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Consider an ideal gas in chemical equilibrium with homogeneous, identical adsorption sites at fixed temperature. Each intact molecule occupies one site; only one monolayer is allowed. Occupation of distinct sites is independent: neighboring bound molecules do not change binding or release rates. The simplest Langmuir adsorption model has one adsorbing chemical species, no splitting of molecules, and no competitors. Irving Langmuir's 1918 paper treats simple adsorption alongside more complicated cases; its scope exceeds this particular model.^[ref]

The derivation is a balance between arrivals finding vacancies and departures from occupied adsorption sites. Let $k_{\mathrm{a}}P$ be the arrival-and-binding rate per vacant site and $k_{\mathrm{d}}$ the departure rate per occupied site. Thus $k_{\mathrm{a}}$ has units of inverse pressure per unit time, whereas $k_{\mathrm{d}}$ has units of inverse time. The mean surface coverage obeysAt chemical equilibrium, setting the two terms equal givesThe product $bP$ is dimensionless. Here $P$ is the adsorbing gas's partial pressure, expressed in modern bar, with $1\mathrm{bar}=10^5\mathrm{Pa}$, and $b$ is expressed in ${\mathrm{bar}}^{-1}$. The pascal is the SI unit of pressure. This dimensional $b$ is not itself a dimensionless thermodynamic equilibrium constant. A larger $b$ gives greater surface coverage at the same $P$ and temperature, but does not specify how quickly chemical equilibrium is reached: multiplying both rate coefficients by the same factor leaves $b$ unchanged.

Choose $b=0.5$, $2$, and $8{\mathrm{bar}}^{-1}$ as illustrative parameters at one fixed temperature, without assigning a particular material. At $P=0.5\mathrm{bar}$, substitution gives, respectively,Solving $\theta =1/2$ gives $P_{1/2}=1/b$: the three half-occupation pressures are $2$, $0.5$, and $0.125\mathrm{bar}$. For $b=2{\mathrm{bar}}^{-1}$, reaching $\theta =0.9$ instead requires $bP=9$, or $P=4.5\mathrm{bar}$. If the complete-monolayer capacity is stipulated to be $q_{\mathrm{m}}=10\mathrm{mmol}{\mathrm{g}}^{-1}$, then $q=q_{\mathrm{m}}\theta$ predicts $q=5\mathrm{mmol}{\mathrm{g}}^{-1}$ at $P=0.5\mathrm{bar}$. This last number is an original hypothetical example, not a measured uptake.

The shape follows directly from the finite number of adsorption sites. For $bP\ll 1$, $\theta \approx bP$: nearly every arriving molecule finds a vacancy. For $bP\gg 1$, $1-\theta =1/(1+bP)\approx 1/(bP)$: vacancies become scarce. More formally,Thus surface coverage rises with diminishing slope and approaches one only as $P\to \infty$. This saturation is a site-capacity limit, not a claim that the gas has reached its condensation pressure. A second layer, different site binding strengths, interactions between bound molecules, or a changing surface invalidate parts of the setup. A good fit alone does not establish identical adsorption sites: a restricted pressure range can hide such differences. The Langmuir adsorption model is useful because it makes the capacity and the pressure scale of filling separate, testable quantities.

In 1918, Irving Langmuir measured adsorption using freshly split mica strips, handled at their edges, separated by iron spacers and held with brass fasteners. A glass bulb contained the bundle; a similar blank contained matching iron and brass but no mica. Evacuated glass connections, Mercury seals, liquid air traps, and McLeod gauges provided controlled gas handling. Calibrated compartment volumes and pressure readings before and after expansion gave free-gas amounts; their deficit indicated uptake. Lowering pressure and measuring released gas tested desorption.^[ref]

Hydrogen blank runs calibrated cooling-related pressure differences. Irving Langmuir pre-cooled the bulbs to reduce thermal delay; he suggested slow diffusion between strips could control the remaining approach to equilibrium. Critically, blank uptake was not subtracted: he estimated the mica-bulb results were 10–30% too high. Liquid air temperature drift was unrecorded. These limits constrain inference of intrinsic binding from the uptake curve. The paper's pressure unit called “bar” was $1\mathrm{dyn}\mathrm{c}{\mathrm{m}}^{-2}=0.1\mathrm{Pa}$, unlike the modern bar used above.^[ref]

Adsorption in which material can form more than one layer above a surface. The number of adsorbed particles per underlying site can exceed one and differs from the fraction of sites having any adsorbate.

Use the ideal Langmuir adsorption model: identical independent sites, at most one atom per site, no lateral interactions, and no additional thermally active adsorbed degrees of freedom beyond the two spin states. Treat the atoms as indistinguishable and localized after adsorption; only site occupancy and spin choices distinguish configurations.

