A Thermodynamically Consistent Formulation of the Fundamental Equation in Inverse Gas Chromatography for an Accurate Determination of Solid Surface Properties

Abstract

The classical theory of inverse gas chromatography (IGC) assumes that adsorbed probe molecules behave as a perfect two-dimensional gas, a simplification that leads to large systematic errors in the surface energy and Lewis acid–base calculations. We introduce a corrected formulation for the Gibbs free energy of adsorption, ΔG _a (T) = ΔG _a (T) + f(T, Σ), where f(T, Σ) accounts for temperature and specific surface area effects. Applied to 17 solid surfaces and 19 solvents, the deviation function proved essential, as the classical models produced errors exceeding 1000%. Errors decreased with an increasing surface area, confirming the physical origin of the deviation. This framework redefines the thermodynamic foundation of IGC, enabling accurate determination of dispersive and polar surface energies and necessitates a reevaluation of decades of published data. Although both van der Waals and Redlich–Kwong equations correct the classical IGC formulation, the Redlich–Kwong model yields the best agreement with experimental data with discrepancies between models diminishing as the solid surface area increases.

Introduction

The accurate determination of surface properties of solids is central to fields ranging from catalysis and polymer science to nanomaterials and pharmaceuticals. Among the methods developed to probe these properties, inverse gas chromatography (IGC) at infinite dilution has, since the 1960s, become a standard tool for quantifying adsorption energetics and surface interactions. − By analyzing the retention of vapor-phase probe molecules on solid surfaces, IGC enables the extraction of adsorption free energies, dispersive surface energy components, and Lewis acid–base parameters. −

The widespread use of IGC has been driven by its versatility: it can be applied to powders, polymers, oxides, carbons, and composite materials under a broad range of conditions. Over five decades, thousands of studies have relied on IGC to characterize surface energies, adhesion parameters, and heterogeneity factors, making it a cornerstone of surface science and engineering. −

Yet, the conceptual foundation of IGC has remained unchanged since its inception. Classical formulations rest on a central assumption: that adsorbed probe molecules behave as a two-dimensional perfect gas at the solid–vapor interface. This simplification, introduced for analytical convenience, implies that probe molecules are noninteracting, occupy a fixed surface area, and exhibit adsorption free energies analogous to those of an ideal gas. On this basis, widely adopted methods such as those based on the Fowkes equation , provided routes to extract London dispersive energies, while extensions by Papirer and Saint-Flour, , Schultz et al., Donnet et al., ,,,, Hamieh et al., − ,− ,− Brendlé et al., , and other scientists enabled quantification of acid–base constants. −

Despite its success, persistent discrepancies have been noted between IGC-derived surface energies and values obtained from independent techniques, such as wettability and adhesion. These differences have typically been attributed to surface heterogeneity or experimental uncertainty rather than to the theoretical framework itself. Recent studies, however, indicate that the classical assumptions fail to capture the real behavior of adsorbed molecules, whose effective surface areas depend on temperature, substrate properties, and intermolecular interactions. − ,− ,−

Here we present a fundamental reformulation of the IGC theory. By replacing the two-dimensional perfect gas hypothesis with a two-dimensional real gas description based on the van der Waals and Redlich–Kwong equations, we establish a thermodynamically consistent framework for adsorption. Application of this new formulation across representative solids and probe molecules reveals that the classical equations systematically overestimate adsorption free energies with relative errors reaching more than 1000%, leading to erroneous London dispersive and acid–base surface energy values. This correction not only resolves long-standing inconsistencies but also calls for a re-evaluation of decades of IGC-derived data, reshaping the thermodynamic basis of surface characterization.

Classic Thermodynamic Development of IGC at Infinite Dilution

In the case of reversible adsorption of gas molecules on a solid substrate, the thermodynamic equilibrium imposes the equality of the chemical potentials of the solute (i) in the gas phase (ig) and in the adsorbed phase (is) onto the solid (eq ).

$${\mu }_{is}={\mu }_{ig}$$ 1

The chemical potential of the solute (i) in the gas phase is given by eq :

$${\mu }_{ig}={\mu }_{ig}^0(T)+RT\ln P_i$$ 2

In infinite dilution, the concentration of the solute (i) on the solid surface named σ _i (in mole.m ^–2) can be related to the partial pressure P _i (in Pa) of the solute in the gas phase by a constant K independent of the pressure (eq ).

$${\sigma }_i=K{P}_i$$ 3

Expressions and allow writing eqs and :

$${\mu }_{is}={\mu }_{ig}^0(T)+RT\ln \frac{{\sigma }_i}{K}$$ 4

$${\mu }_{is}={\mu }_{ig}^0(T)+RT\ln {\sigma }_i-RT\ln K$$ 5

For a gas molecule adsorbed on a solid surface, one can formulate in a two-dimensional state equation identical to that of a perfect gas (eq ):

$$\pi \mathrm{\Sigma }=n_{a}RT$$ 6

where n _a is the mole number of adsorbed gas, Σ the surface area of the solid surface (in m²) and π (in N/m or J/m ²) the two-dimensional pressure or simply the surface pressure of the adsorbed gas.

Using the relation n _a = σΣ, eq can be deduced:

$$\pi =\sigma RT$$ 7

This led to the standard free energy ΔG _i of adsorption of a solute I (eq ):

$$\mathrm{\Delta }{G}_i^{0*}=-RT\ln \left(KRT\frac{P_0}{{\pi }_0}\right)$$ 8

Now in IGC technique, the adsorption of a probe on a solid material filling the chromatographic column is characterized by a retention time t _r (giving a retention volume V _r ) which is related to mole numbers of the probe in the adsorbed n _a and gas n _g phases by eq using the equilibrium constant K _e :

$$K_e=\frac{n_a}{n_g}=\frac{V_n}{V_{dt}}=\frac{t_r}{t_{dt}}-1$$ 9

where t _dt is the time for an unretained compound to travel through the column (dead time) and V ₀ the corresponding retention volume.

If s is the specific surface area of the solid, m (Σ = ms) is its mass, and C _g is the molar concentration of the gas phase, eq can then be then written:

$$K_e=\frac{\sigma }{C_g}\frac{sm}{V_{dt}}$$ 10

Using the equation of perfect gas (eq ):

$$C_g=\frac{P}{RT}$$ 11

and σ the surface concentration (mol m^–2), one obtains eq :

$$K_e=KRT\frac{sm}{V_{dt}}=\frac{V_n}{V_{dt}}$$ 12

With eqs and , one writes:

$$KRT=\frac{V_n}{sm}$$ 13

Combining eqs and , one can write the final classic, eq :

$$\mathrm{\Delta }{G}_a^0=-RT\ln \left(\frac{V_n{P}_0}{sm{\pi }_0}\right)$$ 14

that can be generally written as

$$\mathrm{\Delta }{G}_a^{0*}=-RT\ln V_n+C(T)$$ 15

with C(T) given by eq :

$$C(T)=RT\ln \left(\frac{sm{\pi }_0}{P_0}\right)$$ 16

Two reference states were used in literature to determine the free energy of adsorption of solvents on a solid substrate: the first one was the Kemball and Rideal reference state given for T ₀ = 0 °C by P ₀ = 1.013 × 10⁵ Pa and π₀ = 6.08 × 10^–5 N m ^–1, while the second was that of De Boer et al. characterized by T ₀ = 0 °C by P ₀ = 1.013 × 10⁵ Pa and π₀ = 3.38 × 10^–4 N m ^–1.

