The Temperature Dependence of the Langmuir Adsorption Model for a Single-Site Metal– Organic Framework Dalton Compton, Nan-Chieh Chiu, Kyriakos C. Stylianou, Nicholas P. Stadie This document is the Accepted Manuscript version of a Published Work that appeared in final form in Langmuir, copyright ©, [include copyright notice from the published article] after peer review and technical editing by the publisher. To access the final edited and published work see [insert ACS Articles on Request author-directed link to Published Work, see ACS Articles on Request ].” Accessibility Disclaimer: For a more accessible version of this document, please submit an accessibility request form through the Montana State University Library website. Made available through Montana State University’s ScholarWorks B_ MONTANA STATE UNIVERSITY LIBRARY This document is confidential and is proprietary to the American Chemical Society and its authors. Do not copy or disclose without written permission. If you have received this item in error, notify the sender and delete all copies. The Temperature Dependence of the Langmuir Adsorption Model for a Single-Site Metal-Organic Framework Journal: Langmuir Manuscript ID la-2025-007577.R2 Manuscript Type: Article Date Submitted by the Author: n/a Complete List of Authors: Compton, Dalton; Montana State University Bozeman, Chemistry Chiu, Nan-Chieh; Oregon State University, Materials Discovery Laboratory (MaD Lab), Department of Chemistry Stylianou, Kyriakos; Oregon State University, Materials Discovery Laboratory, Department of Chemistry Stadie, Nicholas; Montana State University System, Chemistry and Biochemistry ACS Paragon Plus Environment Langmuir SCHOLARONE"' Manuscripts The Temperature Dependence of the Langmuir Adsorption Model for a Single-Site Metal-Organic Framework Dalton Compton† , Nan-Chieh Chiu‡ , Kyriakos C. Stylianou‡ , Nicholas P. Stadie†* † Department of Chemistry and Biochemistry, Montana State University, Bozeman, Montana, 59717, United States ‡ Materials Discovery Laboratory (MaD Lab), Department of Chemistry, Oregon State University, Corvallis, Oregon, 97331, United States *Email: nstadie@montana.edu ABSTRACT: The single-site Langmuir adsorption model, also known as the Langmuir isotherm equation, is one of the simplest possible descriptions of adsorption phenomena, and yet finds widespread applicability across a range of disciplines. In its simplest form, it is deployed to treat adsorption equilibria at constant temperature (i.e., along isotherms); however, at the heart of its derivation is a more general class of models that each incorporate an explicit temperature dependence, subject to assumptions about the spatial/translational degrees of freedom of the adsorbed species. In this work, measurements of the temperature dependence of supercritical adsorption of H2 on a single-site metal-organic framework (MOF) are presented, and fitted using a range of Langmuir models with distinct treatments of degrees of freedom in the adsorbed phase. Surprisingly, all of the models can be used to adequately represent the measured data (to within 0.0003 mmol g-1 per point), despite yielding significantly different values for binding energy and the temperature dependence of the isosteric enthalpy of adsorption (i.e., the isosteric heat, qst). However, a critical finding of this work is that the mean-temperature isosteric enthalpy of adsorption remains consistent across all models within experimental error (±0.1% or <0.1 kJ mol-1), highlighting its reliability for evaluating adsorption thermodynamics. Keywords: physisorption, metal-organic framework, isosteric enthalpy of adsorption, binding energy, thermodynamics, hydrogen, porous materials, energy storage Page 1 of 21 ACS Paragon Plus Environment Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 mailto:nstadie@montana.edu INTRODUCTION: In the theory of adsorption, defined as the densification of one phase at the interface (or phase boundary) of another, the equations derived by Irving Langmuir in 1916-1918 have served as foundational models that can accurately describe certain simple systems based on an atomistic picture of site-by-site and layer- by-layer interactions.1-3 The simplest Langmuir-type adsorption model, also referred to as the single-site Langmuir (SL) equation, is constructed based on a minimalist description of adsorption of a fluid