Analytic expressions for the isosteric heat of adsorption from adsorption isotherm models and two‐dimensional
 SAFT‐VR
 equation of state

Funding information DAIP-University of Guanajuato (Mexico), Grant/Award Numbers: CIIC 1143/2016, CIIC 348/2019; Royal Academy of Engineering Newton Research Collaboration Program (3)-NRCP1516/1/124 Abstract The isosteric heat of adsorption is an important thermodynamic property used to characterize and optimize adsorption processes. In this work, analytic expressions for isosteric heats of adsorption are derived for a collection of commonly used isotherm models and a two-dimensional molecular equation of state based on the SAFT-VR approach. The use of these expressions is presented with an example of adsorption of nitrous oxide, N2O, on biochar, which is a waste biomass charcoal that exhibits high adsorption potential. The results show that accurate fitting of the adsorption isotherms leads to consistent results obtained with different approaches; however, the predicted isosteric heat of adsorption exhibits strong variations in the regions where experimental data is insufficient such in the region of low pressure/low coverage. Convergence on the prediction of the isosteric heat of adsorption by the different models is only observed in the region where no extrapolation of experimental data is needed.


| INTRODUCTION
Adsorption is a ubiquitous separation technique in science and engineering and, in many cases, is preferred over other alternatives for separation of gases due to its low economic cost compared to separation techniques such as cryogenic distillation and membrane separation that requires low temperatures conditions and pumping at high pressures, respectively. 1 Theoretical understanding of adsorption is challenging due to the many factors that affect the process such as roughness of the adsorbent, pore size and shape distributions, as well as local fluid-fluid and fluid-solid interactions. 2 The isosteric heat of adsorption, also known as differential enthalpy of adsorption, is a valuable thermodynamic property used for the design and optimization of adsorption processes as it provides information about adsorbate-adsorbate and adsorbate-adsorbent interactions. In homogeneous adsorbents, for example, the isosteric heat of adsorption remains constant at low adsorbate loadings indicating that the interactions between adsorbed molecules do not contribute to the adsorption energy. At high adsorbate loading, however, lateral adsorbateadsorbent interactions become significant, thus causing an increase on the isosteric heat of adsorption. 3,4 A different behavior is observed in energetically heterogeneous adsorbents in which the isosteric heat of adsorption decreases as the adsorbate loading increases as molecules tend to adsorb on high-energy sites even at low pressures. 4 The isosteric heat of adsorption for pure components can be obtained from calorimetric measurements or from adsorption isotherms measured at different temperatures. Under this approach, a Clapeyron-type relationship is used to determine the isosteric heat of adsorption Q st : where n is the amount adsorbed, p and T are the pressure and absolute temperature of the system, and R is the universal gas constant. Details for the derivation of Equation (1) can be found elsewhere. 5 Equation (1) is only valid for pure components and perfect gases and has led to some confusion when working with mixtures or high pressure systems. Therefore, Myers 6 suggested to use the concept of differential enthalpy of adsorption, Δ h a i , defined as the partial molar enthalpy of adsorption: where ΔH a is the integral enthalpy of adsorption, h a i and h i are the molar enthalpies of the adsorbed phase and in the perfect gas reference state; f i is the fugacity of component i, and n a i is the amount adsorbed of component i. Other authors have also used the concept of partial molar enthalpies of adsorption to study adsorption of liquids and mixtures. 7 Q st and Δ h a i are the same for pure components at low pressures (where the ideal gas approximation is valid). Analytic expressions for Q st or Δ h a are readily available for pure components using simple isotherm models or two-dimensional (2D) fluids, 5 but not necessarily for those that accurately represent experimental data. In this article, we derive a collection of expressions that can be used to determine Q st , or equivalent Δ h a , for a variety of commonly used adsorption isotherms: extensions of the Langmuir model, [8][9][10][11][12][13][14] virial-type adsorption isotherms, 15, 16 Freundlich, 17 Sips, 18 and Toth 19 isotherms, as well as a 2D equation of state (EoS) based on perturbation theory, within the SAFT-VR approach framework. [20][21][22][23] As an example, we apply the derived expressions to the adsorption of N 2 O on biochar (BC) to compare the results obtained from using common isotherms and the molecular-based 2D-SAFT-VR EoS.