To date, eqs and are used in all chromatographic measurements by admitting the hypothesis of 2D and 3D perfect gas. Our new methodology (that will be developed later) is based on the two- and three-dimensional states of real gases.

Limitations of the Classical IGC Framework

The quantitative description of adsorption thermodynamics in inverse gas chromatography (IGC) was commonly expressed by the IGC fundamental relation obtained from eqs and :

$$\mathrm{\Delta }{G}_a^{0*}=-RT\ln \left(\frac{V_n{P}_0}{sm{\pi }_0}\right)$$ 17

where ΔG _a is the Gibbs free energy of adsorption of a probe molecule considered as a perfect gas, while P ₀ and π₀ are respectively the reference pressure and two-dimensional pressure given by the reference state. ,

This formulation directly links measurable chromatographic parameters with molecular adsorption energetics and thus provides a robust framework for probing surface–molecule interactions. The theoretical foundations of this equation (detailed in the previous section) were laid in the late 1960s by Kiselev and Yashin, who established the thermodynamic treatment of gas–solid chromatography. Their approach introduced the concept that retention volumes could be rigorously interpreted in terms of adsorption free energies, thereby opening the way to quantitative surface energetics analysis, enabling quantitative evaluation of London dispersive and specific surface free energies, acid–base parameters, and interfacial interaction potentials across a wide range of materials.

However, the classical IGC equation above at infinite dilution assumes that adsorbed probe molecules behave as a 2D perfect gas. This means that probes are treated as noninteracting particles, each occupying a fixed surface area, and that their adsorption free energy can be described with an ideal-gas-like treatment. This simplified hypothesis was convenient and allowed elegant analytical derivations. But it was also a drastic oversimplification of real adsorption phenomena.

In reality, adsorbed molecules are subject to intermolecular interactions, solid-specific perturbations, and temperature effects. Their effective cross-sectional area is not constant, − , but depends on both the probe’s nature and the substrate properties. For decades, discrepancies between IGC-derived surface energies and independent measurements (e.g., by wettability or adhesion) were reported, but these were often attributed to experimental variability or surface heterogeneity rather than to the foundational equation itself.

New Thermodynamic Formulation of IGC at Infinite Dilution

The classic fundamental IGC equation used the perfect gas hypothesis to deduce the free energy of adsorption. This simple model assumed that there is no interaction between the gaseous molecules. However, the lateral interactions between molecules remain present even at infinite dilution; we then propose the correction of the classical IGC model using the van der Waals equation in both the adsorbed and gaseous phases. Therefore, eq relative to the two-dimensional state of gas was replaced by eq :

$${\sigma }_i=\frac{1}{RT}({\pi }_i+\frac{\alpha (2D)}{{\mathrm{\Sigma }}^2})(1-\frac{{\mathrm{\Sigma }}_0}{\mathrm{\Sigma }})$$ 18

where Σ₀ is the surface area of adsorbed molecules that defines the recovery fraction θ = Σ₀/Σ and α (2D) is the corresponding 2D interaction parameter given in Table S1.

Equation then led to the following relation:

$$\mathrm{\Delta }{G}_i^0=-RT\ln \left(KRT\frac{P_0}{{\pi }_0}\right)+RT\ln \left[(1+\frac{\alpha (2D)}{{\mathrm{\Sigma }}^2}\frac{1}{{\pi }_0})(1-\frac{{\mathrm{\Sigma }}_0}{\mathrm{\Sigma }})\right]$$ 19

Whereas eq was replaced by the following 3D van der Waals equation:

$$C_g=\frac{1}{RT}(P+\frac{a}{V^2})(1-\frac{b}{V})$$ 20

where a and b are the van der Waals coefficients of solvents given in Table S1

This gave expression :

$$V_n=KRT\frac{sm{P}_0}{(P_0+\frac{a}{V_0^2})(1-\frac{b}{V_0})}$$ 21

Therefore, the IGC equation can be written as follows:

$$\mathrm{\Delta }{G}_a^0=-RT\ln \left(\frac{V_n}{Sm}\frac{P_0}{{\pi }_0}\right)-RT\ln \left[(1+\frac{a}{P_0{V}_0^2})(1-\frac{b}{V_0})\right]+RT\ln \left[(1+\frac{\alpha (2D)}{{\mathrm{\Sigma }}^2}\frac{1}{{\pi }_0})(1-\theta )\right]$$ 22

The recovery fraction θ of solvents in infinite dilution is very small compared to 1, while the calculated values of $\frac{a}{P_0{V}_0^2}$ and $\frac{b}{V_0}$ for all solvents are neglected in front of 1 (shown in Table S3) and therefore the values of $-RT\ln \left[(1+\frac{a}{P_0{V}_0^2})(1-\frac{b}{V_0})\right]$ given in Table S4 do not exceed 0.300 kJ/mol for the various adsorbed probes and for the various temperatures.

This led to the final IGC equation:

$$\mathrm{\Delta }{G}_a^0=-RT\ln \left(\frac{V_n}{sm}\frac{P_0}{{\pi }_0}\right)+RT\ln \left[\right(1+\frac{\alpha (2D)}{(\mathrm{sm}{)}^2}\frac{1}{{\pi }_0}\left)\right]$$ 23

where ms = Σ representing the surface area of the solid.

The work presented here establishes a new theoretical foundation for inverse gas chromatography. At its origin, IGC assumed that the standard free energy of adsorption could be derived by treating adsorbed molecules as an ensemble of perfect gases, constrained in either two or three dimensions. This simplification permitted a tractable link between retention volumes and surface energetics but also imposed a severe and untested approximation. In fact, adsorption involves dense molecular layers, where finite molecular size and intermolecular interactions cannot be neglected. By reformulating the IGC equation within the framework of the van der Waals equation of state, in both two and three dimensions, we explicitly incorporate excluded volume effects and real attractive interactions into the description of the adsorbed phase. This new formalism eliminates the long-standing inconsistency of the classical model and provides retention–energy relations that correctly account for the physics of adsorption. When applied to a broad series of nonpolar and polar probes interacting with diverse solids, the reformulated equation yields adsorption energies that deviate by more than an order of magnitude from those predicted by classical theory. The new foundation therefore resolves the critical flaw of the classic equation and establishes a physically meaningful basis for using IGC to quantify dispersive and Lewis acid–base surface properties.

Putting

$$f(T,sm)=RT\ln \left[\right(1+\frac{\alpha }{(\mathrm{sm}{)}^2}\frac{1}{{\pi }_0}\left)\right]$$ 24

with Σ = ms, eqs and led to

$$\mathrm{\Delta }{G}_a^0(T)=\mathrm{\Delta }{G}_a^{0*}(T)+f(T,\mathrm{\Sigma })$$ 25

where f(T, Σ) is a correction function representing the deviation from ideality that was applied in the classic IGC theory.