or dissolved species at the surface of a rigid adsorbent containing a fixed number of adsorption sites, where three important assumptions are held: i. every adsorption site is distinguishable and identical, ii. every adsorption site can host up to one adsorbate molecule, and iii. there are no interactions between adsorbate molecules, in the bulk fluid nor in the adsorbed phase. When the bulk fluid is a pure ideal gas and the adsorbent is a solid, the SL equation is derived in terms of the pressure of the gas: 𝑛𝑛𝑎𝑎 [𝑃𝑃] = 𝑛𝑛𝑠𝑠 𝐾𝐾𝑃𝑃 1 + 𝐾𝐾𝑃𝑃 Eq. 1 where 𝑛𝑛𝑎𝑎 is the amount adsorbed (typically per unit mass of the solid adsorbent), 𝑛𝑛𝑠𝑠 is the maximum amount adsorbed or total number of adsorption sites (also, typically per unit mass of solid adsorbent), 𝑃𝑃 is the pressure of the ideal gas, and 𝐾𝐾 is the so-called Langmuir “constant.” This equation has variously been referred to as the single-site Langmuir model, the monolayer Langmuir model, or simply the Langmuir isotherm equation. However, it is important to note that this was not the only equation to be proposed by Langmuir (e.g., Langmuir also developed a multilayer variant in 1918 that was later improved upon and has become widely known as the BET model).3,4 Beyond Langmuir’s contributions, other adsorption equations have been proposed and develobed.5 A distinct advantage of the Langmuir- type models is that they can be derived from first principles using statistical mechanics,6 enabling a direct method for determining the inherent temperature dependence of the Langmuir equation. All in all, the SL equation is a practical starting point for experimental adsorption analysis, from simple data interpolation to determination of key metrics such as the number of adsorption sites per unit mass (which can be directly related to the gas-accessible surface area, when it can be assured that only a monolayer of adsorbate is formed). The SL equation and the more complex Langmuir models related to it are widely used for gas adsorption analysis, particularly to assist in the determination of thermodynamic properties such as the isosteric enthalpy of adsorption.7 The latter analysis requires the measurement of multiple adsorption isotherms at different temperatures, fitting the collection of isotherms to a single model, and extracting relevant thermodynamic properties. Given this process, the inherent temperature dependence of the model plays a critical role. However, this topic is rarely explored. Instead, many researchers simply fit each of their measured isotherms independently with a separate Langmuir equation for each one, rather than using a global model that fits all of the measured isotherms to one set of fitting parameters. This results in a large superset of resulting parameters that require additional analysis to reconcile into meaningful materials properties. This raises an important question: why do researchers avoid the use of an explicit temperature- dependent model? One explanation is that not all adsorption systems, even those that follow the SL model very closely, exhibit the same temperature dependence. In fact, the temperature dependence can vary across different condition regimes and is strongly influenced by the inherent properties of the system8 . Furthermore, Page 2 of 21 ACS Paragon Plus Environment Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 multiple atomistic pictures of the adsorption site can lead to the same temperature dependence but for different reasons, complicating the interpretation of the T-dependence. Ultimately, an entire subset of adsorption equations underlies the above general SL equation, each attributing a unique temperature dependence to the adsorption system. While efforts have been made to systematically organize these equations,9,10 a comprehensive framework remains lacking. In this work, we employ a simple, empirical approach to elucidate the T-dependence of the SL equation for a model adsorption system by fitting a diverse range of equations to an experimental data set in order to better understand which model is most accurate. We present a systematic list of physically justifiable models, each rooted in a simple microscopic interpretation of the adsorption site. All models satisfy the above general Langmuir equation while providing insights into the temperature dependence of K according to: 𝑛𝑛𝑎𝑎 [𝑇𝑇 , 𝑃𝑃] = 𝑛𝑛𝑠𝑠 𝐾𝐾 [𝑇𝑇 ]𝑃𝑃 