| Langmuir model
Equation (1) is easily applied when the adsorption isotherm is written with the amount adsorbed as independent variable, and the pressure as calculated variable. Take as an example the Langmuir adsorption isotherm given by where n is the amount adsorbed, n sat is the saturation capacity, b is the adsorption affinity, and p is the pressure of the system. The Langmuir isotherm can also be written as: Following the assumptions of the Langmuir model where the saturation capacity n sat is independent of temperature, but the adsorption affinity b depends on temperature, the isosteric heat of adsorption for the Langmuir model can be written as: Various temperature dependences have been suggested for the adsorption affinity. Assuming that the temperature dependence of the adsorption affinity follows an Arrhenius expression 6 then where the pre-exponential factor b 0 is related to the entropy of saturation, and E to the differential enthalpy of adsorption, where it is understood that E < 0. Therefore, the isosteric heat of adsorption for the Langmuir model is simply given by assuming that b 0 is independent of temperature. Formally, the preexponential factor b 0 in the adsorption affinity is a function of temperature, 5,24 and is related to the vibrational partition function of the adsorbed molecule and the ideal gas contribution. In the classical limit, the temperature dependence of the pre-exponential factor can be written as: where h is the Planck's constant, k is the Boltzmann constant, m is the mass of the particle, and ν is the frequency of vibration of atoms adsorbed on the substrate, therefore and the resulting isosteric heat can be written as: The Langmuir model assumes that the saturation capacity n sat is independent of temperature, but adsorption may not take place on well defined sites in practical applications and as a consequence, there is a temperature dependence of the saturation capacity as suggested by Hill. 5 Then, the isosteric heat of adsorption for the Langmuir model with temperature-dependent n sat is be given by A functional form for the temperature dependence of the saturation capacity can be given by where Γ is a measure of the adsorbed phase thermal expansion coefficient. Therefore, the expression for the isosteric heat of adsorption for the modified Langmuir isotherm that follows is:

| Multisite Langmuir model
An analytic expression for the isosteric heat of adsorption is not necessarily straight forward when the adsorption isotherms become more complex. An expression for the isosteric heat of adsorption for the dual-site Langmuir model has been reported, 25 where an explicit expression for the pressure p as a function of the amount adsorbed n is obtained from a quadratic equation and used in Equation (1).
Starting from the dual-site Langmuir equation: the pressure is given as: where: and The expression for calculating the isosteric heat of adsorption assuming that the number of sites is independent of temperature is given by: where dα dT = n sat,1 + n sat,2 −n ð and An alternative approach to obtain the isosteric heat of adsorption is to differentiate implicitly any adsorption isotherm where the pressure appears as independent variable. For the dual-site Langmuir model in Equation (15) with the number of sites independent of temperature, we have: that leads to: resulting in an expression for the isosteric heat of adsorption as: This approach is easily generalized to a multisite Langmuir isotherm, with i number different types of adsorption sites 10 : where It is easy to show that Equation (26) reduces to Equation (23) for the case when only two types of sites are present. Then, the isosteric heat of adsorption for the multisite-Langmuir model with number of sites independent of temperature is given by: A more general case could also consider that the number of sites or maximum amount adsorbed depends on temperature, as shown in Equation (13). Then, the isosteric heat of adsorption becomes: or in terms of the adsorbed phase expansion coefficient: A generalization of the Langmuir model 11 captures the variation in the asymptotic amount adsorbed at different temperatures by including the exponents s and t in the following equation: where u, η, and n sat are constants and b has the temperature dependence given in Equation (6). Then, the isosteric heat of adsorption is given by: assuming that both s and t are independent of temperature.

| Virial equation
The virial equation to describe adsorption of fluids in porous materials can be written as a polynomial expansion of the form 15 : where the Henry's constant b is given as lnb = − c 0 and the temperature dependence for the other constants has been suggested to have the following form: where a i and b i are adjustable parameters.
For this isotherm, the isosteric heat of adsorption is given as: The virial isotherm given in Equation (32) is unable to describe adsorption isotherms with well defined saturation capacities. An alternative formulation using a Langmuir type saturation capacity is sometimes used 16 : where the isosteric heat of adsorption is given by:

| Freundlich and Sips models
Freundlich proposed a simple and empirical model with two adjustable parameters that is often used to describe adsorption of organic vapors on activated carbon (AC) and other heterogeneous surfaces, that is, materials with different types of adsorption sites with different energies of adsorption. 17 The Freundlich isotherm is given by where b is the Freundlich constant that denotes that adsorption capacity of the substrate and t is an empirical exponent that indicates the adsorption intensity of the substrate.
The isosteric heat of adsorption for this isotherm, assuming that both adjustable parameters are temperature dependent is given as: An extension of the Freundlich model, proposed by Sips, 18 is often referred to as the Langmuir-Freundlich model, where the adsorption isotherm is given by: and the isosteric heats of adsorption assuming that all adjustable parameters are temperature dependent is given by:

| Toth
The Freundlich and Sips adsorption isotherm models can describe well experimental data but they have an important disadvantage: the slope of the adsorption isotherm at the limit of zero pressure is not defined.
The isotherm model proposed by Toth 19 overcomes this problem and is given by where n sat , b, and t are adjustable parameters. The isosteric heat of adsorption for this model is given by:

| BET
The BET equation 26 where v is the volume adsorbed, v m is the volume of the monolayer, p is the pressure, p 0 is the saturation pressure at the temperature of the isotherm, and c is an adjustable parameter, related to the difference in the energy of adsorption of the first layer and the other layers.
Assuming that v m and c are independent of temperature, the isosteric heat of adsorption is given by: A more general expression can be obtained when considering that v m and c are temperature dependent, then the isosteric heat of adsorption is given as: where and