There are $\left(\genfrac{}{}{0ex}{}{M}{N}\right)$ ways to select $N$ occupied sites. For any such selection, independent spin sums contribute $z^N$. ThusWith $q_{\mathrm{ads}}=Mz$ as defined in part (b), this isEquivalently, starting from labelled particles gives $M!/(M-N)!$ distinct assignments to different sites; division by $N!$ removes particle relabellings. The $M^{-N}$ removes the site multiplicity already included in each full one-particle function.

If $q_{\mathrm{ads}}$ denotes the one-site internal factor instead, $f(M,N)=\left(\genfrac{}{}{0ex}{}{M}{N}\right)$ and $Q_N=\left(\genfrac{}{}{0ex}{}{M}{N}\right)q_{\mathrm{ads}}^N$. These are identical physical results, not different counting models. In particular, the complete-function convention gives $f(M,1)=1$ and hence $Q_1=q_{\mathrm{ads}}$.

At adsorption equilibrium, ${\mu }_{\mathrm{ads}}={\mu }_g$. The expressions from part (d) giveDefineSolving the occupation ratio yieldsThe coefficient $b$ has units of inverse pressure, so $bp$ is dimensionless. The two spin states contribute a factor two on both sides of the adsorption equilibrium and cancel; retaining the surface factor but dropping the bulk degeneracy would incorrectly double $b$.

At chemical equilibrium, adsorption and desorption balance:Canceling $M$ and rearranging givesThus $K_{\mathrm{ads}}=b$ when the kinetic and thermodynamic descriptions refer to the same ideal surface and energy convention. In particular,The ratio fixes the adsorption isotherm, while the individual rate constants also determine relaxation toward equilibrium. Here $k_a$ has units of inverse pressure per time and $k_d$ of inverse time, so $K_{\mathrm{ads}}$ is dimensional. This equivalent-site arrival/departure argument is the simple case of the Langmuir adsorption model.^[ref]

Amine nitrogen binds the carbon end of CO${}_2$; Mg${}^{2+}$ binds through oxygen. The polarized C=O bonds make carbon electrophilic and oxygen electron rich. An amine nitrogen lone pair therefore donates toward carbon, potentially giving a carbamate-like adduct after proton transfer. The hard Lewis acid Mg${}^{2+}$ accepts an oxygen lone pair, either from coordinated CO${}_2$ or from an oxygen of a captured carbamate species. Simply reversing these assignments because nitrogen and oxygen both have lone pairs would put two electron-pair donors together without accounting for the electrophilic site.

The local binding pattern does not by itself determine the full proton-transfer state or whether cooperative chains form; those depend on the framework and amine arrangement. A documented diamine-appended framework shows cooperative carbon-dioxide insertion and ammonium carbamate chain formation, illustrating one possible implementation.^[ref]

To regenerate the material, lower the CO${}_2$ chemical potential using reduced pressure or a CO${}_2$-free purge, and, where binding is exothermic, heat the framework to favour desorption. The temperature must remain compatible with framework stability. This reverses adsorption or reversible chemisorption; it does not require permanently destroying the metal–organic framework.

Event a probes electron transfer between the semiquinone and dianion. Benzoquinone's smaller framework localizes charge and undergoes a greater change in bonding and solvation on this reduction; a larger effective reorganization barrier can make its heterogeneous transfer insufficiently fast for Nernst equilibrium on the $50\mathrm{mV}{\mathrm{s}}^{-1}$ time scale. Anthraquinone's extended framework distributes the added charge and can support the more reversible response seen here.

This is a kinetic interpretation of quasi-reversibility, not a claim that benzoquinone cannot be reduced. Follow-up chemistry or adsorption can also distort peak separation and return-current ratios; scan-rate-dependent measurements would distinguish them from slow electron transfer. A single trace does not uniquely identify the cause.

Within the stated ECEC model and with CO${}_2$ genuinely in large excess, its concentration is effectively constant. Both capture steps are then pseudo-first-order in the quinone species, and at the same scan rate the faradaic current is linear in bulk AQ concentration.

The sketch therefore retains the peak potentials, sweep directions and relative features of the $200\mathrm{mV}{\mathrm{s}}^{-1}$ trace, including the positive-potential return feature III, but reduces the faradaic current scale by a factor of ten. Charging/background current need not scale with AQ concentration and should be shown separately. A changed normalized shape would indicate effects outside this simple scaling limit, such as CO${}_2$ depletion, adsorption or reactions between quinone species.