1 + 𝐾𝐾 [𝑇𝑇 ]𝑃𝑃 Eq. 2 The results of this systematic approach reveal new insights into the role of the T-dependence of the Langmuir model in determining thermodynamics properties of experimental adsorption systems such as the binding strength (or binding energy) and isosteric heat of adsorption. THEORY: Gibbs Excess Adsorption All experimental adsorption measurements have the simple issue of not being able to directly assess the “absolute” (actual) adsorbed amount. J. Willard Gibbs realized this and hence the experimental quantity measured is referred to as the Gibbs “excess” amount: the amount beyond what would be expected in the same volume if no adsorbent were present.11 The difference between the absolute and excess amounts is significant at higher pressures and colder temperatures, where the bulk adsorbate fluid density approaches that of the adsorbed phase. The relationship between the excess and absolute amounts adsorbed is: 𝑛𝑛𝑒𝑒 = 𝑛𝑛𝑎𝑎 − 𝜌𝜌𝑔𝑔 𝑉𝑉𝑎𝑎 Eq. 3 where 𝑛𝑛𝑒𝑒 is the experimentally measured excess adsorbed amount, 𝑛𝑛𝑎𝑎 is the actual or “absolute” adsorbed amount, 𝜌𝜌𝑔𝑔 is the density of the bulk adsorbate fluid, and 𝑉𝑉𝑎𝑎 is the volume of the adsorbed phase. It is usually important to distinguish between 𝑛𝑛𝑒𝑒 and 𝑛𝑛𝑎𝑎 since the difference could be significant. However, when the conditions of adsorption are sufficiently dilute in the gas phase (i.e., 𝜌𝜌𝑔𝑔 ≈ 0 or 𝜌𝜌𝑎𝑎 ≫ 𝜌𝜌𝑔𝑔 ), it is convenient to be able to ignore the role of 𝑉𝑉𝑎𝑎 since it is not experimentally measurable, and to thereby assume that 𝑛𝑛𝑒𝑒 ≈ 𝑛𝑛𝑎𝑎 . In order to assess whether the difference between the experimentally measured adsorbed amount and the actual adsorbed amount is significant, a simple test can be performed. If the crystals of solid (porous) adsorbent are reasonably large, an approximate upper limit of 𝑉𝑉𝑎𝑎 is the total pore volume, 𝑉𝑉𝑡𝑡𝑡𝑡 𝑡𝑡 . By multiplying 𝑉𝑉𝑡𝑡𝑡𝑡 𝑡𝑡 and the maximum gas-phase density of the adsorbate under the conditions explored, a maximum correction term to the absolute adsorbed amount can be approximated (see Supporting Information). For the experimental data collected in this work, that value was determined to be within the experimental error and hence the correction term is negligible under all conditions and 𝑛𝑛𝑒𝑒 ≈ 𝑛𝑛𝑎𝑎 . Henceforth, the raw (as-measured) excess adsorption equilibria will be treated as absolute adsorption quantities for all further analysis herein. Page 3 of 21 ACS Paragon Plus Environment Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Langmuir Theory Following Langmuir’s approach, the three measured quantities at each point of adsorption equilibrium (adsorbed amount, 𝑛𝑛𝑎𝑎 , pressure, 𝑃𝑃 , and temperature, 𝑇𝑇 ) are related by a collection of fixed parameters that describe the nature of the adsorbent surface. Once known, these parameters can then be used to derive any desired thermodynamic information related to the experimental adsorption system. The simplest possible case is a surface comprising a collection of identical and independent binding sites, each hosting a single adsorbate molecule, giving rise to the general SL adsorption model given by Equation 2. A simpler form defines the concept of fractional site occupancy, 𝜃𝜃, as: 𝜃𝜃 [𝑇𝑇 , 𝑃𝑃 ] = 𝑛𝑛𝑎𝑎 𝑛𝑛𝑎𝑎 + 𝑛𝑛𝑒𝑒𝑒𝑒𝑒𝑒 𝑡𝑡𝑒𝑒 = 𝑛𝑛𝑎𝑎 𝑛𝑛𝑠𝑠 = 𝐾𝐾 [𝑇𝑇 ] 𝑃𝑃 1 + 𝐾𝐾 [𝑇𝑇 ] 𝑃𝑃 Eq. 4 The Langmuir constant, 𝐾𝐾 [𝑇𝑇 ], has both an exponential “Boltzmann factor” term (related to the energy of adsorption, which is identical at each site, 𝜀𝜀 )5,6 as well as a possible temperature dependence in the pre- exponential term, referred to herein as 𝐴𝐴[𝑇𝑇 ]: 𝐾𝐾 [𝑇𝑇 ] = 𝐴𝐴 [𝑇𝑇 ] 𝑒𝑒 − 𝜀𝜀 𝑅𝑅𝑅𝑅 Eq. 5 where 𝑅𝑅 is the gas constant. Langmuir Pre-Factor The precise form of 𝐴𝐴[𝑇𝑇 ] can be derived using statistical mechanics in the grand canonical formalism12 and varies depending on the treatment of the degrees of freedom of the adsorbate on the adsorption site. For simplicity, we briefly treat two extreme cases (all intermediate cases are thoroughly treated in the Supporting Information): “fixed” adsorption (where the adsorbate loses all translational degrees of freedom upon adsorption) and the “Einstein crystal” adsorbed phase (where each adsorbate is bound to its site by harmonic oscillator type bonds in all three spatial dimensions). In the former case, the final result for the pre-factor is: 𝐴𝐴𝐹𝐹𝐹𝐹 𝐹𝐹 [𝑇𝑇 ] = ℎ3 (2𝜋𝜋 𝑚𝑚� )1.5 (𝑘𝑘𝐵𝐵 𝑇𝑇 )2.5 ∝ 𝑇𝑇 −2.5 Eq. 6 where ℎ is Planck’s