| 2D-SAFT-VR EoS
Most adsorption models such as the Langmuir and the BET isotherms use the concept of adsorption sites in which equilibrium is described as the balance between the energy of the molecules adsorbed on these sticky sites and the gain of translational entropy from the desorption of the molecules from the substrate. 27 An alternative description of adsorption is introduced from the use of 2D molecularbased equations of state where the adsorbed phase is represented as a 2D fluid in equilibrium with a 3D bulk phase. 28,29 Within this approximation and the framework of the Barker and Henderson perturbation theory, 30 it is possible to model the adsorption of particles interacting via a square-well (SW) pair potential. 31  The 2D-SAFT-VR approach considers that bulk and adsorbed phases are formed by spherical particles of diameter σ interacting via SW potentials with energy well depths and attractive ranges parameters (ϵ bulk , λ bulk ) and (ϵ ads , λ ads ) for both phases, respectively; interactions between bulk-adsorbed fluid particles are neglected whereas surface-adsorbed-particles interaction is also modeled with an SW potential with parameters (ϵ w , λ w ). The expression for the isosteric heat of the 2D-SAFT-VR EoS is given by: where N 0 is Avogadro's number, γ and g hd are the 2D hard-disks packing fraction and radial distribution function, respectively, and γ eff and Φ are parameters of the mean-attractive energy of the adsorbed particles. Details of the theory and the derivation of Equation (48)  We begin our discussion with the use of the Langmuir isotherm  Table 1 for pressures in the range 0.03 to 0.66 atm. The values of these parameters are the same as those originally reported by Cornelissen et al. 40 As expected, both n sat and b decrease as the temperature decreases indicating a reduction of the adsorption capacity of the material. The Langmuir isotherms using the parameters reported in Table 1 are shown in Figure 1, where it is observed that the fitting of the model to the experimental adsorption data is good, particularly at high temperatures. The error δ X on the fitting of property X for all the isothermal models presented in this work is quantified using the average absolute deviation given by where X e and X c are the experimental and computed values of variable X, respectively, and N p is the number of experimental points. For the Langmuir isotherm, the error on the fitting of the amount adsorbed δ n is lower than 3% for all temperatures reported in the range 263-313 K.
A more general form of representing experimental data for different systems and temperatures is by obtaining the dependence of the Langmuir parameters as function of temperature for a specific system.
The affinity parameter, for example, can be represented using an Arrhenius-type model given in Equation (6), in which the preexponential factor b 0 as well as activation energy E can be determined using experimental data at different temperatures. For the case of the saturation capacity n s , the temperature dependence is obtained through the calculation of Γ defined in Equation (13). Using again the experimental data reported by Cornelissen et al. 40 for different types of ACs and BCs, the fitted values for b 0 , E, and Γ are reported in Table 2.
The reported experimental data were also fitted with other models described in this work: first-and second-order virial, Freundlich, Sips, Toth, and dual site Langmuir isotherms. The results for these models to represent the adsorption isotherms of N 2 O on BC-60 are presented in Figure 1. The constants obtained from the T A B L E 1 Langmuir isotherm parameters, n sat and b, for the representation of the adsorption of N 2 O on biochar BC-60 at different temperatures reported by Cornelissen et al. 40 The fitting error δ n on the amount adsorbed n is also reported   (6)). Γ corresponds to the adsorbed phase thermal expansion coefficient defined in Equation (13). Experimental adsorption isotherms used for the fitting are taken from Cornelissen et al 40 Material T A B L E 3 Parameters for the virial isotherm truncated to first order (two constants) for the adsorption of N 2 O on BC-60. Experimental adsorption isotherms used for the fitting are taken from Cornelissen et al. 40 The fitting error δ p on the pressure p is also reported. The gradients of the parameters −lnb and c 1 with respect to the temperature T evaluated at T = 293.15 K are reported in the last row. The units of the parameter b are (cm 3 g −1 atm −1 )  40 The fitting error δ p on the pressure p is also reported. The gradients of the parameters −lnb, c 1 , and c 2 with respect to the temperature T evaluated at T = 293.15 K are reported in the last row. The units of the parameter b are (cm 3 g −1 atm −1 )  40 The fitting error δ n on the amount adsorbed n is also reported. The gradients of the parameters b and t with respect to the temperature T evaluated at T = 293.15 K are reported in the last row All the isotherm models previously discussed have in common that the adsorption equilibrium is described as the balance between the energetic contribution due to molecules adsorbed in active sites and the gain in entropy from the molecules escaping from the sub- strate. An alternative interpretation to the adsorption of fluids on solid substrates corresponds to the equilibrium of a 2D fluid phase, representing the adsorbed phase, with a 3D bulk phase. In this context, the so-called 2D-SAFT-VR EoS has been previously reported to describe the adsorption of gases on solids. Here, this molecular-based EoS has been used to represent the adsorption of N 2 O on BC-60. The SW parameters in the 2D-SAFT-VR equation for bulk N 2 O gas (ϵ bulk , λ bulk , and σ) have been obtained from fitting to the critical values of density, temperature and pressure, according to the data reported by NIST. 45 The SW bulk parameters are very similar to those used by other authors with similar molecular-based equations of state. 46 The adsorbed phase parameters have been obtained following the procedure used in previous 2D-SAFT-VR studies, 22,23 assuming that the size of the particles and the energy-well depth of the SW interaction are not modified by the adsorption process, that is, σ ads = σ bulk and ϵ ads = ϵ bulk , respectively. The effect of the substrate on the adsorbed fluid is only taken into account in the range of the SW potential of the T A B L E 8 Parameters for the dual-site Langmuir isotherm for the adsorption of N 2 O on BC-60. Experimental adsorption isotherms used for the fitting are taken from Cornelissen et al. 40 The fitting error δ n on the amount adsorbed n is also reported. The gradients of the parameters n sat,1 , b 1 , n sat,2 , and b 2 with respect to the temperature T evaluated at T = 293.15 K are reported in the last row. The units of the parameter n sat,1 and n sat,2 are (cm 3 g −1 ), while the units of the parameters b 1 and b 2 are (atm −1 ) T A B L E 6 Parameters for the Sips isotherm for the adsorption of N 2 O on BC-60. Experimental adsorption isotherms used for the fitting are taken from Cornelissen et al. 40 The fitting error δ n on the amount adsorbed n is also reported. The gradients of the parameters n sat , b, and t with respect to the temperature T evaluated at T = 293.15 K are reported in the last row  40 The fitting error δ n on the amount adsorbed n is also reported. The gradients of the parameters n sat , b, and t with respect to the temperature T evaluated at T = 293.15 K are reported in the last row  ratios of the Ising model 47 and the computer simulated values for the Lennard-Jones system. 48 Since the low-limit of the isosteric heat is the particle-surface SW energy ϵ w was taken to reproduce Q st in this limit, according to the experimental value at a temperature of 293.15 K (see Table 3 in Reference 40). The remaining parameters, the range of the surface-particles attractive interaction λ w , that also gives an estimation of the size of the adsorbed layer, and the surface area S were fitted to reproduce all the selected adsorption isotherms.
The surface area was also fitted and reproduced according to the