The Langmuir adsorption model assumes equivalent independent sites, a maximum of one adsorbate per site, no lateral adsorbate interactions, reversible adsorption-desorption equilibrium and non-dissociative adsorption here. Surface coverage obeys $\theta =bC/(1+bC)$ for the solution analogue.

BET adsorption allows multiple layers: occupation of one layer creates sites for further adsorption above it, and higher-layer binding is treated differently from first-layer binding, commonly using a condensation-like energy. Langmuir saturation is monolayer saturation; the BET model is not simply Langmuir with a different numerical equilibrium constant.

The inferred saturation density is physically implausible for a single molecular layer in the printed units:That is far smaller than an atomic cross-section, let alone a butyric-acid molecule. The near-linear reciprocal plot therefore does not validate the physical monolayer interpretation. Possible issues include reported units or normalization, inaccessible/internal area not accounted for correctly, multilayer adsorption or other uptake processes. The numbers must not silently be relabelled as micromoles to obtain a more comfortable molecular area.

Electric-dipole sum-frequency generation spectroscopy is a coherent second-order optical probe. It is sensitive to appropriately oriented, symmetry-allowed molecular responses at the interface; randomly oriented populations can cancel in the orientational average, and centrosymmetric arrangements give no net electric-dipole second-order response.

Thus an ordered fraction of surface-bound carboxylate or hydrocarbon groups may dominate the measured spectrum, while disordered, differently oriented or weakly active adsorbates contribute little. The observed vibrational intensity is not a direct count of every molecule retained in an adsorption measurement. Selection rules, polarization and interference between contributions also matter.

For simultaneous competitive adsorption, use the butyric-acid constant inferred in part (c), $b_b\simeq 1.21{\mathrm{m}}^3\mathrm{mmo}{\mathrm{l}}^{-1}$, and $C_b=10^6\mathrm{mmol}{\mathrm{m}}^{-3}$:The propionic-acid rate is therefore approximatelyThe abundant, more strongly adsorbing butyric acid occupies most common sites, reducing propionic conversion by roughly a factor of $35$. This calculation follows the requested competitive Langmuir model despite the physical concern about the fitted saturation density noted earlier.

$0.1\mathrm{M}=100\mathrm{mol}{\mathrm{m}}^{-3}=10^5\mathrm{mmol}{\mathrm{m}}^{-3}$. ThereforeThe surface is almost saturated, so the rate approaches the supplied limiting constant. This assumes rapid adsorption equilibrium, negligible product coverage, no competing species and unchanged substrate area/photochemical conditions.

For competing reactant and product species on shared equivalent sites, add each $b_j{C}_j$ term to the denominator:Rapid product desorption makes product coverages small, reducing competitive site blocking and simplifying the rate to reactant coverage alone. Without that assumption, product adsorption can inhibit conversion and make the apparent rate evolve as products accumulate.

Nitrogen-down adsorption places the positive hydrogen end farther from the metal, producing an outward molecular electric dipole moment and the observed decrease in work function. Its maximum normal component, however, is only $1.42$ D. Tilting alone can only reduce that component; it cannot explain an effective $2.20$ D.

The adsorbed molecule and metal therefore undergo appreciable electron density. Lone-pair donation into the metal, polarization of the substrate, image response and distortion of the adsorbate can change the effective interface dipole. The inferred increase relative to an upright isolated molecule is about $0.78$ D. For scale, transferring about $0.16$ elementary charges over $1$ Å would produce that dipole increment:This scale illustrates a plausible interfacial polarization; the work-function measurement alone does not uniquely determine a transferred charge or adsorption geometry.

At higher coverage, neighboring outward dipoles generate a local field opposing each other's polarization. This dipole depolarization reduces the incremental dipole per added molecule, so the work function becomes less sensitive to additional adsorption.

Crowding can also tilt the molecules, alter their coordination or make them occupy less strongly donating sites. Any of these changes can decrease the normal dipole increment. A reduced slope is consistent with coverage-dependent depolarization and geometry changes; it is not evidence that coverage has stopped increasing.

The same wave vectors now differ in energy by only 0.070 eV, so the curvature is $A^{\prime }=0.14A=3.64756\mathrm{eV}{\mathring{\mathrm{A}}}^2$. ThereforeThe nearly free-electron in-plane mass and high unoccupied energy suggest an image-potential-derived state, with an electron confined outside the film by its image attraction and surface barrier but relatively free parallel to it. Relative to the supplied clean-platinum vacuum level, its minimum is 0.84 eV below vacuum, a plausible scale for an image state. gold adsorption can change the work function, so 0.84 eV is conditional on using the supplied clean-surface value; an adsorbate work-function measurement is needed for a quantitative assignment. The dispersion alone supports the interpretation without proving it uniquely.