constant, 𝑚𝑚� is the mass of the adsorbate, and 𝑘𝑘 𝐵𝐵 is Boltzmann’s constant. This form of 𝐴𝐴 [𝑇𝑇 ] does not contain any “loose” fitting parameters and so is simply a constant for a given adsorbate at a given temperature. The temperature dependence is significant: ∝ 𝑇𝑇 −2.5 . Hence this model is also referred to as “model −2.5”. On the other hand, in the “Einstein crystal” case (the other extreme of cases investigated in this work), the final result for the pre-factor is: 𝐴𝐴𝐸𝐸 𝐹𝐹 𝐸𝐸 [𝑇𝑇 ] = (2𝜋𝜋 )1.5 (𝑘𝑘𝐵𝐵 𝑇𝑇 )0.5 𝑚𝑚� 1.5 𝜔𝜔𝑠𝑠 3 ∝ 𝑇𝑇 +0.5 Eq. 7 where 𝜔𝜔𝑠𝑠 is the natural frequency of the harmonic oscillation (taken to be identical in all three spatial dimensions), a “loose” fitting parameter that is temperature invariant but must be determined experimentally for a given system. The temperature dependence is significantly less and inversely correlated compared to the fixed model: ∝ 𝑇𝑇 +0.5 . Hence this model is also referred to as “model +0.5”. A third important case is where the adsorbate retains three translational degrees of freedom upon adsorption but only two of which are harmonic oscillator type motion, while the third is constrained by an Page 4 of 21 ACS Paragon Plus Environment Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 infinite square-well potential. This case does not have an easily justifiable physical picture, but has an important outcome for the pre-exponential factor (see Supporting Information for more details): 𝐴𝐴0 [𝑇𝑇 ] = 2𝜋𝜋 𝐿𝐿𝑠𝑠 𝑚𝑚� 𝜔𝜔𝑠𝑠 2 ∝ 𝑇𝑇 0 Eq. 8 where 𝜔𝜔𝑠𝑠 is the natural frequency of the harmonic oscillation (taken to be identical in both of the first two dimensions of freedom) and 𝐿𝐿𝑠𝑠 is the length of the potential well in the third dimension of freedom. This form of 𝐴𝐴 [𝑇𝑇 ] is independent of temperature: ∝ 𝑇𝑇 0 . Hence this model is also referred to as “model 0”. Different physical pictures and the corresponding degrees of freedom of each lead to different temperature dependencies of 𝐴𝐴 [𝑇𝑇 ] at intervals of 𝑇𝑇 0.5 from 𝑇𝑇 −2.5 to 𝑇𝑇 +0.5 . Likewise, each leads to a different high-temperature heat capacity of the adsorbed phase, as expected based on the equipartition theorem.12 All such models are simplistic in nature, much like the Einstein model of monatomic crystals, and may not be realistic for any experimental system; nevertheless, they provide a toolbox for exploring the effects of the temperature-dependent pre-factor on the resulting Langmuir adsorption model. The explicit forms of 𝐴𝐴 [𝑇𝑇 ] for each case explored herein are given in the Supporting Information (Table S1) and the properties of a subset of highest importance are summarized in Table 1. Table 1. Six important adsorption models explored herein, and their temperature dependence of the pre- exponential factor in the Langmuir constant, high-temperature limit heat capacity, and number of degrees of freedom. Further details are provided in Table S1. Physical Picture Adsorption Site Partition Function (x × y × z) Adsorbate Degrees of Freedom High T Heat Capacity (kBT) 𝐴𝐴 [𝑇𝑇 ] Fixed DF × DF × DF 0 0 + 0 + 0 = 0 𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐 𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐 𝑇𝑇 2.5 1D Ideal Gas SP × DF × DF 1 0.5 + 0 + 0 = 0.5 𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐 𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐 𝑇𝑇 2 2D Ideal Gas SP × SP × DF 2 0.5 + 0.5 + 0 = 1 𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐 𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐 𝑇𝑇 1.5 3D Ideal Gas SP × SP × SP 3 0.5 + 0.5 + 0.5 = 1.5 𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐 𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐 𝑇𝑇 1 2D Lattice Gas QP × QP × DF 2 1 + 1 + 0 = 2 𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐 𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐 𝑇𝑇 0.5 (no physical picture) QP × QP × SP 3 1 + 1 + 0.5 = 2.5 𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐 𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐 Note: potential function descriptions along each spatial dimension (x, y, or z) are abbreviated as: DF for delta function, SP for infinite square-well potential, and QP for quadratic potential. Page 5 of 21 ACS Paragon Plus Environment Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 EXPERIMENTAL METHODS: Materials Synthesis A metal-organic framework (MOF) of composition [Ni3(pzdc)2(ade)2(H2O)4]·2.18 H2O (where “pzdc” is pyrazole-3,5-dicarboxylic acid and “ade” is adenine) was synthesized according to the previously reported procedure.13 The material was degassed under high vacuum at 130 ˚C for 12 h to obtain the dried composition of Ni3(pzdc)2(ade)2(H2O)4 prior to further analyses (referred to herein simply as “MOF”, shown in Figure 1); the MOF powder is sky blue prior to activation and turns lavender when fully dried. The phase purity of the sample was confirmed using powder X-ray diffraction (Figure S1). Materials Characterization Powder X-ray diffraction was measured under ambient conditions using a benchtop X-ray diffractometer (D2 Phaser, Bruker Corp.) with Cu Kα1,2 radiation in reflection geometry. Nitrogen (99.999%) adsorption/desorption uptake was measured using an automated Sieverts apparatus (3Flex, Micromeritics Corp.); the sample (dry mass: 0.344 g, skeletal density: 2.5 g mL-1) was held in a liquid nitrogen bath (the boiling temperature of N2 is 75.9 K in Bozeman, Montana). The surface area was estimated using the Brunauer-Emmett-Teller (BET) model, employing the Rouquerol consistency criteria.14 The pore size distribution was determined using a non-local density functional theory (NLDFT) slit pore model (implemented using Micromeritics MicroActive software). Hydrogen Adsorption Measurements Hydrogen (99.9999%) adsorption/desorption uptake was measured using an automated Sieverts apparatus (3Flex, Micromeritics Corp.). The sample (dry mass: 0.344 g, skeletal density: 2.5 g mL-1) was held in the cold well of a closed-cycle helium refrigerated cryostat (CH-104, ColdEdge Technologies). The sample temperature was measured using a silicon diode (DT-670C, Lake Shore Cryotronics) and calibrated (indicating an offset of 1.4 K) using boiling liquid nitrogen (75.9 K) as a reference (Figure S2). Structure Analysis Structural modeling of the MOF pore network was performed using two dedicated software packages: CrystalMaker (v. 10.8.2) and Zeo++ (v. 0.3). The probe radius used to determine the accessible volume was 1.2 Å. Page 6 of 21 ACS Paragon Plus Environment Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 RESULTS AND DISCUSSION: Model System Selection To properly investigate the temperature dependence of the Langmuir model, it is essential to use a homogeneous adsorption system that fulfills the fundamental assumptions inherent to the SL equation. Hydrogen (H2) is an ideal candidate for this purpose, as it is one of the weakest interacting molecular adsorbates (with the exception of helium, which presents significant experimental challenges since it is used as the de facto free space probe in most laboratories). This ensures the smallest perturbation to the porous adsorbent structure upon binding, satisfying a critical aspect of Langmuir’s theory that the underlying structure of the adsorbent remains unchanged during binding. The ideal adsorbent for such studies must also exhibit rigidity, stability, and lack of any strong (chemisorption) binding sites. Most importantly the ideal adsorbent material should possess distinct, periodic, and disparate binding sites that validate the key assumptions of distinguishable, identical, and non-interacting sites, respectively. A crystalline MOF with one binding site per cage and per unit cell is a desirable adsorbent for applications of the SL equation. This material should have narrow pores to prevent multilayer adsorption and disparate binding sites to prevent cooperative binding. A porous material is particularly desirable since it provides a high adsorption capacity relative to the mass of the adsorbent, enhancing the signal to noise ratio in the measurements. Previous reports of Xe adsorption on SBMOF-1 meet these criteria,15 but Xe is significantly more polarizable than H2. Many traditional porous adsorbents fail to qualify as model materials for H2 due to the heterogeneity of the surface toward such a small molecular adsorbate. For example, even platinum, a classic surface for H2 adsorption, exhibits heterogeneity with two distinct sites depending on the facet exposed.16 In this work, a nickel-based MOF with ~7 Å pores and ~200 m2 g-1 surface area (Ni3(pzdc)2(ade)2(H2O)4) was identified as an ideal porous material that satisfies the assumptions of the SL equation toward H2 adsorption.13 The MOF features distinct, homogeneous binding sites that can