# How does it work? > This document is a plain-text mirror of the **How does it work?** section > of the *Unbound Mean Field model of Catalysis* web application > (). It exists so that crawlers and > language-model agents that do not execute JavaScript can still read the > theory in full, with the math preserved as raw LaTeX rather than rendered > CSS spans. The interactive version lives at > . A short tour of the Unbound Mean Field (UMF) framework: one universal kinetic matrix $\mathcal{K}$, three domain-specific driving forces $\vec{\mu}_\xi$, and a single closed-form determinant ratio that delivers reaction orders, Tafel slopes, and apparent activation energies. The framework is developed in two companion manuscripts — the determinant-ratio formalism in [^KO] and the corrected Tafel analysis in [^TC] — both cited inline below where each result first appears. ## 1. One object, three observables Apparent reaction orders, Tafel slopes, and apparent activation energies are routinely measured with completely different instruments — yet algebraically they are the *same* object. Pick a dimensionless perturbation variable $\xi$ and define the **generalised apparent reaction order**: $$ \gamma_\xi \;=\; \left[\frac{\mathrm{d}\ln r_{\text{net}}}{\mathrm{d}\xi}\right]_{\xi'=0\;\forall\;\xi'\neq\xi} \tag{1} $$ The three classical descriptors fall out by choosing $\xi$ appropriately: $$ \xi_m = \ln\!\left(\frac{c_m}{c_m^{\mathrm{eq}}}\right),\qquad \xi_\eta = \frac{F\eta}{RT},\qquad \xi_T = \frac{T_{\mathrm{eq}}}{T} - 1 \tag{2} $$ yielding, respectively, the concentration order $\gamma_m$, the apparent transfer coefficient $\gamma_\eta$ (with Tafel slope $b = b_0/\gamma_\eta$ and $b_0 = RT\ln(10)/F$), and a dimensionless apparent activation energy $\gamma_T = -E_A^{\mathrm{app}}/(RT_{\mathrm{eq}})$. The unification of all three observables into the single object $\gamma_\xi$ is the central thesis of [^KO]; the reinterpretation of the Tafel slope as the special case $b = b_0/\gamma_\eta$ — and the consequences this has for half-century-old Tafel analyses — is the subject of [^TC]. Locally, integrating the definition of $\gamma_\xi$ turns it into a single predictive power-law form in which every $\gamma$ appears as an explicit exponent: $$ r_{\text{net}} \;\propto\; \prod_m \!\Bigl(\tfrac{c_m}{c_m^{\mathrm{eq}}}\Bigr)^{\!\gamma_m} \;\cdot\; e^{\,\gamma_\eta\, F\eta/(RT)} \;\cdot\; e^{\,\gamma_T\,(T_{\mathrm{eq}}/T \,-\, 1)} \tag{3} $$ One product per perturbation knob, one exponent per $\gamma$. The rest of this page is the story of how those exponents come from a single matrix calculation on the reaction network. ## 2. The elementary rate law Before we talk about observables, we need the rate of each individual step. In the UMF framework every elementary step $k$ obeys an equilibrium-referenced rate law built from transition-state theory and a linear free-energy relationship. The net rate is the difference of a forward and a reverse contribution, each with the same algebraic skeleton: $$ r_k \;=\; r_{\mathrm{fwd},k} \;-\; r_{\mathrm{rvs},k} \tag{4} $$ $$ \begin{aligned} r_{\mathrm{fwd},k} &= \kappa_{0,k}\,\prod_i\!\Bigl(\tfrac{a_i}{a_i^{\mathrm{eq}}}\Bigr)^{\!\nu_{\mathrm{rct},i,k}}\, e^{-\beta_k\,\tfrac{\sum_i \Delta\nu_{i,k}\sum_j [g_{j,i}(\Gamma_j)-g_{j,i}(\Gamma_j^{\mathrm{eq}})]}{RT}}\, e^{\beta_k\, z_k F \eta/(RT)} \\[6pt] r_{\mathrm{rvs},k} &= \kappa_{0,k}\,\prod_i\!\Bigl(\tfrac{a_i}{a_i^{\mathrm{eq}}}\Bigr)^{\!