Predicting the adsorption and isosteric heat of pure gases in active carbons with the slit-pore model, MC simulation and DFT

We describe a slit-pore model and a fast density functional theory (a 'slab-DFT') that predict gas adsorption and the isosteric heat in active carbons. The DFT parameters are fitted to reproduce adsorption isotherms of each pure gas in graphitic slit pores generated by Monte-Carlo simulation. A novel feature of this work is that gas surface interactions are calibrated to a high surface area carbon, rather than a low surface area carbon as in all previous work. We also discuss the isosteric method, correcting some errors and confusion that persist in the literature. We present predictions for the adsorption of carbon dioxide, methane, nitrogen and hydrogen up to reasonably high pressure in several active carbons at a range of temperatures based on an analysis of a single carbon dioxide adsorption isotherm. These results demonstrate that our models are accurate for relatively simple gases at near-critical or supercritical temperatures. Introduction Activated carbons are used for a variety of purposes, and on an industrial scale many of them involve the separation of fluid mixtures. To design a separation process it is useful to have a phase diagram describing how fluid mixtures are adsorbed at equilibrium by a given material. However, rapid and accurate prediction of the phase behaviour of adsorbed fluid mixtures is problematic because of the number of degrees of freedom. In practice, this problem is often solved by use of a theory that predicts mixture adsorption on the basis of the adsorption of each pure component. This strategy is the basis of ideal adsorbed solution theory (IAST). In our recent work 3 we take a further step, i.e. we predict gas mixture adsorption in active carbons given a single ‘probe’ adsorption isotherm as input. This is achieved by predicting the adsorption of each pure component from analysis of this probe isotherm, and then using a novel ‘slab-DFT’ to make predictions for the mixture. Here we describe the first stage of this approach, i.e. prediction of the adsorption of pure gases up to relatively high pressure over a range of temperatures based on analysis of one ‘probe’ gas adsorption isotherm. We will describe the connection between the temperature variation of adsorption isotherms and the ‘isosteric heat’, and also present results for adsorption over a range temperatures using our models. The second stage of our approach is dealt with in another contribution. Our approach is based on the polydisperse independent ideal slit-pore model, Monte-Carlo simulation and a ‘slab-DFT’. We use these models to predict the adsorption of carbon dioxide, methane, nitrogen and hydrogen, in several active carbons up to reasonably high pressure over a range of temperatures. The results demonstrate that these models are quite accurate for these gases at near-critical or supercritical temperatures. However, we are cautious in advocating these methods for significantly subcritical, strongly polar or complex molecular gases. Further work beyond what is presented here is needed to establish whether more detailed models are required to describe adsorption of these gases in active carbons. A great deal of work already exists in the literature on the subject of predicting pure 13 and mixed gas adsorption at a range of temperatures on the basis of a single probe gas adsorption isotherm, using either DFT or MC simulation. However, all this work, we feel, is limited because in every case gas – surface interactions are calibrated to low surface area carbons such as graphite, Sterling or Vulcan. The most significant contribution of our work is that it shows that much greater accuracy can be obtained if gas – surface interactions are calibrated to a reference high surface area active carbon. The premise here is that the surfaces of active carbons are more similar to each other than to low surface area carbons. Some thermodynamics and the isosteric method In experiments it is excess, not absolute, quantities that are measured. Excess quantities are defined with respect to a particular pore volume, Vp, as follows ex p b a p b a ex x V x x V X X X = − = − = ) ( (1) We show that a popular definition of the isosteric method in terms of a ‘differential enthalpy’ rather than ‘reversible heats’ is appropriate only when excess adsorption is calculated. When absolute adsorption is studied, as is often the case in idealised models, the isosteric method can no longer be expressed purely in terms of enthalpies, but it can still be expressed in terms of reversible heats. So expression of the isosteric method in terms of reversible heats, or the ‘isosteric heat’, is more general than in terms of enthalpies. This is important for all theoretical models of adsorption. Active carbons have been modelled successfully as rigid adsorbent so a natural ensemble with which to study their adsorption is the grand canonical ensemble, in which we have (for a pure adsorbed fluid) μ ω Nd SdT d Vp − − = (2) where ω, S, Τ, N and μ are the average grand potential density, entropy, temperature, number of particles and chemical potential respectively of fluid in the pore space. The ‘isosteric heat’, qst, is defined via the isosteric method, which consists of analysing the variation of pressure with temperature at fixed adsorbed amount. Together with the above fundamental relation we write ex p ex p N V b N V b st T T Ts T P T q , ,       ∂ ∂ + =       ∂ ∂ = μ ρ (3) since -P is the grand potential density and sb is the entropy per particle of the bulk gas. Then, using (2) together with F = ωVp + μN gives dT S dN dV dF ex ex p ex ex − + = μ ω (4) from which we obtain the following Maxwell relation