accommodate a single H2 molecule per site (Figure 1). Importantly, the adsorption mechanism is purely physisorptive, with minimal contribution from the Ni metal centers.17 The synthesis and characterization of the MOF were performed as previously reported, and a homogeneous pore-size distribution was confirmed (Figure 1d). Figure 1. (a-c) Crystal structure of the single-site MOF (Ni3(pzdc)2(ade)2(H2O)4) showing the one- dimensional channels in yellow that are perpendicular to the one-dimensional Ni-pzdc-ade chains. (d) N2 adsorption/desorption isotherms at ~77 K showing its NLDFT pore size of ~7 Å and BET surface area of 195 m2 g-1 . Page 7 of 21 ACS Paragon Plus Environment Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 a b C d 2.0 00000 t 195 m2 g ·l ~ 1.S [J E :;I :; I .§. 11.0 :; I :, -£ 0.5 0 10 15 20 [010) [001) Pore Width {A) 0.0 0 0.2 0.4 0.6 0.8 Pressure (bar) Hydrogen Adsorption Uptake Excess H2 uptake equilibria (measured in mmol g-1) as a function of pressure (in bar) and temperature (in K) were measured along four isotherms between 70-100 K using the volumetric technique (Figure 2). The excess uptake quantities were converted into absolute uptake by neglecting the contribution of the correction term (see the Supporting Information for the justification of this approach under the conditions explored herein). Adsorption equilibria for further analysis were selected by truncating each isotherm at near the completion of a monolayer, beyond which adsorption occurs primarily on the outer surfaces of the particles. These selected adsorption equilibria were tabulated, imported into a Python package (Realist, v. 0.31418), and globally fitted to a series of SL equations of the general form shown in Equations 4-5. Specifically, the temperature dependence of the pre-exponential factor in Equation 5 (referred to herein as 𝐴𝐴 [𝑇𝑇 ]) was varied by powers of 0.5 from -2.5 to 0 (some additional experiments are also described in the Supporting Information). This analysis culminated in six total fits, each characterized by a unique set of fitting parameters. The results of the fitting procedure, including the best-fit parameters and the goodness of fit (determined herein as the root of the sum of the squared residuals per data point, or RMSE), are summarized in Figure 3 and tabulated in Table 2. Figure 2. Equilibrium excess adsorption uptake of H2 on Ni3(pzdc)2(ade)2(H2O)4 between 70-100 K and 0-1 bar with: (a) normal and (b) logarithmic pressure axes. Page 8 of 21 ACS Paragon Plus Environment Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 2.5 a ;;--.. 2.0 ] 1.5 .§. b. 1.0 :, £ 0.5 0.0 0 ............. ::::== ••• : : : : : : : : : : ... .... . ·• . .. ·•• . . 0.2 .. 0.4 ..... ... • 70 K • 77 K • 87 K • 100 K 0.6 0.8 Pressure (bar) 2.5 b 2.0 bo 0 E 1.5 .§. C. 1.0 :, £ 0.5 0.0 10·5 ..... / ... 10·4 10·3 10·2 Pressure (bar) 10·1 Figure 3. Equilibrium excess uptake (circles) of H2 on Ni3(pzdc)2(ade)2(H2O)4 between 70-100 K and 0-1 bar fitted by a series of SL equations (lines) with varying temperature dependencies: 𝐴𝐴[𝑇𝑇 ] ∝ (a) 𝑇𝑇 −2.5 to (f) 𝑇𝑇 0 . Goodness of fit (RMSE) is indicated. Isotherm temperature is indicated by color from red (70 K) to blue (100 K). The optimized fits of the experimental adsorption equilibria to the six models exhibited nearly identical goodness of fit, despite each model representing a different atomistic description of the single binding site. In general, all models provided a good fit to the experimental data, as confirmed by both statistical analysis and visual inspection (Figure 3). The goodness of fit systematically improved as the T- dependence of the pre-exponential factor varied from a large negative power (i.e., 𝑇𝑇 −2.5 ) to T- independent (i.e., 𝑇𝑇 0 ) (Table 2), but this was only a very small effect. In contrast to the small difference in goodness of fit between each model, the respective binding energies varied systematically and significantly from ~10 to ~8 kJ mol-1 , from 𝑇𝑇 −2.5 to 𝑇𝑇 0 , respectively. Hence, the results of this study imply that a nearly identical goodness of fit can be achieved while attributing widely varying binding energies to the adsorption event; a difference of 2 kJ mol-1 is experimentally and computationally meaningful, at high levels of theory19 . The meaning of the fitting parameters also differs widely depending on the model chosen (see Supporting Information). Interestingly, all best-fit values of the various fitting parameters were found to be physically reasonable for H2 on Ni3(pzdc)2(ade)2(H2O)4, further preventing any strong discrimination against any of the models used herein. In other words, no single model with a specific temperature dependence could be unambiguously declared as the “best model” to describe H2 adsorption on the single-site MOF at between 70-100 K. Page 9 of 21 ACS Paragon Plus Environment Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 a =···· b =···· C =···· 2.0 ... 