\nu_{\mathrm{prd},i,k}}\, e^{(1-\beta_k)\,\tfrac{\sum_i \Delta\nu_{i,k}\sum_j [g_{j,i}(\Gamma_j)-g_{j,i}(\Gamma_j^{\mathrm{eq}})]}{RT}}\, e^{-(1-\beta_k)\, z_k F \eta/(RT)} \end{aligned} \tag{5} $$ The two halves are mirror images of one another: $\nu_{\mathrm{rct}} \leftrightarrow \nu_{\mathrm{prd}}$ in the activity product, the symmetry factor flips $\beta_k \to 1-\beta_k$ in both the interaction-pressure exponential and the electric-work term, and the latter changes sign with the direction of charge flow. At equilibrium $r_{\mathrm{fwd},k} = r_{\mathrm{rvs},k}$ for every step and the net rate vanishes by construction. The symbols: - $a_i$ — activities: external concentrations / partial pressures for gas-phase or dissolved species, coverages $\Gamma_i$ for surface intermediates. - $\nu_{\mathrm{rct},i,k},\,\nu_{\mathrm{prd},i,k}$ — stoichiometric coefficients (the equation editor parses them); $\Delta\nu_{i,k} = \nu_{\mathrm{prd},i,k} - \nu_{\mathrm{rct},i,k}$. - $g_{j,i}(\Gamma_j)$ — lateral-interaction free energies. Only their logarithmic derivatives $\Psi_{j,i}$ (§9) survive in the end. - $\beta_k \in [0,1]$ — symmetry factor (reactant-like at 0, product-like at 1). - $z_k$ — electrons transferred in step $k$; $\eta$ — overpotential (zero for purely thermal steps). - $\kappa_{0,k}$ — equilibrium prefactor, which carries the elementary activation energy through the transition-state relation $\kappa_{0,k}\propto e^{-E_{A,k}/(RT)}$. **Everything downstream is calculus on this one equation.** The universal balance equation, the kinetic matrix $\mathcal{K}$, and the determinant-ratio expression for $\gamma_\xi$ all come from taking $\mathrm{d}\ln r_k/\mathrm{d}\xi$ of the expression above. The driving-force vector $\vec{\mu}_\xi$ simply records *which* factor is being differentiated: reactant activities for concentration, the electric-work exponential for potential, and the temperature-dependent prefactor $\kappa_{0,k}$ for temperature. ## 3. The universal balance equation Logarithmically differentiating the elementary rate law above for every step $k$ and collecting the results into a single matrix equation gives the central object of the UMF framework: $$ \gamma_\xi\,(\vec{1} - \vec{\chi}) \;=\; \vec{\mu}_\xi \;+\; \mathcal{K}\,\vec{x}_\xi \tag{6} $$ Here $\vec{\chi} \in [0,1]^{n_{\text{steps}}}$ collects the step reversibilities ($\chi_k = r_{\text{rvs},k}/r_{\text{fwd},k}$), $\vec{\mu}_\xi$ is the **driving-force vector** (the only quantity that depends on which knob you turn), and $\vec{x}_\xi$ is the response of the surface coverages: $x_{\xi,j} = \mathrm{d}\ln \Gamma_j/\mathrm{d}\xi$. **The remarkable point:** the matrix $\mathcal{K}$ is identical across all three domains. It encodes the topology of the reaction network and the current kinetic state. Only $\vec{\mu}_\xi$ changes when you switch from a concentration sweep to a Tafel sweep to a temperature sweep. The derivation of this universal balance equation from the elementary rate law of §2 is given in [^KO]. ## 4. The kinetic matrix $\mathcal{K}$ — topology in one block $\mathcal{K}$ is built from the stoichiometry matrices $\mathcal{V}_{\text{rct}}, \mathcal{V}_{\text{prd}}$ (which the editor parses for you), the interaction-pressure matrix $\Psi$, and the per-step vectors $\vec{\chi}$ and $\vec{\beta}$ (construction derived in [^KO]): $$ \mathcal{K} \;=\; \underbrace{\mathcal{V}_{\text{rct}}^{\mathsf{T}} - \operatorname{diag}(\vec{\chi})\,\mathcal{V}_{\text{prd}}^{\mathsf{T}}}_{\text{stoichiometric part}} \;+\; \underbrace{\bigl(\operatorname{diag}(\vec{\beta})\operatorname{diag}(\vec{\chi}) - \operatorname{diag}(\vec{\beta}+\vec{\chi})\bigr)\,\Delta\mathcal{V}^{\mathsf{T}}\,\Psi^{\mathsf{T}}}_{\text{interaction