Experimental Section

Nineteen nonpolar and polar probes were used in this study. Whereas 17 solid materials were selected as chromatographic adsorbents. The following solid surfaces were previously characterized ,,,,,,,,, by IGC using the classic eq : oligoacenaphthylene, oligoacenaphthylene/p-Hydroxyphenylacetic acid (p-HPA) composite, carbon nanotube (CNT) materials such as the nitrogen doped multiwalled carbon nanotubes (N-MWCNT) with carboxyl methyl cellulose (CMC) materials, silica, beta-zeolite, mesoporous silica doped with cerium and terbium zinc oxide (Ce–Ag/MS and Tb–Ag/MS), zinc oxide, titanium dioxide, , Amberlite XAD-7 acrylic-ester-resin, MOF materials (UiO-66), , two 2D single-crystalline and polycrystalline covalent organic frameworks (COF), , graphene, graphene oxide, and reduced graphene oxide. Table S1 gave, for the different solvents, the van der Waals coefficients a and b, the two-dimensional coefficient α (2D), the corrected donor DN ^’ and acceptor AN ^’ numbers of electrons, the enthalpy of vaporization, and the boiling point. While Table S2 gave the specific surface area s and mass m of the various adsorbents used in chromatographic experiments.

The chromatographic measurements were previously carried out by a Focus GC Chromatograph equipped with a high-sensitivity flame ionization detector of high sensitivity. Dried nitrogen was used as the carrier gas. The gas flow rate was set between 20 and 40 mL/min, depending on the examined solid surfaces. Whereas the injector and detector temperatures were maintained at 400 K during the experiments. All columns used in this study were prepared using a stainless-steel column with a 2 mm inner diameter and with a length of 20 to 60 cm. The column temperatures were 300 to 480 K, varied in 10 or 20 °C steps. Each solvent injection was repeated three times, and the average retention time, t _R , was used for the determination of the net retention volume V _n . The standard deviation was less than 1% in all of the measurements.

Results and Discussion

Failure of the Classical Model

The values of different thermodynamic parameters of solvents and solid surfaces given in Tables S1–S4 were used to confirm the validity of the new thermodynamic eq (or ) were obtained by applying the two- and three-dimensional VDW state equations.

The experimental IGC measurements carried out by adsorption of solvents on the solid materials allowed us to obtain the values of the free energy of adsorption at different temperatures using the classic eq and the new eq . The results given in Tables S5–S72 for the adsorption of solvents on the chosen solid substrates showed a large deviation between the free energy of adsorption of adsorbed probes calculated by applying the classic and newest equations and highlighted the important effect of the interaction of solvent – solid surfaces on the free energy of adsorption.

Two extreme cases were presented in Figures and clearly showing the large effect of the surface area of solid materials on the free energy of adsorption. The first case given in Figure a-d concerns zinc oxide material exhibiting a smaller surface area Σ _ZnO = 35 m² and the second given in Figure a-d is relative to silica surfaces Σ _Silica = 157.5 m². In the case of ZnO, a large difference between the values of the free energy of adsorption was observed, and the relative error, depending on the solvent nature, the temperature, and the interaction with ZnO, varied from 62% for n-nonane at 323.15K to 1092% for the adsorption of dichloromethane at 383.15K. However, this deviation decreased in the case of silica where a relative error between 4.7% (for chloroform at 323.15K) and 45.1% (for n-pentane, at 483.15). The results showed that when the available surface area Σ of solid material increases, the deviation between the classic eq and the new eq decreases and tends toward several percentages in the case of highest surface area such as beta-zeolite or single crystalline surfaces shown in Figures S65–S72.

1.

Comparison between the classic and new chromatographic equations in the case of ZnO: (1.a.) Free energy of adsorption of solvents on ZnO using the classic eq . (1.b.) Free energy of adsorption of solvents on ZnO using the new eq . (1.c.) Deviation function f(T, Σ) of solvents adsorbed on ZnO. (1.d.) Relative error (in %) on the free energy of adsorbed solvents on ZnO surfaces as a function of temperature T (K), defined by $\left|\frac{\mathrm{\Delta }{G}_a^{0*}(T)-\mathrm{\Delta }{G}_a^0(T)}{\mathrm{\Delta }{G}_a^0(T)}\right|\times 100$ .

2.

Comparison between the classic and new chromatographic equations in the case of silica: (2.a.) Free energy of adsorption of solvents on silica using the classic eq . (2.b.) Free energy of adsorption of solvents on silica using the new eq . (2.c.) Deviation function f(T, Σ) of solvents adsorbed on silica. (2.d.) Relative error (in %) on the free energy of adsorbed solvents on silica surfaces as a function of temperature T (K), defined by $\left|\frac{\mathrm{\Delta }{G}_a^{0*}(T)-\mathrm{\Delta }{G}_a^0(T)}{\mathrm{\Delta }{G}_a^0(T)}\right|\times 100$ .

Discussion and Implications for Surface Energy and Acid–Base Characterization

The correction introduced by the new formulation of the Gibbs free energy of adsorption has profound consequences for the thermodynamic determination of the surface energy components of solids. In the classical framework of inverse gas chromatography (IGC), the London dispersive surface energy (γ _s ) and the Lewis acid–base parameters are computed directly from the apparent adsorption free energy, ΔG _a , derived under the assumption of a two-dimensional perfect adsorbed gas. This assumption, embedded in the classic equation, implicitly neglects molecular packing effects, interfacial stress, and curvature contributions that arise from the local finite density and heterogeneity of the real adsorbed phase.

Our results demonstrate that the additional corrective term f(T, Σ), which depends explicitly on temperature and specific surface area, and then on the surface area Σ of solid substrate, introduces a non-negligible shift in the absolute magnitude and temperature dependence of the measured ΔG _a . Because the dispersive and acid–base components of the surface energy are obtained from the slope and intercept of linear relations involving ΔG _a , even modest deviations in ΔG _a propagate as significant multiplicative errors in the derived surface energy values. Across the 17 studied solid materials, we observed that the difference between dispersive surface energies calculated from the classical and corrected formulations ranges from 5% for high-surface-area solids (such as beta-zeolite shown in Tables S65–S68) to more than 1000% for solid materials with small Σ = sm such as oligoacenaphthylenes (Tables S5–S12), graphene (Figure S13–S16), or zinc oxide (Tables S17–20).

The present analysis confirms that the effective molecular area and the free energy of adsorption are coupled through the local surface structure, electronic density, and deformation field of the solid, as reflected in f(T, S). Consequently, the London dispersive surface energy γ _s (T) must be re-evaluated using the corrected free energy term. Indeed, the free energy of n-alkanes C _n adsorbed on solid materials is expressed as follows (eq ):

$$\mathrm{\Delta }{G}_a^0(T,C_n)=2\mathcal{N}{a}_{C_n}(T)\sqrt{{\gamma }_l^d(T,C_n){\gamma }_s^d(T)}$$ 26

where a _{C n} (T) is the n-alkane’s contact area and γ _l (T) the London dispersive surface energy of the n-alkanes.