2.0 . ... . .. 2.0 . ... . .. .... . .. . . . . .. . ... . . . . ... . . . . . . . . .. ;:;-- . . t 1.5 bo bo 1.5 0 1.5 0 E .. E .. E . . .s . . .s . . .s . . . . . 1.0 . 1.0 . 1.0 . E. A[T] ex: r -2-5 E. A[T] ex: r -2 E. A[T] ex: r -1.s ::, ::, ::, £ £ £ 0.5 RMSE = 0.00380 0.5 RMSE = 0.00372 0.5 RMSE = 0.00365 0.0 0.0 0.0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 Pressure (bar) Pressure (bar) Pressure (bar) d e f =···· =···· =···· 2.0 ... . . . 2.0 ... ... 2.0 . .. . .. . ... . . . . ... . . . . . . .. . ... .. . .. . .. ;:;-- . . . . . 1.5 . bo .. bo . . 0 1.5 0 1.5 E .. E .. E . . .s . . .s . . .s . . . . . . . . 1.0 1.0 1.0 a. A[T] ex: r - 1 C. A[T] ex: r-o.s C. A[T] ex: T0 ::, ::, ::, £ £ £ 0.5 RMSE = 0.00360 0.5 RMSE = 0.00355 0.5 RMSE = 0.00352 0.0 0.0 0.0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 Pressure (bar) Pressure (bar) Pressure (bar) Thermodynamic Analysis An important practical metric for adsorption systems that is widely reported in the adsorption literature is the isosteric enthalpy (or “heat”) of adsorption (−∆𝑎𝑎𝑎𝑎𝑠𝑠 𝐻𝐻 or 𝑞𝑞𝑠𝑠𝑡𝑡 ). This value is often used as a proxy to describe the average interaction energy or strength of binding of the adsorption system, often as a function of surface occupancy. In some cases, 𝑞𝑞𝑠𝑠𝑡𝑡 , is also reported as a function of both temperature and surface occupancy,20,21 requiring a more sophisticated modeling approach with an explicit temperature dependence built into the fitting model. Although the isosteric enthalpy can, in principle, be measured directly using calorimetry, such measurements are far less common than indirect determination via fitting measured adsorption equilibria. In this work, the isosteric enthalpy of adsorption is calculated as a function of both temperature and surface occupancy for each adsorption model using the widely employed Clausius-Clapeyron equation (which requires two assumptions to be validated, see the Supporting Information) as shown below: 𝑞𝑞𝑠𝑠𝑡𝑡 [𝑇𝑇 , 𝑃𝑃] = −∆𝑎𝑎𝑎𝑎𝑠𝑠 𝐻𝐻 [𝑇𝑇 , 𝑃𝑃 ] = 𝑅𝑅 𝑇𝑇 2 𝑃𝑃 � 𝜕𝜕 𝑃𝑃 𝜕𝜕 𝑇𝑇 � 𝜃𝜃 Eq. 9 The explicit form of the partial derivative in Equation 9 is directly influenced by the T-dependence inherent in the chosen adsorption model, as shown in the Supporting Information. This is the same relation employed in almost all efforts to deduce 𝑞𝑞𝑠𝑠𝑡𝑡 from adsorption uptake equilibria, whether an explicit model is employed or not. In the present work, the isosteric enthalpy of adsorption is invariant with respect to surface occupancy (𝜃𝜃 ) owing to two key assumptions: (1) a homogeneous binding site distribution on the adsorbent surface and (2) the treatment of the bulk adsorbate as an ideal gas. However, depending on the model chosen (i.e., the treatment of degrees of freedom in the binding site itself), the isosteric enthalpy of adsorption can be anything from invariant with temperature to highly sensitive to temperature. Two extreme cases are the “fixed” adsorption site model (where the adsorbate loses all translational degrees of freedom upon adsorption and 𝐴𝐴𝐹𝐹𝐹𝐹𝐹𝐹 [𝑇𝑇 ] ∝ 𝑇𝑇 −2.5 ) and a looser-bound adsorption site (where each adsorbate can access 3 degrees of freedom and 𝐴𝐴0 [𝑇𝑇 ] ∝ 𝑇𝑇 0 ). In the former case, the final result for the isosteric heat of adsorption is: 𝑞𝑞𝑠𝑠𝑡𝑡 ,𝐹𝐹𝐹𝐹 𝐹𝐹 [𝑇𝑇 ] = −(𝜀𝜀 − 2.5𝑅𝑅𝑇𝑇 ) Eq. 10 On the other hand, in “model 0” the final result for the isosteric heat of adsorption is: 𝑞𝑞𝑠𝑠𝑡𝑡 ,0 [𝑇𝑇 ] = −𝜀𝜀 Eq. 11 The isosteric heats of all models can be described by a simple relation: 𝑞𝑞𝑠𝑠𝑡𝑡 ,𝐹𝐹 [𝑇𝑇 ] = −(𝜀𝜀 − 𝑥𝑥𝑅𝑅𝑇𝑇 ) Eq. 12 where 𝑥𝑥 is the (positive) exponent of 𝑇𝑇 in the denominator of 𝐴𝐴 [𝑇𝑇 ]. All of the above equations are valid in the ideal gas limit only. The isosteric heats of adsorption of H2 on Ni3(pzdc)2(ade)2(H2O)4, according to the results of six different SL models, are shown in Figure 4. Despite nearly