part}} \tag{7} $$ - The **stoichiometric part** is what classical Langmuir–Hinshelwood kinetics uses: it accounts for how the rate of each step responds to coverages through the bare reaction equation. - The **interaction part** is the UMF-specific addition. It propagates lateral interactions globally: a high coverage of *j* raises the bound-state free energy of every species, and the linear free-energy relationship (parameterised by $\vec{\beta}$) feeds that change back into the activation barriers of every step that creates or consumes those species. When $\Psi = 0$ the second block vanishes and $\mathcal{K}$ reduces to its purely stoichiometric skeleton — the ideal 2-D gas / non-interacting limit. The opposite limit, $\Psi \to \infty$, sharpens the asymptotic self-limitation into an *effective* hard exclusion. This still differs from Langmuir / Frumkin kinetics in a substantive way: there, the site balance enters *only locally*, as the activity $\theta_\ast = 1 - \textstyle\sum_j \Gamma_j$ of the free-site species, and only in the rate of steps that have $\ast$ as a stoichiometric reactant — i.e. the adsorption / formation steps. Internal surface steps that never touch $\ast$ — diffusion, recombination, intramolecular shuffles — feel nothing of it. UMF, by contrast, routes coverage shifts through the LFER into the activation barriers of *every* step that creates or consumes the affected species, with the full pair-dependent matrix $\Psi$ in place of a single scalar cap (see §9). ## 5. Three driving forces, one network All the domain-specific physics lives in the driving-force vector $\vec{\mu}_\xi$. Setting parallel-path occurrence shifts to zero ($\vec{\varsigma}_\xi = \vec{0}$): | Domain | Variable $\xi$ | Observable | Driving force $\vec{\mu}_\xi$ | | ------------- | ---------------------------- | --------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------- | | Concentration | $\ln(c_m/c_m^{\mathrm{eq}})$ | $\gamma_m$ | $\vec{\nu}_{\text{rct},m} - \operatorname{diag}(\vec{\chi})\,\vec{\nu}_{\text{prd},m}$ | | Potential | $F\eta/(RT)$ | $b = b_0/\gamma_\eta$ | $\bigl(\operatorname{diag}(\vec{\beta}+\vec{\chi}) - \operatorname{diag}(\vec{\beta})\operatorname{diag}(\vec{\chi})\bigr)\,\vec{z}$ | | Temperature | $T_{\mathrm{eq}}/T - 1$ | $E_A^{\mathrm{app}} = -RT_{\mathrm{eq}}\,\gamma_T$ | $-\operatorname{diag}(\vec{1}-\vec{\chi})\,\vec{E}_A/(RT_{\mathrm{eq}})$ | Notice the temperature row carries an explicit $(\vec{1}-\vec{\chi})$ prefactor: only steps that are not quasi-equilibrated contribute to the apparent activation energy — a textbook result that here drops out of the matrix algebra automatically. ## 6. Closing the system: a determinant ratio One species is singled out as the **reference species** and indexed by the symbol $\ast$, borrowed from Langmuir/Frumkin notation. The UMF boundary condition fixes the corresponding component of the response vector to zero, $\vec{e}_{\ast}^{\mathsf{T}}\,\vec{x}_\xi = 0$, which makes the system square and pins down a unique solution. Combining that constraint with the balance equation and applying Cramer's rule produces a closed-form, domain-agnostic expression — the central result of [^KO]: $$ \gamma_\xi \;=\; \frac{\det\!\bigl[\,\vec{\mu}_\xi\,\vec{e}_{\ast}^{\mathsf{T}} + \mathcal{K}\,(\mathcal{I} - \vec{e}_{\ast}\vec{e}_{\ast}^{\mathsf{T}})\bigr]}{\det\!