| CONCLUSIONS
In this work, we have presented a collection of equations to calculate the isosteric heat of adsorption for gases derived from some of the most common models for adsorption. The use of simple isotherm models is very common in the literature, despite of having evidence that the experimental data does not follow the assumptions of the selected model. More complex isotherms can adjust better experimental data, but often require more parameters to fit, as has been reviewed by Foo and Hameed. 49 The use of a more complex approach like the 2D-SAFT-VR EoS becomes attractive because less adjustable parameters are required than the simpler isotherm models, and also because it is possible to improve the molecular description through a more realistic intermolecular interaction, as recently reported using a Mie intermolecular potential model. 50 It is impossible to say which model discussed in this work is better or more accurate to extrapolate the isosteric heats of adsorption, but this article highlights that care must be taken in the extrapolations of the heats of adsorption, even if the model fits well the adsorption isotherms. As our results demonstrate very different behavior, even some unrealistic behavior, is observed for both low and high coverage since these regions have been extrapolated.

ACKNOWLEDGMENTS
The authors are grateful for the support of the Royal Academy of

APPENDIX
The 2D-SAFT-VR method 21,22 considers that molecules in both the adsorbed and bulk phases interact via SW potentials u sw (r;σ, ϵ, λ) where σ is the particle's diameter, ϵ is the energy well-depth, and λ is the attractive range of the potential. The model assumes that the bulk and adsorbed particles have the same size σ, but the attractive interaction is modified due to the effect of the substrate, and the values of the ϵ and λ potential parameters corresponding to the bulk and adsorbed phases are different. We will denote by (ϵ bulk , λ bulk ) and (ϵ ads , λ ads ) the bulk and adsorbed SW particle-particle interactions parameters, respectively.
Typically, ϵ ads ≈ 0.8ϵ bulk and λ ads < λ bulk in order to reproduce the experimental ratio of the critical temperatures of the adsorbed and bulk fluid phases R c = T ads c =T bulk c ≈0:4 . 22,29,51 However, just in order to reduce the number of molecular parameters that requires the model, we can consider the effect of the substrate into the adsorbed particle-particle interaction only in the range of the potential and to assume that ϵ ads = ϵ bulk . This approximation has given accurate predictions of adsorption isotherms of CO 2 in AC with the Mie intermolecular potential 50 and we have used this approximation in this work.
The interaction potential u pw exerted on the adsorbed particles by a planar structureless surface is also given by an SW potential with parameters (ϵ w , λ w ), written as a function of the orthogonal coordinate distance z between the particle and the surface. If z = 0 is set at the position of contact of the particle with the surface then u pw is given by: The Helmholtz free energy of the adsorbed fluid can be written as. 22,23 A ads where η ads is the number of moles in the adsorbed phase, λ B is the de Broglie's thermal wavelength, and N 0 is Avogadro's number. Details of the Helmholtz free energy of the bulk fluid are given in References 22 and 23.
The thermodynamic properties of the adsorbed phase are determined by the equilibrium condition where μ bulk = ∂A bulk /∂η bulk and μ ads = ∂A ads /∂η ads are the chemical potentials of the bulk and adsorbed phases, respectively. From this equation, it is then possible to give the bulk pressure, p, at low values of η bulk , where ρ ads = η ads /S and Δμ 2D are the molar density and excess chemical potential of the adsorbed fluid, respectively, and S is the surface area. According to Equations (1) and (A5), the isosteric heat of the 2D-SAFT-VR is given by For nonassociating fluids, the first-order-perturbation term for Δμ 2D is given by 23 : where α vdW is the van der Waals energy constant, γ is the 2D packing fraction γ = N 0 ρ ads πσ 2 /4, g hd (σ, γ eff ) is the contact value of the harddisks radial distribution function given in terms on an effective packing fraction γ eff , and Φ(γ eff ) is a function determined by g hd and its first density derivative. Following previous work, we use the Henderson harddisks EoS, 52 and with d 1 = 1:4215− 0:405625λ ads − 0:03869981λ 2 ads ðA10Þ d 2 = 1:5582−1:89768λ ads + 0:405215λ 2 ads ðA11Þ and α vdW = πϵ ads σ 2 λ 2 ads −1 The corresponding expression for Φ(γ eff ) is The expressions for the augmented vdW EoS can be obtained from Equation (A7) assuming g hd (σ, γ eff ) = 1, and consequently Φ(γ eff ) = 1.