A simple van’t Hoff law for calculating Langmuir constants in clathrate hydrates Azzedine Lakhlifi a,⇑ , Pierre Richard Dahoo b , Sylvain Picaud a , Olivier Mousis a,c a Institut UTINAM-UMR 6213 CNRS, Université de Franche-Comté, Observatoire de Besançon, 41 bis avenue de l’Observatoire, BP 1615, 25010 Besançon Cedex, France b Université de Versailles-Saint-Quentin-en-Yvelines, Sorbonne Universités, Laboratoire Atmosphères Milieux Observations Spatiales, CNRS, UMR 8190, Observatoire de Versailles Saint-Quentin-en-Yvelines, 11 Bd d’Alembert, F-78820 Guyancourt, France c Aix Marseille Université, CNRS, LAM (Laboratoire d’Astrophysique de Marseille) UMR 7326, 13388 Marseille, France article info Article history: Received 10 October 2014 In final form 5 January 2015 Available online 13 January 2015 Keywords: Clathrate hydrates Langmuir constants Atom–atom interaction potential energy abstract This work gives a van’t Hoff law expression of Langmuir constants of different species for determining their occupancy in clathrate hydrates. First, a pairwise site–site interaction potential energy model is used to calculate the Langmuir constants in an otherwise anisotropic potential environment, as a function of temperature. The results are then fitted to a van’t Hoff law expression to give a set of parameters that can be used for calculating clathrates compositions. The van’t Hoff law’s parameters are given for eigh- teen gas species trapped in the small and large cavities of structure types I and II. The accuracy of this approach is based on a detailed comparison with available experimental and/or previously calculated data for ethane, cyclo-propane, methane and carbon dioxide clathrate hydrates. A comparison with the analytical cell method is also carried out to better understand the importance of asymmetry and possible limitations of the van’t Hoff temperature dependence. Ó 2015 Elsevier B.V. All rights reserved. 1. Introduction A clathrate is an ice-like crystalline solid consisting of water molecules forming a cage structure around smaller guest mole- cules under suitable conditions of low temperature and high pres- sure. On Earth, it is considered that clathrate hydrates are the most important reservoirs of fossil energy [1,2], and that favorable con- ditions for gas hydrate formation exist in about 25% of the earth’s land mass. Moreover, the thermodynamics conditions of pressure and temperature prevailing in the oceans are such that hydrates should easily be formed in about 90% of the ocean or sediments. The most common guest molecules in terrestrial clathrates are of organic aliphatic nature like methane, ethane, propane or butane, but other small inorganic molecules like nitrogen, carbon dioxide, and hydrogen sulfide can also be trapped in the cages of clathrates [3–9]. In the advent of global warming, these clathrates can enhance the temperature rise when the trapped species are released. Clathrate hydrates are also suspected to be extensively present on several planets, satellites and comets of the Solar Sys- tem. Planetologists are thus concerned with the possible clathrate impact on the distribution of the planet’s volatiles and on the mod- ification of their atmosphere’s compositions [10]. Hence, it is of great interest to correctly determine the amount of species poten- tially trapped in the cages of clathrates, i.e. the fractional occu- pancy of guest species under the thermodynamic conditions (pressure and temperature) prevailing in the regions where clath- rates might form. From a theoretical point of view, the thermodynamics of the formation or dissociation of clathrates is most often based on the model developed by van der Waals and Platteeuw [11] following the same hypotheses under which was developed the adsorption theory of Langmuir [12]. The Langmuir isotherms of adsorbed mol- ecules on a surface are determined from the calculation of the Langmuir constant, which is also the main parameter to be consid- ered in the determination of the amount of species trapped in the clathrate cages as a function of pressure and temperature. To calculate these Langmuir constants, most of the models are based on a molecular description of the guest-water interactions using a Lennard-Jones or Kihara potential form. The parameters of these potentials are usually empirically obtained from experi- mental data of phase equilibrium. Such models most often neglect interactions of the guest molecules with water beyond a few cages only and are therefore questionable [13–18]. Moreover, it is generally assumed that the environment of the cage in which a gas molecule is trapped in clathrates is of spherical symmetry. Whereas this assumption may be justified for molecules such as CH 4 or NH 3 , it is certainly not well-suited for http://dx.doi.org/10.1016/j.chemphys.2015.01.004 0301-0104/Ó 2015 Elsevier B.V. All rights reserved. ⇑ Corresponding author. E-mail address: azzedine.lakhlifi@obs-besancon.fr (A. Lakhlifi). Chemical Physics 448 (2015) 53–60 Contents lists available at ScienceDirect Chemical Physics journal homepage: www.elsevier.com/locate/chemphys