\bigl[\,(\vec{1}-\vec{\chi})\,\vec{e}_{\ast}^{\mathsf{T}} + \mathcal{K}\,(\mathcal{I} - \vec{e}_{\ast}\vec{e}_{\ast}^{\mathsf{T}})\bigr]} \tag{8} $$ The denominator is the same in every domain — it is the irreversibility vector $(\vec{1}-\vec{\chi})$ projected through the kinetic topology. The numerator merely swaps in the domain-specific driving force $\vec{\mu}_\xi$. So **$\gamma_\xi$ is the ratio of driving-force capacity to kinetic-bottleneck capacity** — a property of the network as a whole rather than of any single elementary step. > **$\ast$ is a gauge, not a free-site count.** > > Despite the suggestive symbol, the reference species $\ast$ is a pure > book-keeping placeholder; UMF carries no site-balance constraint at > all. The right multiplication > $\mathcal{K}\,(\mathcal{I} - \vec{e}_\ast\vec{e}_\ast^{\mathsf{T}})$ > annihilates the column of $\mathcal{K}$ corresponding to $\ast$, and > the rank-one term $\vec{\mu}_\xi\,\vec{e}_\ast^{\mathsf{T}}$ (or > $(\vec{1}-\vec{\chi})\,\vec{e}_\ast^{\mathsf{T}}$ in the denominator) > overwrites it with the boundary-condition data. Consequently the > stoichiometric coefficients of $\ast$ enter *only* that masked column > and cannot affect the predicted $\gamma_\xi$: any chemically sensible > choice (typically the one that makes each elementary step look > chemically balanced) gives the identical determinant ratio. No *a priori* classification of steps as irreversible or quasi-equilibrated is required: the continuous $\chi_k \in [0,1]$ sliders parametrise everything in between, including the smooth transitions where classical kinetics cannot say anything quantitative. ### The most general result: a pseudo-inverse covector Cramer's rule, and therefore the determinant-ratio form above, presupposes that $\mathcal{K}$ is **square** — exactly one independent elementary step per surface species (counting the reference). This is the **single-path setting** in which the occurrence numbers are constant ($\sigma_k = 1$, $\vec{\varsigma}_\xi = \vec{0}$) and the network admits a unique steady-state pathway. As soon as the mechanism contains **parallel channels**, distinct elementary steps share intermediates, the number of steps exceeds the number of species ($N > n_{\text{sp}}$), and $\mathcal{K}$ becomes rectangular: the master equation §3 is then *overdetermined* in $\vec{x}_\xi$ until supplemented by an explicit distribution of flux among the branches (the occurrence-number sensitivities $\vec{\varsigma}_\xi$ encoded in $\vec{\mu}_\xi$). In that regime the determinant simplification fails outright. What survives — and is in fact the **most fundamental algebraic statement of the framework** — is obtained by left-multiplying the balance equation by the **Moore–Penrose pseudo-inverse** $\mathcal{K}^{+}$ and applying the boundary condition $\vec{e}_{\ast}^{\mathsf{T}}\,\vec{x}_\xi = 0$ directly, without going through Cramer's rule [^KO]: $$ \gamma_\xi \;=\; \frac{\vec{e}_{\ast}^{\mathsf{T}}\,\mathcal{K}^{+}\,\vec{\mu}_\xi}{\vec{e}_{\ast}^{\mathsf{T}}\,\mathcal{K}^{+}\,(\vec{1}-\vec{\chi})} \tag{9} $$ This **inverse-image form** is the universal, fully general representation of the generalised reaction order. In the square single-path case $\mathcal{K}^{+} = \mathcal{K}^{-1}$ and it is algebraically identical to the determinant ratio above. In the rank-deficient / rectangular parallel-path case it continues to hold whenever $\vec{\mu}_\xi$ and $(\vec{1}-\vec{\chi})$ lie in the column space of $\mathcal{K}$ — which is exactly the consistency condition for the augmented master equation — while the determinant ratio no longer collapses to a single closed form. Two structural facts that the determinant version hides become explicit here: 1. **The reference species is purely a row selector.