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Calculation of Chemical Potential and Activity Coefficient of Two Layers of Co2 Adsorbed on a Graphite Surface

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Calculation of Chemical Potential and Activity Coefficient of Two Layers of Co2 Adsorbed on a Graphite Surface Physical Chemistry Chemical Physics CALCULATION OF CHEMICAL POTENTIAL AND ACTIVITY COEFFICIENT OF TWO LAYERS OF CO2 ADSORBED ON A GRAPHITE SURFACE Journal: Physical Chemistry Chemical Physics Manuscript ID: CP-ART-08-2014-003782.R1 Article Type: Paper Date Submitted by the Author: 07-Nov-2014 Complete List of Authors: Trinh, Thuat; NTNU, Department of Chemistry Bedeaux, Dick; NTNU, Chemistry Simon, Jean-Marc; Universit� de Bourgogne, Chemistry Kjelstrup, Signe; Norwegian University of Science and Technology, Natural Sciences Page 1 of 8 Physical Chemistry Chemical Physics PCCP RSC Publishing ARTICLE CALCULATION OF CHEMICAL POTENTIAL AND ACTIVITY COEFFICIENT OF TWO LAYERS Cite this: DOI: 10.1039/x0xx00000x OF CO2 ADSORBED ON A GRAPHITE SURFACE T.T. Trinh,a D. Bedeaux a , J.-M Simon b and S. Kjelstrup a,c,* Received 00th January 2014, Accepted 00th January 2014 We study the adsorption of carbon dioxide at a graphite surface using the new Small System DOI: 10.1039/x0xx00000x Method, and find that for the temperature range between 300K and 550K most relevant for CO separation; adsorption takes place in two distinct thermodynamic layers defined according www.rsc.org/ 2 to Gibbs. We calculate the chemical potential, activity coefficient in both layers directly from the simulations. Based on thermodynamic relations, the entropy and enthalpy of the CO 2 adsorbed layers are also obtained. Their values indicate that there is a trade-off between entropy and enthalpy when a molecule chooses for one of the two layers. The first layer is a densely packed monolayer of relatively constant excess density with relatively large repulsive interactions and negative enthalpy. The second layer has an excess density varying with the temperature, an activity coefficient which also indicates repulsion, but to a much smaller degree than in the first layer. Results for activity coefficients, entropies and enthalpies can be used to model transport through and along graphitic membranes for carbon dioxide separation purposes. INTRODUCTION constructed by Schnell et al using Hill’s thermodynamics for small systems.15, 16 An ensemble of small systems of varying Thanks to the definitions of excess variables by Gibbs already sizes was examined, and scaling laws for the small system were in 1877,1, 2 we are able to describe surfaces as thermodynamic derived from statistical mechanics. Fluctuation data were systems. With the surface thickness integrated out, they are sampled, scaled appropriately and analyzed to find two-dimensional systems. Phase transformations,3 formation of thermodynamic limit values. The method has now been used nanostructures,3, 4 metastable phases 5 or agglomerates 6 become successfully to obtain Kirkwood – Buff integrals for mixtures special at surfaces or interfaces. Due to this a surface can pose a and electrolytes,16 reaction enthalpy 17 and accurate Fick major barrier to transport 7, 8 and catalyze chemical reactions.9 diffusion coefficients in binary 15 and ternary mixtures.18 More In this entire context thermodynamic data are needed for recently Collell and Galliero reported the thermodynamic modelling and understanding. correction factor of Lenard-Jones fluid in confinement using the Thermodynamic knowledge of surfaces cannot be extrapolated Small System Method and a Langmuir isotherm model.19 from knowledge of homogeneous systems.3 It is common to use We shall now demonstrate how the method can be used to find the ideal Langmuir theory, or empirical variations of this surface thermodynamic data, like the chemical potential, the theory, and Henry’s law to describe adsorption of gas at enthalpy, the entropy, and the activity coefficient of a gas internal or external surfaces.10-12 Current experiments 3 do not adsorbed to a surface, expanding on a first short 20 give easy access to thermodynamic data for real surfaces. With communication. the accuracy now obtainable for atomic force fields one may The adsorption of CO 2 on a graphite surface is relevant to expect quantitative information from computational tools, graphitic membranes, which are promising cheap membranes 13, 14 21 among them Monte Carlo techniques. To the best of our for CO 2 separation purposes. As a first case, we study the knowledge, the chemical potential or activity coefficient of a physisorption process component in a surface have not yet been obtained from a single, simple molecular simulation. CO2 (g)+ graphite CO 2 (s) (1) We shall show in this