If γ _l (T) represents the classic London dispersive surface area of solids obtained by the approximation of a 2D perfect adsorbed gas, the following expression can be written:

$${\gamma }_s^d(T)={\gamma }_s^{d*}(T){\left[\frac{\mathrm{\Delta }{G}_a^0(T,C_n)}{\mathrm{\Delta }{G}_a^{0*}(T,C_n)}\right]}^2$$ 27

or

$${\gamma }_s^d(T)={\gamma }_s^{d*}(T){[1+\frac{f(T,\mathrm{\Sigma })}{\mathrm{\Delta }{G}_a^{0*}(T,C_n)}]}^2$$ 28

The analysis of deviation function f(T, Σ) demonstrates a strong inverse correlation with the specific surface area of the solid. For solids with very large surface areas, f(T, Σ) → 0, and the corrected free energy of adsorption ΔG _a (T) converges toward its classical counterpart ΔG _a (T). Under these conditions, the London dispersive surface energy obtained from the reformulated model, γ _s (T), becomes indistinguishable from the classical value γ _s (T), validating the 2D perfect gas approximation only in this limiting case. However, for solids with moderate or low surface areas, this approximation breaks down. The relative error ε(T, Σ) on γ _s is given by the following relation:

$$\epsilon (T,\mathrm{\Sigma })=\left|\frac{{\gamma }_s^d(T)-{\gamma }_s^{d*}(T)}{{\gamma }_s^d(T)}\right|\times 100=|1-{[1+\frac{f(T,\mathrm{\Sigma })}{\mathrm{\Delta }{G}_a^{0*}(T,C_n)}]}^{-2}|\times 100$$ 29

When the magnitude of the deviation function f(T, Σ) becomes comparable to the classical free energy ΔG _a (T, C _n ), the corrected dispersive surface energy reaches γ _s (T) = 4 γ _s (T), corresponding to a relative error of 75%. This regime represents a complete collapse of the classical IGC assumption, where surface–molecule interactions are no longer adequately described by idealized adsorbed gas behavior. Experimentally, this situation was observed for oligoacenaphthylene, where errors ranging from 65% to 73% were recorded, depending on the probe alkane and temperature. Similar but smaller deviations were found for ZnO (≈55%) and silica (≈32%), with the relative error decreasing systematically as the specific surface area increased. These results confirm that the thermodynamic inaccuracy of the classical IGC equation is intrinsically controlled by the surface area of the solid and that large-surface-area materials mask, rather than eliminate, the fundamental flaw in the ideal gas approximation.

The new refined relation then restores thermodynamic consistency and eliminates the artificial linearities introduced by ideal-gas simplification.

The implications extend equally to the determination of the Lewis acid–base parameters. In the various chromatographic methodologies, the specific or polar enthalpy and free energy of adsorption of polar probes are referenced to those of nonpolar homologues, assuming the same geometric adsorption environment.

To further validate the thermodynamic implications of the new formulation, we extended the analysis by providing explicit experimental examples comparing the different free energy components of adsorption, the corresponding London dispersive surface energy, and Lewis acid–base parameters obtained from both the classical and corrected IGC equations.

To assess the magnitude and physical origin of the deviation function f(T, Σ), two limiting cases in specific surface area (Σ) were examined: oligoacenaphthylene with a smaller Σ = 1.418 m² and a high-area oxide (silica): Σ = 157.5 m². These solids were selected to represent the two extreme regimes of surface density in inverse gas chromatography (IGC) and to test the generality of the newly derived thermodynamic formulation:ΔG _a (T) = ΔG _a (T) + f(T, Σ). The comparison between the classical IGC equation and the corrected model revealed systematic, surface-area-dependent discrepancies in all thermodynamic quantities.

These results, summarized in Tables –, show that the polar enthalpy of adsorption varies only slightly with temperature and surface area, so that the derived Lewis acid and base constants (K _A and K _D ) remain nearly unchanged, while the main correction introduced by the deviation function f(T, Σ) also remains very small tending toward zero. Indeed, the results in Tables and showed comparable results for polar enthalpy ΔH _a and moderate changes for polar entropy ΔS _a by comparing the classic and new methods.

1. Values of the Free Energy of Adsorption ΔG _a (T) (kJ/mol) of Solvents on Oligoacenaphthylene Using Classic and New Equations with Relative Error Percentages.

ΔG a (T) (kJ/mol) using the classic method
	Temperature (K)
	343.15	353.15	363.15	373.15	383.15	% error
n-Hexane	34.422	34.646	34.611	35.035	34.900	1611.423
n-Heptane	36.352	35.916	36.111	36.235	36.630	1023.234
n-Octane	38.322	37.256	36.851	36.965	36.830	1119.339
n-Nonane	40.192	39.036	39.241	39.235	39.030	706.630
n-Decane	43.162	42.516	42.241	41.835	41.630	485.862
Dichloromethane	38.492	38.666	38.921	38.875	38.760	430.960
Chloroform	38.962	38.756	39.021	38.695	38.930	298.213
Ethyl acetate	41.292	40.536	40.151	39.505	39.370	475.352
Diethyl ether	36.362	35.866	36.021	36.215	36.340	749.391
Acetone	39.542	39.846	39.621	39.735	39.370	432.335
THF	38.772	38.696	38.721	38.725	38.560	495.484

ΔG a (T) (kJ/mol) using the new method
	Temperature (K)
	343.15	353.15	363.15	373.15	383.15
n-Hexane	4.992	4.359	3.465	3.032	2.039
n-Heptane	6.467	5.160	4.484	3.737	3.261
n-Octane	8.042	6.094	4.806	4.038	3.020
n-Nonane	9.570	7.522	6.834	5.936	4.839
n-Decane	12.242	10.695	9.519	8.212	7.106
Dichloromethane	10.316	9.670	9.103	8.236	7.300
Chloroform	12.852	11.885	11.389	10.302	9.776
Ethyl acetate	12.160	10.556	9.322	7.827	6.843
Diethyl ether	7.647	6.315	5.633	4.991	4.278
Acetone	10.906	10.376	9.316	8.596	7.396
THF	10.037	9.124	8.311	7.478	6.475

8. Values of the Dispersive Free Energy of Adsorption ΔG _a (T) (kJ/mol) of Solvents on Silica Using Classic and New Equations with the Relative Error Percentages.