** Gauge fixing reduces to picking one row, $\vec{e}_{\ast}^{\mathsf{T}}$, of the network's pseudo-inverse $\mathcal{K}^{+}$. The rest of the algebra does not know which species was singled out. 2. **A single covector controls every observable.** Define the dual covector $\vec{v}^{\mathsf{T}} \equiv \vec{e}_{\ast}^{\mathsf{T}}\,\mathcal{K}^{+}$. Then for *any* domain $\xi$, $\gamma_\xi = \vec{v}^{\mathsf{T}}\vec{\mu}_\xi \,/\, \vec{v}^{\mathsf{T}}(\vec{1}-\vec{\chi})$ is a single **fractional-linear function** of the driving force at fixed network state $(\vec{\chi},\vec{\beta},\Psi)$. Concentration, potential, and temperature observables are three projections of the same network covector $\vec{v}^{\mathsf{T}}$ onto three different driving forces. The single-path determinant ratio is what the calculator currently evaluates and is the most efficient closed form when it applies; the pseudo-inverse expression is the principle from which it descends and the only form that survives in genuinely multi-path networks. ## 7. Reading the outputs Locally the rate is a power law in the perturbation variable, $r_{\text{net}} \propto e^{\gamma_\xi \,\xi}$, so each $\gamma_\xi$ is the instantaneous exponent of an Arrhenius / Tafel / power-law fit: - **Concentration:** $r_{\text{net}} \propto \prod_m (c_m/c_m^{\mathrm{eq}})^{\gamma_m}$ — $\gamma_m = +1$ means doubling $c_m$ doubles the rate; $\gamma_m = 0$ means the rate is insensitive. - **Potential:** $r_{\text{net}} \propto e^{\gamma_\eta\,F\eta/(RT)}$ — the Tafel slope is $b = RT\ln(10)/(F\,\gamma_\eta)$, recovering the familiar 60 mV / dec at $\gamma_\eta=1, T=298\,\text{K}$. The systematic deviation between this $\gamma_\eta$-corrected slope and the classical Butler–Volmer $\alpha$-only reading, and the re-interpretation of the published Tafel literature this implies, is the subject of [^TC]. - **Temperature:** $r_{\text{net}} \propto e^{-E_A^{\mathrm{app}}/(RT)}$ with $E_A^{\mathrm{app}} = -RT_{\mathrm{eq}}\,\gamma_T$. In the asymptotic regimes — every step either fully irreversible or fully quasi-equilibrated, $\Psi \to 0$, no parallel-path shifts — the determinant ratio becomes a constant and $\mathrm{d}\gamma_\xi/\mathrm{d}\xi = 0$: a clean power law, a straight Tafel line, a pure Arrhenius exponential. The UMF framework reproduces these classical plateaus and, more importantly, predicts the smooth transitions between them where $\gamma_\xi$ changes continuously with $\xi$. ## 8. Coverage response $\vec{x}_\xi$ — the mechanistic story Solving the same balance equation for $\vec{x}_\xi$ (rather than $\gamma_\xi$) gives the coverage response: $x_{\xi,j} = \mathrm{d}\ln \Gamma_j/\mathrm{d}\xi$ tells you how each surface intermediate's coverage shifts when you change the external knob. This decomposition is what makes the apparent reaction order interpretable: $\gamma$ emerges from the interplay of **(1)** direct stoichiometric participation, **(2)** coverage shifts of intermediates, and **(3)** lateral interactions through $\Psi$. The "Coverage response" panel in the results section visualises exactly these $x_{\xi,j}$ values. ## 9. The interaction-pressure matrix $\Psi$ All dependence on adsorbate–adsorbate interaction free energies $g_{j,i}(\Gamma_j)$ enters the kinetic equations through their dimensionless logarithmic derivative: $$ \Psi_{j,i} \;=\; \frac{\Gamma_j}{RT}\,\frac{\mathrm{d} g_{j,i}}{\mathrm{d}\Gamma_j} \tag{10} $$ $\Psi_{j,i}$ measures how strongly the coverage of intermediate *j* penalises the binding of species *i*. Crucially, this is a global perturbation: rising $\Gamma_j$ shifts the bound-state free energy of *i*, and the linear free-energy relationship feeds that shift into the activation barrier of *every* elementary step that creates or consumes *i*, not only the step that produces or consumes *j*. ### A geometric reading: effective molar footprints Dimensional inspection of $\Psi_{j,i}$ isolates a factor with units of molar area — an *effective molar footprint* or energetic exclusion zone $A^{\text{eff}}_{j,i}$ by which intermediate *j* penalises the binding of species *i*: $$ A^{\text{eff}}_{j,i} \;=\; \frac{1}{RT}\,\frac{\mathrm{d} g_{j,i}(\Gamma_j)}{\mathrm{d}\Gamma_j}\,,\qquad \Psi_{j,i} \;=\; \Gamma_j\,A^{\text{eff}}_{j,i} \tag{11} $$ Because $g_{j,i}$ encodes the specific electronic and steric relation between *j* and *i*, this footprint is intrinsically **pair-dependent**: one species can project a large self-exclusion area that forces sparse packing among identical neighbours, while simultaneously projecting a much smaller footprint toward a chemically dissimilar partner — enabling dense co-adsorption of the two. ![Three-panel illustration of the pair-dependent interaction-pressure footprints: (a) sparse packing of identical adsorbates forced by a large self-exclusion area; (b) dense co-adsorption of a second species fitting into the gaps via a smaller cross-footprint; (c) A_eff for self-interaction is larger than A_eff toward a chemically distinct partner.](https://reactionorders.energy.aau.dk/figures/interaction_pressure_geom.png) *(a) Self-interaction: a species projects a large self-exclusion footprint $A^{\text{eff}}_{i,i}$, forcing sparse packing of identical adsorbates. (b) Cross-interaction: the same species projects a substantially smaller footprint toward a chemically distinct partner, $A^{\text{eff}}_{j,i} < A^{\text{eff}}_{i,i}$, enabling dense co-adsorption. (c) In a mixed adsorbate layer the attainable coverage is set by the overlapping, pair-dependent footprints of the instantaneous composition rather than by a single universal site-balance bound.)* The instantaneous saturation capacity of the catalyst is therefore *not* a static scalar property of the underlying lattice but a *dynamic, compositional* limit set by the overlapping energetic footprints of the current adsorbate mixture. A single, coverage-independent constraint of the form $\textstyle\sum_j \Gamma_j \le \Gamma_{\max}$ collapses the pair-dependent structure into one scalar and is generically inconsistent with the energetics of multi-species networks. The UMF closure keeps the full matrix $\Psi$ alive and replaces the hard cap with an asymptotic, pair-aware self-limitation. The geometric reading of $\Psi$ as a matrix of effective molar footprints, and the resulting critique of scalar site-balance closures, is developed in [^KO]. ### Limits - **$\Psi = 0$** — ideal 2-D lattice gas. Adsorbates are blind to one another; coverages and rates decouple (classical non-interacting mean-field limit). - **finite $\Psi$** — soft repulsion. Coverage-dependent activation barriers emerge naturally; this is the physically realistic regime for most surfaces. - **$\Psi \to \infty$** — the asymptotic self-limitation becomes very steep, so coverages effectively saturate. This is still **not** the same as classical Langmuir–Hinshelwood: there, the site count enters only locally, as the