
$$
The <catalyst> speeds up both directions, so it changes how quickly <chemical equilibrium> is approached without changing this <equilibrium constant>. For a general <chemical reaction>, the thermodynamic relation is $K=\exp[-\Delta_rG^\circ/(RT)]$ with consistent <standard states>; an arbitrary multistep <reaction mechanism> need not permit $K=k_f/k_r$ for two measured <rate constants>. At the same <temperature>, <pressure>, and <thermodynamic activities>, a <catalyst> also leaves the actual overall reaction change in <Gibbs free energy> unchanged. See <Chemistry 2e by Paul Flowers and others/13.3 Shifting Equilibria: Le Châtelier’s Principle> (§13.3: a catalyst does not shift equilibrium). To drive an unfavourable overall conversion, a process such as <energy coupling> must change the combined transformation.
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