work that this situation may change with the advent of the Small System Method. The method was This journal is © The Royal Society of Chemistry 2014 Phys. Chem. Chem. Phys ., 2014, 00 , 1-3 | 1 Physical Chemistry Chemical Physics Page 2 of 8 ARTICLE Journal Name where (g) means gas phase, and (s) means gas adsorbed at the We determine the activity coefficient by integrating equation surface. We shall see that new details emerge for the state (6) from zero adsorption: denoted (s): Adsorption takes place in two distinct layers with γ s Cs different enthalpy and entropy in the temperature range ∫dlnγ s= ∫ ( Γ s − 1) d lnC s (7) considered (300K-500K). The first layer has low entropy with 1 0 less mobile molecules, while the second layer has higher When C s → 0 , there is no interaction between particles, entropy contains less bound molecules. The layers are not filled meaning that γ s =1. sequentially, but maintain states which are in equilibrium with For the gas phase the chemical potential is: each other. P µg=+ µg,0RTaln g =+ µ g,0 RT ln (8) P0 METHODS where ag is the activity of the gas and P is the pressure of the gas, which is taken to be ideal. If the gas is non-ideal P must be Thermodynamic relations replaced by the fugacity. We know the standard state value for The chemical potential for a gas adsorbed to a surface in layer the gas phase from standard tables, and want to determine the no i (i= 1,2) is: value for the adsorbed state for a chosen standard state. We choose an ideal state with full coverage. This is a hypothetical C s µµss=+,0RTln a ss =+ µ ,0 RT ln γ s i (2) state with activity coefficient of unity. The state is obtained by ii ii i C s,0 i extrapolating Henry’s law to the state of full coverage. Here µs,0 is the standard chemical potential, a s is the We will see that the second layer behaves according to Henry’s (dimensionless) activity of the adsorbed phase, C s is the surface law for the whole pressure range (See SI). This makes it excess concentration, and γ s is the activity coefficient. The convenient to use the second layer in the definition of the standard state is normally chosen as the hypothetical ideal state surface standard state. The first layer has a much smaller having γs=1 at layer saturation C s ,0 (given as particles per pressure range where Henry’s law applies, and the (nm 2)). Subscript i refers to the layer number. The entropy extrapolation becomes uncertain. Henry’s law can be written: s0 s follows from the standard relation S=-(dµ/d T): CP2 C 2 P 2 KH =s,0, s ,0 = K H 0 (9) s CP2 2 C 2 P s s,0 s Ci Si= S i − R ln γ i s,0 (3) Ci Where K H is the temperature dependent, Henry’s law constant, s,0 The enthalpy of layer i is accordingly: and C2 is a monolayer coverage of the surface. The law s assumes that γ i = 1for low adsorptions. For adsorbed CO 2 in s s s Hi=µ i + TS i (4) layer 2 in equilibrium with the gas, we have under ideal conditions We shall determine the variables in these equations for s adsorption (1). Following Gibbs, the total adsorption C is Cs P s s,02 s ,0 2 (10) defined as the total surface excess concentration. For the layer i, µ2=+ µ 2RTlns,0 =+ µ 2 RTK ln H 0 C2 P we use two surfaces α and β between the two bulk phases. The total adsorption is then the sum of the adsorptions in the layers With equilibrium between the layers we have α β Cs=+= C s C s Czdz() + Czdz () (5) µs= µ s = µ g (11) 1 2 ∫ ∫ 1 2 0 α where C(z) is the concentration of the CO 2 molecules. For the The last equality means that definition of α and β, we refer to the results and discussion P P section. µg,0+RTln2 = µ s ,0 + RT ln K 2 (12) 2P0 2 H P 0 A thermodynamic state is characterized, once the standard state s,0 g,0 and activity coefficients of components are known as a function µ2= µ 2 − RTln K H (13) of temperature, cf. equation (2-5). The thermodynamic correction factor, easily accessible by the Small System This expression enables us to calculate the standard state Method, can be used to find this information. For constant chemical potential of the surface as a function of temperature, temperature T and surface area A the thermodynamic factor is in terms of known quantities. When the standard state is known defined by: we can use the results for the activity coefficient and s concentration to calculate the chemical potential under real s ∂ln γ Γ =1 + (6) conditions. ∂ln C s T, A 2 | Phys. Chem. Chem. Phys ., 2014, 00 , 1-3 This journal is © The Royal Society of Chemistry 2012 Page 3 of 8 Physical Chemistry Chemical Physics Journal Name ARTICLE The Small System Method applied to surface adsorption The interaction between CO 2 and the C-atoms of graphite was In the Small System Method, we make use of the relation similarly described by a LJ potential.
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