activity $\theta_\ast = 1 - \textstyle\sum_j \Gamma_j$ of the free-site species, and only in steps that have $\ast$ as a stoichiometric reactant — i.e. the adsorption / formation steps. Internal surface steps that never touch $\ast$ see no coverage-dependent barrier at all. UMF instead routes the resulting free-energy shifts through the LFER into *every* step that touches the affected species, with the full pair-dependent footprint matrix $\Psi$ rather than a single scalar $\theta_\ast$ activity. --- ## 10. Building on previous work The notion that the three apparent descriptors are connected has a long history. We summarise it here following the introduction of [^KO], which identifies three threads of prior work that the present framework explicitly builds on. Full bibliographic entries are listed in the references block below. **Domain-specific frameworks for the three observables.** Each descriptor is routinely measured and interpreted within its own specialised theoretical framework: power-law rate expressions and Langmuir–Hinshelwood kinetics for reaction orders (Bard & Faulkner, 2001); the Butler–Volmer framework for Tafel slopes (Bockris & Reddy, 1973; Guidelli *et al.*, IUPAC 2014); and Arrhenius / Eyring analysis for activation energies (Eyring, 1935; modern transition-state theory review: Truhlar, Garrett & Klippenstein, 1996). The apparent activation energy reported by this tool follows the IUPAC convention of Laidler (1996), which identifies $E_A^{\mathrm{app}}$ with the log-derivative of the rate at a reference temperature. **Algebraic / graph-determinant lineage.** On the mathematical side, Christiansen (1953), Temkin (1979), and King & Altman (1956) showed that steady-state reaction networks admit closed-form solutions through graph- and determinant-based algebra; this lineage continues today in master-equation microkinetic modelling, where matrix inversions replace numerical integration to extract kinetic descriptors cleanly (Motagamwala & Dumesic, 2021). The determinant ratio for $\gamma_\xi$ in §6 is the logarithmic-derivative analogue of those determinants: a closed-form descriptor that lives one differentiation above the steady-state rate law itself. **Sensitivity analysis.** On the analytical side, the language of normalised logarithmic sensitivities $\mathrm{d}\ln r/\mathrm{d}\xi$ — in which every $\gamma_\xi$ in this work is written — was systematised in chemical-kinetic sensitivity analysis (Rabitz, Kramer & Dacol, 1983; Turányi, 1990). **Degree of Rate Control: the closest antecedent.** On the physical side, the Degree of Rate Control (DRC) framework, introduced by Campbell (2017) and developed by Stegelmann, Andreasen & Campbell (2009) to identify rate-controlling transition states and intermediates, was extended by Baz & Holewinski (2021) to show that apparent reaction orders, Tafel slopes, and apparent activation energies can all be written as sensitivity-weighted sums over the same set of microscopic DRC coefficients, establishing their physical kinship. The present contribution is structural rather than re-unifying: the three sensitivities are not merely sums over a common set of coefficients but the same fractional linear functional of a single domain-independent operator $\mathcal{K}$ driven by a single domain-specific vector $\vec{\mu}_\xi$, with the generalised DRC sums recovered as the perturbative expansion of Eq. (10) around the asymptotic regimes where the reversibilities saturate. ### What this unification buys you Collapsing all three observables onto