ΔG a (T)(kJ/mol) using the classic method
	Temperature (K)
	323.15	343.15	363.15	383.15	% error
n-Pentane	15.729	13.918	12.471	11.213	45.12
n-Hexane	19.603	17.514	15.832	14.361	37.33
n-Heptane	23.365	20.960	18.997	17.245	32.89
n-Octane	27.213	24.480	22.221	20.168	29.52
n-Nonane	30.994	27.920	25.349	22.979	27.03
Carbon tetrachloride	18.699	16.665	15.031	13.592	38.80
Dichloromethane	10.371	9.029	8.010	7.184	73.12
Chloroform	14.133	12.479	11.182	10.079	50.45
Chloroform	13.885	12.251	10.973	9.888	51.40
Diethyl ether	11.170	9.762	8.684	7.799	66.24
Tetrahydrofuran	13.690	12.072	10.808	9.737	52.19
Ethyl acetate	7.545	6.438	5.628	5.010	124.14
Acetone	4.447	3.598	3.016	2.626	1835.09
Acetonitrile	17.896	15.928	14.354	12.974	40.28
Toluene	15.437	13.674	12.281	11.082	46.15

ΔG a (T)(kJ/mol) using the new equation
	Temperature (K)
	323.15	343.15	363.15	383.15
n-Pentane	12.788	10.796	9.166	7.726
n-Hexane	16.311	14.018	12.132	10.458
n-Heptane	19.766	17.138	14.952	12.977
n-Octane	23.337	20.363	17.864	15.572
n-Nonane	26.870	23.540	20.714	18.089
Carbon tetrachloride	15.497	13.262	11.426	9.793
Dichloromethane	7.814	6.312	5.132	4.150
Chloroform	11.285	9.451	7.976	6.699
Chloroform	11.056	9.244	7.788	6.531
Diethyl ether	8.551	6.979	5.736	4.691
Tetrahydrofuran	10.876	9.081	7.640	6.398
Ethyl acetate	5.207	3.953	2.996	2.235
Acetone	2.348	1.368	0.655	0.136
Acetonitrile	14.756	12.591	10.819	9.248
Toluene	12.488	10.539	8.961	7.582

3. Values of the Polar Enthalpy ΔH _a (kJ/mol) and Entropy ΔS _a (J K^–1 mol^–1) of Solvents Adsorbed on Oligoacenaphthylene Using Classic and New Equations in the Temperature Range 323.15–383.15K.

	IGC method
	Classic method		New method
Polar solvents	ΔS a	ΔH a	ΔS a	ΔH a
Dichloromethane	24.9	17.834	24.2	17.971
Chloroform	22.3	15.186	17	16.265
Ethyl acetate	54.8	28.314	55.3	28.212
Diethyl ether	21.2	12.215	20.8	12.281
Acetone	39.7	25.377	40.4	25.227
THF	31	19.583	31.2	19.536

7. Values of the Polar Enthalpy ΔH _a (kJ/mol) and Entropy ΔS _a (J K^–1 mol^–1) of Solvents Adsorbed on Silica Using Classic and New Equations in the Temperature Range 323.15–383.15 K.

	IGC method
	Classic method		New method
Polar solvents	ΔS a	ΔH a	ΔS a	ΔH a
CCl4	–3.6	5.5839	–4.1	5.5654
CH2Cl2	29.5	32.001	29.4	31.989
CHCl3	20	26.192	15.8	26.177
Diethyl ether	76.2	51.5	76.1	51.485
THF	125.7	76.064	126.4	76.052
Ethyl acetate	24.7	12.531	25.6	12.517
Acetone	46.5	25.594	47.8	25.584
Acetonitrile	66.4	38.139	69	38.132
Toluene	28.8	26.612	29.4	26.594
Benzene	21.8	12.67	22	12.654

A strong inverse correlation between the relative error and the surface area of the solids was observed. For oligoacenaphthylene (Σ = 1.418 m²) the classical equation underestimates adsorption strength dramatically:

• ΔG _a : 400%–1700% error (Table )

• ΔG _a : 450%–3600% error, with extreme dispersive outliers reaching 30 700% (ethyl acetate), 73 600% (diethyl ether), and 100 340% (chloroform) (Table )

• ΔG _a : modest error, 1.7%–33% (Table )

• ΔH _a : 0%–7%; ΔS _a : 0.6%–32% (Table )

In contrast, for silica (Σ = 1.418 m²), the deviation is far smaller:

• ΔG _a : 6%–50% (Table )

• ΔG _a : 25%–125%, with one dispersive outlier (acetone, 1835%) (Table )

• ΔG _a : 0%–10% (Table )

• ΔH _a : 0%–0.33%; ΔS _a : 0%–12.1% (Table )

4. Values of the Dispersive Free Energy of Adsorption ΔG _a (T) (kJ/mol) of Solvents on Oligoacenaphthylene Using Classic and New Equations, with Relative Error Percentages.

ΔG a (T) (kJ/mol) using the classic method
	Temperature (K)
	343.15	353.15	363.15	373.15	383.15	% error
n-Hexane	34.422	34.646	34.611	35.035	34.900	1611.42
n-Heptane	36.352	35.916	36.111	36.235	36.630	1023.23
n-Octane	38.322	37.256	36.851	36.965	36.830	1119.34
n-Nonane	40.192	39.036	39.241	39.235	39.030	706.63
n-Decane	43.162	42.516	42.241	41.835	41.630	485.86
Dichloromethane	28.847	29.394	29.579	30.432	30.727	2804.87
Chloroform	31.016	31.303	31.431	32.104	32.320	100342.19
Ethyl acetate	30.762	31.079	31.213	31.908	32.133	30739.54
Diethyl ether	30.875	31.179	31.310	31.995	32.216	73629.95
Acetone	27.219	27.962	28.189	29.177	29.532	1567.16
THF	29.310	29.801	29.974	30.788	31.067	3602.75

ΔG a (T) (kJ/mol) using the new equation
	Temperature (K)
	343.15	353.15	363.15	373.15	383.15
n-Hexane	4.992	4.359	3.465	3.032	2.039
n-Heptane	6.467	5.160	4.484	3.737	3.261
n-Octane	8.042	6.094	4.806	4.038	3.020
n-Nonane	9.570	7.522	6.834	5.936	4.839
n-Decane	12.242	10.695	9.519	8.212	7.106
Dichloromethane	0.310	0.024	–0.620	–0.604	–1.136
Chloroform	2.099	1.541	0.830	0.655	0.032
Ethyl acetate	1.889	1.363	0.660	0.507	–0.105
Diethyl ether	1.983	1.443	0.736	0.573	–0.044
Acetone	–1.033	–1.115	–1.709	–1.548	–2.013
THF	0.691	0.348	–0.311	–0.335	–0.887

2. Values of the Polar Free Energy of Adsorption ΔG _a (T) (kJ/mol) of Solvents on Oligoacenaphthylene Using Classic and New Equations with Relative Error Percentages.

ΔG a (T )(kJ/mol) using the classic method
	T (K)
	323.15	343.15	363.15	373.15	383.15	% error
Dichloromethane	9.645	9.272	9.342	8.443	8.032	4.78
Chloroform	7.945	7.454	7.590	6.591	6.610	32.16
Ethyl acetate	10.530	9.457	8.937	7.597	7.237	4.17
Diethyl ether	5.487	4.688	4.711	4.220	4.124	4.59
Acetone	12.323	11.884	11.432	10.559	9.838	4.56
THF	9.462	8.895	8.747	7.937	7.493	1.77

ΔG a (T)(kJ/mol) using the new method
	T (K)
	323.15	343.15	363.15	373.15	383.15
Dichloromethane	10.006	9.645	9.723	8.840	8.436
Chloroform	10.752	10.344	10.558	9.648	9.744
Ethyl acetate	10.271	9.193	8.661	7.320	6.948
Diethyl ether	5.665	4.872	4.897	4.418	4.322
Acetone	11.939	11.490	11.024	10.144	9.409
THF	9.345	8.776	8.622	7.814	7.362

5. Values of the Free Energy of Adsorption ΔG _a (T) (kJ/mol) of Solvents on Silica Using Classic and New Equations with the Relative Error Percentages.