identical goodness of fit across all models, it is evident that the choice of model is of great importance to the interpretation of the adsorption thermodynamics. For example, the model with the best fit by a slight margin (“model 0”, RMSE = 0.00352) would suggest that the isosteric heat is temperature invariant (Figure 4f) while the model with the marginally worst fit Page 10 of 21 ACS Paragon Plus Environment Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 --- -- (“model −2.5”, RMSE = 0.00380) would suggest that the isosteric heat increases by 0.6 kJ mol-1 between 70-100 K (Figure 4a). While this +0.6 kJ mol-1 variance is small within the 30 K temperature range explored in this work, this would increase to a discrepancy of +4.8 kJ mol-1 over a wider range of 70-298 K, a temperature range of importance for applications. Whether the isosteric heat changes by 5 kJ mol-1 within this temperature range or not is still a subject of ongoing investigation, but clearly this would be of major technological significance. It is clear from this work that the experimental data alone cannot unambiguously be used to determine the temperature dependence of the isosteric heat since all models give a similar goodness of fit. Table 2. Best-fit parameters for six models of H2 adsorption on Ni3(pzdc)2(ade)2(H2O)4 according to SL equations with different T-dependence of the pre-exponential factor, 𝐴𝐴 [𝑇𝑇 ]. 𝑛𝑛𝑠𝑠 (mmol g-1) 𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐𝑐𝑐𝑐𝑐 𝑛𝑛𝑐𝑐 (Kx bar-1) 𝜀𝜀 (kJ mol-1) ∆𝑎𝑎𝑎𝑎𝑠𝑠 𝐻𝐻 at 82 K (kJ mol-1) RMSE Fixed (𝑇𝑇 −2.5 ) 2.19 408 –7.97 –9.71 0.00380 1D Ideal Gas (𝑇𝑇 −2 ) 2.19 27.0 –8.31 –9.72 0.00372 2D Ideal Gas (𝑇𝑇 −1.5 ) 2.19 1.79 –8.66 –9.72 0.00365 3D Ideal Gas (𝑇𝑇 −1 ) 2.19 0.119 –9.01 –9.72 0.00360 2D Lattice Gas (𝑇𝑇 −0.5 ) 2.19 0.00785 –9.35 –9.75 0.00355 QPQPSP* (𝑇𝑇 0 ) 2.19 0.000520 –9.70 –9.74 0.00352 *Note: there is no name or reasonable “physical picture” for this model (see Supporting Information) Average Temperature Isosteric Heat The calculated isosteric enthalpy of adsorption varies significantly across the different models explored in this study (Figure 4), despite that there is no significant difference between the goodness of fit for the different models. Notably, however, the calculated isosteric enthalpy of adsorption at the average temperature of measurement (defined herein as the thermodynamic average, or the average of 1/T) remains consistent across all the models explored. This effect is shown in Figure 5, where the isosteric enthalpy of adsorption at the average temperature of 82 K for the four isotherms is identical for all models. This finding is an important result as it demonstrates that, regardless of the specific model employed, an unambiguous outcome of adsorption modeling is the value of the isosteric heat at the thermodynamic average temperature over the temperature range explored. We recommend that researchers seeking to determine the isosteric enthalpy of adsorption at a given process temperature collect a large set of data around their temperature of interest, ensuring that this temperature corresponds Page 11 of 21 ACS Paragon Plus Environment Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 to the average of the experimental temperature set. While the binding energy reported by the best fit will not be unambiguously true, nor will the temperature dependence of 𝑞𝑞𝑠𝑠𝑡𝑡 , the value of the 𝑞𝑞𝑠𝑠𝑡𝑡 at the average temperature will be a reliable metric for further analysis. For H2 adsorption on Ni3(pzdc)2(ade)2(H2O)4, the isosteric heat of adsorption (which we expect would be corroborated by calorimetry) at 82 K is found to be 9.7 kJ mol-1 . Interestingly, this “average temperature” heat of adsorption is the same deliverable that is possible to achieve using non-temperature-dependent methods such as individual isotherm fitting, but with far fewer fitting parameters in this case. Figure 4. Isosteric enthalpy of adsorption of H2 on Ni3(pzdc)2(ade)2(H2O)4 between 70-100 K and 0-1 bar based on a series of SL equations with varying temperature dependencies: 𝐴𝐴[𝑇𝑇 ] ∝ (a) 𝑇𝑇 −2.5 to (f) 𝑇𝑇 0 . Isotherm temperature is indicated by color from red (70 K) to blue (100 K). Page 12 of 21 ACS Paragon Plus Environment Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 a 12 b 12 C 12 A[T] oc r - 2.s A[T] oc r - 2 A[T] oc r -1.s 11 11 11 ::- ::- 0 0 0 E E E g 10 g 10 g 10 I I I