one operator $\mathcal{K}$ and one driving-force vector $\vec{\mu}_\xi$ has four concrete consequences that distinguish UMF from a sensitivity-sum reformulation of any one classical framework: - **Closed-form predictions in all three domains** from the same mechanism input — no numerical sensitivity sweep over microscopic parameters is required. - **No a priori labelling** of which steps are irreversible or quasi-equilibrated. The reversibilities $\chi_k$ sit inside $\mathcal{K}$ as continuous parameters and the determinant ratio handles every regime smoothly, with the asymptotic plateaus emerging in the appropriate limits (§7). - **Globally propagated lateral interactions through $\Psi$.** Because UMF couples the coverage of every intermediate $j$ through the LFER to the transition state of *every* elementary step that forms or breaks species $i$ (§9), lateral effects propagate globally rather than locally — in contrast with Langmuir/Frumkin-type closures, where coverage enters only through the free-site activity $\theta_\ast = 1 - \textstyle\sum_j \Gamma_j$ in those steps that have $\ast$ as a stoichiometric reactant, leaving the activation energies of internal surface steps untouched. - **A single multi-domain inverse problem.** The same parameter set $\Theta = \{\vec{\chi},\vec{\beta},\Psi,\vec{E}_A\}$ governs all three domains, so kinetic data measured in concentration, potential, and temperature can be fitted jointly to one mechanism rather than to three disconnected ones. The "Reaction orders", "Tafel slope", and "Apparent activation energy" panels of this tool are three views of exactly the same evaluation of $\mathcal{K}$ on three different driving-force vectors $\vec{\mu}_\xi$. --- ## References The two manuscripts derive every result above. Both are currently unpublished; DOIs will be appended here as soon as they are assigned. For the canonical citation block see also the *How to cite* section of the in-app [Legal Notice](https://reactionorders.energy.aau.dk/legal). ### Historical context (cited in §10) All entries below are taken from the bibliography of [^KO]; nothing is cited that is not also cited there. *Domain-specific frameworks for the three observables.* Bard & Faulkner, *Electrochemical Methods*, 2nd ed., Wiley (2001); Bockris & Reddy, *Modern Electrochemistry* (1973); Guidelli *et al.*, IUPAC *Pure Appl. Chem.* **86**, 245–258 (2014); Eyring, *J. Chem. Phys.* **3**, 107–115 (1935); Truhlar, Garrett & Klippenstein, *J. Phys. Chem.* **100**, 12771–12800 (1996); Laidler, IUPAC *Pure Appl. Chem.* **68**, 149–192 (1996). *Algebraic / graph-determinant lineage.* Christiansen, *Adv. Catal.* **5**, 311–353 (1953); King & Altman, *J. Phys. Chem.* **60**, 1375–1378 (1956); Temkin, *Adv. Catal.* **28**, 173–291 (1979); Motagamwala & Dumesic, *Chem. Rev.* **121**, 1049–1076 (2021). *Sensitivity analysis.* Rabitz, Kramer & Dacol, *Annu. Rev. Phys. Chem.* **34**, 419–461 (1983); Turányi, *J. Math. Chem.* **5**, 203–248 (1990). *Degree of Rate Control (closest antecedent).* Stegelmann, Andreasen & Campbell, *J. Am. Chem. Soc.* **131**, 8077–8082 (2009); Campbell, *ACS Catal.* **7**, 2770–2779 (2017); Baz & Holewinski, *J. Catal.* **397**, 233–244 (2021). [^KO]: Brandes, B. A. & Vegge, T., *On the Universal Algebraic Structure of Kinetic Observables in Catalysis*, 2026 (in preparation; DOI to be added). [^TC]: Brandes, B. A., Hansen, H. A., Vegge, T. & Chatzichristodoulou, C., *Correcting the Foundations of Tafel Analysis in Electrocatalysis*, 2026 (in submission; DOI to be added).