ΔG a (T)(kJ/mol) using the classic method
	Temperature (K)
	323.15	343.15	363.15	383.15	% error
n-Pentane	15.729	13.918	12.471	11.213	45.124
n-Hexane	19.603	17.514	15.832	14.361	37.327
n-Heptane	23.365	20.960	18.997	17.245	32.890
n-Octane	27.213	24.480	22.221	20.168	29.520
n-Nonane	30.994	27.920	25.349	22.979	27.034
Carbon tetrachloride	25.451	23.475	21.912	20.560	21.538
Dichloromethane	32.862	30.876	29.279	27.900	11.994
Chloroform	33.886	31.782	30.107	28.625	6.684
Diethyl ether	40.724	37.714	34.775	32.202	11.664
Tetrahydrofuran	46.677	42.549	39.119	35.706	10.456
Ethyl acetate	18.255	16.087	14.338	12.816	40.184
Acetone	18.157	16.047	14.331	12.826	34.761
Acetonitrile	21.182	18.901	17.032	15.364	29.452
Toluene	35.225	32.653	30.521	28.572	16.250
Benzene	21.077	18.844	17.026	15.410	30.416

ΔG a (T) (kJ/mol) using the new method
	Temperature (K)
	323.15	343.15	363.15	383.15
n-Pentane	12.788	10.796	9.166	7.726
n-Hexane	16.311	14.018	12.132	10.458
n-Heptane	19.766	17.138	14.952	12.977
n-Octane	23.337	20.363	17.864	15.572
n-Nonane	26.870	23.540	20.714	18.089
Carbon tetrachloride	22.378	20.212	18.458	16.917
Dichloromethane	30.342	28.200	26.447	24.912
Chloroform	32.373	30.176	28.407	26.831
Diethyl ether	37.887	34.701	31.587	28.839
Tetrahydrofuran	43.826	39.522	35.915	32.326
Ethyl acetate	15.157	12.797	10.856	9.142
Acetone	15.367	13.084	11.195	9.517
Acetonitrile	18.233	15.771	13.719	11.868
Toluene	31.857	29.076	26.736	24.578
Benzene	18.046	15.625	13.620	11.816

6. Values of the Polar Free Energy of Adsorption ΔG _a (T) (kJ/mol) of Solvents on Silica Using Classic and New Equations with the Relative Error Percentages.

ΔG a (T)(kJ/mol) using the classic method
	T (K)
	323.15	343.15	363.15	383.15	% error
Carbon tetrachloride	6.752	6.810	6.881	6.968	2.19
Dichloromethane	22.490	21.846	21.269	20.716	0.22
Chloroform	19.752	19.304	18.925	18.546	7.88
Diethyl ether	26.838	25.462	23.802	22.314	0.03
Tetrahydrofuran	35.506	32.787	30.435	27.908	0.99
Ethyl acetate	4.566	4.015	3.530	3.079	12.20
Acetone	10.612	9.608	8.703	7.816	7.33
Acetonitrile	16.734	15.304	14.016	12.738	8.57
Toluene	17.330	16.724	16.168	15.598	1.75
Benzene	5.640	5.170	4.745	4.328	2.23

ΔG a (T)(kJ/mol) using the new method
	T (K)
	323.15	343.15	363.15	383.15
Carbon tetrachloride	6.881	6.950	7.032	7.124
Dichloromethane	22.528	21.888	21.315	20.762
Chloroform	21.088	20.725	20.431	20.132
Diethyl ether	26.830	25.457	23.799	22.307
Tetrahydrofuran	35.275	32.543	30.179	27.635
Ethyl acetate	4.281	3.716	3.215	2.744
Acetone	10.160	9.130	8.199	7.282
Acetonitrile	15.885	14.403	13.064	11.733
Toluene	17.101	16.484	15.917	15.329
Benzene	5.558	5.086	4.659	4.234

These results confirm that the magnitude of f(T, Σ) decreases rapidly with increasing Σ, approaching zero for extended, high-area solids, in agreement with the theoretical limit f(T, Σ → ∞) → 0.

In the classical IGC framework, the adsorbed layer is treated as a two-dimensional perfect gas confined to the solid surface. This simplification neglects the finite volume of the adsorbed molecules and their lateral van der Waals (VDW) coupling. The introduction of f(T, Σ) effectively reintroduces these three-dimensional corrections, accounting for both the excluded volume and the intermolecular attractions via a van der Waals-type term (a/V ²).

For low-surface-area solids such as oligoacenaphthylene, molecular confinement is strong and lateral VDW interactions between neighboring adsorbates become significant. The energy stored in these lateral fields, neglected by the classical model, manifests as the deviation function f(T, Σ). As a consequence, the free energy and its derived components are systematically underestimated when the classical assumption is applied.

The corrected results also show that the dispersive free energy (ΔG _a ) is far more sensitive to f(T, Σ) than the polar free energy (ΔG _a ). On oligoacenaphthylene, dispersive errors reach several thousand percent (Table ), while the polar term varies only by 1.7%–33% (Table ). This contrast arises because:

• 1.Applying eq to the measured dispersive free energies ΔG _a (T) of n-alkanes adsorbed on oligoacenaphthylene and on silica, we quantified the relative error in the London dispersive surface energy γ _s . For oligoacenaphthylene, the relative error in γ _s lies between 68%–73% across all n-alkanes and temperatures examined. For silica, the corresponding error is 25%–37%. (The relative error is defined with respect to the corrected value, (γ _s – γ _s )/γ _s ).
• 2.Polar interactions are governed by localized acid–base forces, less affected by lateral coupling, and their enthalpic contribution ΔH _a remains almost unchanged (0%–7%). However, the entropy term ΔS _a varies until 31%, demonstrating that f­(T, Σ) is predominantly entropic in origin, reflecting configurational compression of the adsorbed layer rather than changes in specific bonding energy. These results show that the new formulation primarily affects the polar and dispersive components of the free energy of adsorption through the correction of the adsorbed-phase thermodynamics. However, the specific acid–base parameters of solid surfaces are not generally very affected by the application of new van der Waals or Redlich–Kwong equations. Indeed, the Lewis acid and base constants (K_A and K_D) exhibit limited numerical variation between the classical and corrected formulations, even for low-surface-area solids. This distinction is physically meaningful: the new model corrects the state behavior of the adsorbed phasedominated by lateral van der Waals coupling and entropywhile preserving the intrinsic energetic nature of localized donor–acceptor interactions. Thus, the formulation refines dispersive surface energetics without altering the fundamental chemical identity of acid–base surface sites.

This finding also explains why ΔG _a remains relatively stable even when ΔS _a fluctuates moderatly: enthalpy and entropy corrections partially compensate in the Gibbs relation ΔG _a = ΔH _a – TΔS _a .

The exceptionally large dispersive errors for ethyl acetate, diethyl ether, and chloroform on oligoacenaphthylene and for acetone on silica correspond to polar probes misclassified as purely dispersive under the classical 2D model (Table ). When f(T, Σ) is ignored, the measured retention volume incorporates both dispersive and polar contributions, leading to severe partitioning of the interaction energy. The corrected equation redistributes this free energy properly between dispersive and polar channels, exposing the underlying polarity that had been mathematically “hidden” in the classical regression schemes.

The clear Σ^–1 scaling of the relative error across all quantities confirms that the surface area is the controlling thermodynamic parameter in the deviation function. Practically:

• The classical equation is invalid for low-area solids (Σ < 100 m²); f­(T, Σ) cannot be neglected, and the error may exceed 1000% for lower surfaces. Even for Σ = 416 m² as in the case of UiO-66 surface, the relative error on the thermodynamic parameters remains still important from 20% to 40%.

• For higher-area solids, the classical equation may provide approximate results for polar interactions but still misestimates dispersive energies.

• The correction term f­(T, Σ) should be explicitly reported or computed for any quantitative IGC analysis, as it unifies adsorption thermodynamics across both regimes.

These results definitively demonstrate that the two-dimensional perfect gas approximationthe cornerstone of IGC theory for more than half a centuryis thermodynamically invalid for real adsorbed systems.

The newly formulated equation not only corrects this foundational error but also identifies the surface-area–dependent nature of the deviation. By restoring physical consistency, the corrected IGC model enables an accurate, predictive determination of dispersive and polar free energies of adsorption, thereby redefining the quantification of surface energetics for both low- and high-area materials.

In summary, the incorporation of f(T, Σ) into the free energy formulation not only rectifies the thermodynamic description of adsorption but also restores the internal consistency of the entire IGC-based surface energy framework. It bridges the gap between macroscopic wetting approaches and microscopic adsorption thermodynamics, providing a reliable quantitative route to the true dispersive and Lewis surface energy components. The corrected theory thereby establishes a new standard for characterizing interfacial properties of solids with direct implications for adhesion science, catalysis, nanomaterial design, and pharmaceutical surface engineering.

Extension of the Study to Other Equations of State: Case of the Redlich–Kwong Equation

Several equations of state have been proposed in the literature following the success of the van der Waals cubic equation. The first modification was introduced by Redlich and Kwong, who incorporated thermal effects through the well-known Redlich–Kwong (R–K) equation in order to correct some deficiencies of the van der Waals equation. Later, Soave significantly improved the predictive capability of the R–K equation by introducing a temperature-dependent α-term, which depends on the acentric factor (ω) to account for the nature of the compounds (Soave–Redlich–Kwong, SRK, equation). A major improvement of the SRK equation was achieved in 1976 by Peng and Robinson (P–R equation). They demonstrated that the accuracy of liquid-phase density predictions obtained with the SRK equation decreases with increasing molecular size.

Considering that the temperature range investigated lies between 300 and 500 K and gas adsorption occurs at infinite dilution, the previous study was extended using the R–K equation in both three-dimensional (3D) and two-dimensional (2D) states.

The Redlich–Kwong equation in the 3D state of gas can be written as

$$C_g=\frac{1}{RT}(P+\frac{a}{V(V+b)\sqrt{T}})(1-\frac{b}{V})$$ 30

while the two-dimensional state of gas was given by

$${\sigma }_i=\frac{1}{RT}(\pi +\frac{\alpha (2D)}{\mathrm{\Sigma }(\mathrm{\Sigma }+\beta (2D))\sqrt{T}})(1-\frac{\beta (2D)}{\mathrm{\Sigma }})$$ 31

Therefore, the IGC equation can be written as follows:

$$\mathrm{\Delta }{G}_a^0=-RT\ln \left(\frac{V_n}{Sm}\frac{P_0}{{\pi }_0}\right)-RT\ln \left[(1+\frac{a}{P_0{V}_0(V_0+b)\sqrt{T}})(1-\frac{b}{V_0})\right]+RT\ln \left[(1+\frac{\alpha (2D)}{\mathrm{\Sigma }(\mathrm{\Sigma }+\beta (2D))\sqrt{T}}\frac{1}{{\pi }_0})(1-\frac{\beta (2D)}{\mathrm{\Sigma }})\right]$$ 32

As proved previously, the values of $RT\ln \left[(1+\frac{a}{P_0{V}_0(V_0+b)\sqrt{T}})(1-\frac{b}{V_0})\right]$ can be neglected behind the other terms and for the different adsorbed molecules.

This led to the final IGC equation:

$$\mathrm{\Delta }{G}_a^0=-RT\ln \left(\frac{V_n}{Sm}\frac{P_0}{{\pi }_0}\right)+RT\ln [(1+\frac{\alpha (2D)}{\mathrm{\Sigma }(\mathrm{\Sigma }+\beta )\sqrt{T}}\frac{1}{{\pi }_0})]$$ 33

where Σ = ms representing the surface area of the solid.

The application of the R-K equation to the different solid materials showed a strong correlation with the surface area of solid surfaces. The results showed a deviation between the VDW and R-K equations when the surface area of the materials increased. The relative error percentage of VDW equation with respect to R-K equation was given in Table for several solid materials with different surface areas.

9. Relative Error Percentage on the Free Energy of Adsorbed Solvents on Different Materials as a Function of the Temperature T; Effect of R-K Equation.

Oligoacenaphthylene (Σ = 1.418 m2)
	Temperature T (K)
	343.15	353.15	363.15	373.15	383.15
n-Hexane	18.7%	21.8%	27.8%	32.1%	48.4%
n-Heptane	15.0%	19.1%	22.3%	27.1%	31.5%
n-Octane	12.5%	16.7%	21.5%	25.9%	35.1%
n-Nonane	10.7%	13.9%	15.5%	18.1%	22.5%
n-Decane	8.5%	9.9%	11.3%	13.3%	15.6%
Dichloromethane	10.3%	11.2%	12.1%	13.5%	15.5%
Chloroform	7.6%	8.3%	8.8%	9.9%	10.5%
Ethyl acetate	8.1%	9.5%	10.9%	13.2%	15.3%
Diethyl ether	13.4%	16.5%	18.8%	21.5%	25.4%
Acetone	9.4%	10.0%	11.3%	12.4%	14.6%
Tetrahydrofuran	9.8%	11.0%	12.2%	13.7%	16.1%

Oligoacenaphthylene-p-HPA (Σ = 1.426 m2)
	Temperature T (K)
	343.15	353.15	363.15	373.15	383.15
n-Hexane	9.1%	9.7%	10.7%	12.7%	13.6%
n-Heptane	8.2%	8.5%	9.6%	12.6%	13.5%
n-Octane	7.7%	8.1%	9.3%	10.9%	11.7%
n-Nonane	7.0%	7.6%	8.7%	10.4%	11.8%
n-Decane	6.2%	6.6%	7.5%	9.5%	10.2%
Dichloromethane	8.1%	9.2%	10.6%	11.3%	12.2%
Chloroform	6.2%	7.0%	7.7%	8.3%	8.6%
Ethyl acetate	7.0%	8.5%	9.7%	11.1%	12.4%
Diethyl ether	7.9%	9.3%	10.1%	12.3%	12.9%
Acetone	7.5%	8.3%	8.7%	9.4%	10.5%
Tetrahydrofuran	6.9%	8.1	8.6	9.8	10.9%

Graphene (Σ = 25 m2)
	Temperature T (K)
	323.15	343.15	363.15	383.15
n-Pentane	12.8%	14.9%	17.6%	21.2%
n-Hexane	9.7%	11.2%	13.0%	15.4%
n-Heptane	8.7%	9.9%	11.4%	13.3%
n-Octane	13.6%	15.1%	16.9%	18.9%
Dichloromethane	10.4%	11.5%	12.8%	14.3%
Ethyl acetate	10.5%	11.6%	12.9%	14.4%
Diethyl ether	13.6%	14.9%	16.4%	18.0%
Tetrahydrofuran	12.4%	13.7%	15.3%	17.1%
Acetonitrile	19.2%	21.1%	23.3%	24.6%

N2-doped MWCNTs-CMC (Σ = 47.63 m2)
	Temperature T (K)
	353.15	363.15	373.15	383.15
n-Hexane	19.6%	21.8%	23.3%	24.6%
n-Heptane	17.4%	18.4%	21.0%	23.2%
n-Octane	16.1%	16.9%	19.1%	21.6%
n-Nonane	14.2%	15.5%	17.4%	19.8%
n-Decane	13.3%	14.4%	16.1%	18.0%
Dichloromethane	16.2%	17.3%	19.1%	21.7%
Chloroform	27.7%	29.4%	33.2%	38.6%
Ethyl acetate	13.6%	14.8%	15.6%	17.4%
Diethyl ether	18.6%	20.0%	21.8%	22.7%
Acetone	18.6%	20.3%	22.5%	23.2%
Tetrahydrofuran	13.4%	14.1%	15.9%	17.8%

Silica (Σ = 157.5 m2)
	Temperature T (K)
	323.15	343.15	363.15	383.15
n-Pentane	13.4%	15.5%	17.8%	20.2%
n-Hexane	11.7%	13.4%	15.3%	17.2%
n-Heptane	10.4%	12.0%	13.6%	15.3%
n-Octane	9.4%	10.7%	12.2%	13.8%
n-Nonane	8.5%	9.8%	11.1%	12.6%
Dichloromethane	8.3%	9.2%	9.9%	10.7%
Chloroform	6.1%	6.7%	7.2%	7.8%
Carbon tetrachloride	8.6%	9.6%	10.5%	11.5%
Benzene	11.2%	12.9%	14.7%	16.6%
Ethyl acetate	12.8%	15.0%	17.2%	19.8%
Diethyl ether	5.6%	6.2%	6.9%	7.6%
Acetone	11.4%	13.2%	15.2%	17.3%
Tetrahydrofuran	4.6%	5.2%	5.8%	6.5%
Toluene	6.4%	7.1%	7.8%	8.5%
Acetonitrile	11.9%	13.7%	15.7%	17.8%

Single-crystalline (Σ = 728 m2)
	Temperature T (K)
	323.15	343.15	363.15	383.15
n-Pentane	4.4%	5.5%	7.2%	10.5%
n-Hexane	3.7%	4.4%	5.3%	6.8%
n-Heptane	3.3%	3.8%	4.5%	5.5%
n-Octane	3.0%	3.5%	4.0%	4.7%
Dichloromethane	5.2%	7.3%	11.8%	16.1%
Ethyl acetate	3.7%	4.5%	5.7%	7.5%
Diethyl ether	4.1%	5.1%	6.9%	10.5%
Tetrahydrofuran	3.6%	4.5%	5.8%	8.1%
Acetonitrile	7.8%	11.5	13.3%	15.5%

The results in Table clearly showed the superiority of the Redlich–Kwong equation compared to the van der Waals equation. It was observed that the relative error percentage is less than 25%, while for Σ = 728 m², the error becomes lower than 5% for most solvents and temperatures.

The comparison between the classic and new equations is shown in Figures in the case of adsorption of n-octane and ethyl acetate on different solid surfaces exhibiting lower and higher surface areas.

The variations of the free energy of n-octane and ethyl acetate adsorbed on the surface materials as a function of temperature plotted in Figures showed that the deviation between the various models decreases at higher surface area of solid materials. The results also proved that the van der Waals and Redlich–Kwong equations both correct the classic chromatographic equation with a net advantage for the R-K equation, which gives the best results.

3.

Comparison between the perfect gas equation, van der Waals, and Redlich–Kwong equations in the case of adsorption of n-octane and ethyl acetate on oligoacenaphthylene, N₂-doped MWCNTs/CMC, silica, and single crystalline: (3.a.) Free energy of adsorption of n-octane on oligoacenaphthylene. (3.b.) Free energy of adsorption of ethyl acetate on oligoacenaphthylene. (3.c.) Free energy of adsorption of n-octane on N₂-doped MWCNTs/CMC. (3.d.) Free energy of adsorption of ethyl acetate on N₂-doped MWCNTs/CMC. (3.e.) Free energy of adsorption of n-octane on silica. (3.f.) Free energy of adsorption of ethyl acetate on silica. (3.g.) Free energy of adsorption of n-octane on single crystalline. (3.h.) Free energy of adsorption of ethyl acetate on single crystalline.

Indeed, incorporating real-gas equations of state into the derivation of the fundamental IGC relation systematically improves the thermodynamic consistency of adsorption free energies extracted from retention data. Both van der Waals and Redlich–Kwong corrections reduce the large systematic bias inherent to the classical ideal model, confirming that the failure of the traditional equation originates from an oversimplified state description of the adsorbed phase. Across polar and nonpolar probes, the Redlich–Kwong formulation provides the best agreement with the measured ΔG _a (T) trends, consistent with its more realistic temperature dependence of cohesive attraction. The model-to-model deviation diminishes as the solid surface area increases, indicating that the thermodynamic weight of nonideality corrections decreases in the high-Σ limit where f(T, Σ) → 0. These results establish that real-gas-based corrections are essential for quantitative IGC, with the Redlich–Kwong equation offering a robust and broadly applicable improvement over the van der Waals equation.

Conclusion

This study demonstrates that the classical IGC equation fails to describe the real adsorption thermodynamics. By introducing the deviation function f(T, Σ), explicitly dependent on temperature and specific surface area, we show that the free energy of adsorption and the derived surface energies must be corrected. The results prove that surface energy is not an intrinsic constant of the solid but depends on the solid–solvent coupling. Neglecting f(T, Σ) yields errors that reach several orders of magnitude, especially for small-surface-area materials. The new formulation restores thermodynamic consistency to IGC and establishes a more accurate framework for determining dispersive and acid–base surface energies.

Overall, the introduction of real-gas equations of state establishes a corrected thermodynamic basis for IGC, with the Redlich–Kwong equation emerging as the most reliable formulation for describing adsorption free energies over broad temperature and surface-area ranges.

All data are available in the main text or the Supporting Information.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/prechem.5c00285.

• Tables presenting the values of the different thermodynamic parameters relative to 19 organic solvents adsorbed on 17 solid surfaces, the theoretical, and experimental results concerning the free energy of adsorption, the deviation function, and the relative error percentage using the classic and the new chromatographic equations (PDF)

The author declares no competing financial interest.