Appunti Principles of Advanced Separations

Formattazione del testo

V L V˜ ˜ PV VL L(conventional one: variation of the chemical potential multiplied by the number of moles). In whichwe performed the integration of the Lewis definition of fugacity from pure V to pure L, assumingalso ideal gas (P is the vapour pressure).s − −∆G = 2Aγ 2Aγ = (γ γ )2A (92)surf SL SV SL SV64this term is not negligible given our particular schematization of the pore structure.At equilibrium: ∆G = ∆G + ∆G = 0 and substituting the Young’s equation we haveV surfAh P s −RT ln + (γ γ )2A = 0SL SV˜ PV L ˜ ˜2V 2VP L Ls −= (γ γ ) = γ cos θ (93)RT ln SV SL LVP h hPisolating :P s ˜P 2V γ cos θs L LVln =P RT h !˜P 2γ V cos θLV L−= exp (94)P (T ) hRTswhich is basically equal to the Kelvin equation, in fact: h = r . At this point we can observeporethat:• ◦ ◦ ⇒cos θ > 0 (0 < θ < 90 ) P

So we might have liquid formation in the pore even if the operating pressure is smaller than the vapour pressure, at the corresponding temperature. This is due to the compatibility between the liquid and the solid: • ◦ ◦ ⇒cos θ > 0 (90 < θ < 180 ) P > P. So only operating with a pressure larger than the vapour pressure, we’ll induce the formation of the liquid in the pore (if we work exactly at the vapour pressure we would not induce the condensation, because the liquid is not compatible with the solid) so, depending on the value of θ we can have condensation at a pressure higher or lower than the vapour pressure. This phenomena explains the trend observed in the adsorption types IV, V:

Until now we considered all isotherms dealing with a mono-component system, but now we’re going to investigate also multi-component ones. So we need to introduce a proper thermodynamic

1. The thermodynamic approach that we'll exploit is the one of Gibbs, which is based on the following assumptions:
   • The adsorbed phase is a 2D region (thin layer, negligible thickness) close to the surface of an "inert" solid. As we previously said, in the adsorbed phase we have an increase of the density, with respect to the one of the bulk, because of weak physical interactions with the solid surface. By thermodynamically inert solid we mean that its properties remain constant during the adsorption process.
   • The surface is equally accessible to all species.
2. To describe the thermodynamic behaviour, we start expressing the differential of the internal energy for the overall system (adsorbent + adsorbed species): CX* * * * -dU = T dS P dV + µ dn + µ dn (95) a i ia i=1*

a = adsorbent), whereasaaCP ∗µ dn is associated to the adsorbed species (the means only that we are considering thei ii=1overall system).Then, by considering adsorbent only (no adsorbed species contribution) and indicating with thelabel ”0” the pure adsorbent: 0 0 0 0−dU = T dS P dV + µ dn (96)aataking the difference between the two, we can compute the contribution of adsorbed species only:∗ 0−dU = dU dU CX∗ ∗ ∗0 0 0− − − −dU = T d(S S ) P d(V V ) + (µ µ )dn + µ dna i ia a i=1∗ ∗ ∗ 00 0 −− − is the change ofµso we can define: S = S S , V = V V and also notice that µ aathe chemical potential associated to the formation of the adsorbed phase onto the adsorbent (finalsystem with adsorbed species on the solid - only solid), we can express this terms as:∂U∗ 0 |− − − (97)Φ = µ µ =

V,S,na a i∂na(variation of the internal energy of the adsorbed phase with respect to the number of moles of solid,keeping constant: volume, entropy, number of moles of adsorbed species).substituting: CX− −dU = T dS P dV Φdn + µ dn (98)i ia i=1

The quantity Φ represents the U change per unit mole of solid due to the “spreading” of the molecules adsorbed on the surface (in other words, the formation of the thin layer of molecules on the adsorbent). The corresponding term can be conveniently re-expressed as:

Φdn = πdĀ (99)awhere π is called the spreading pressure, defined as:

∂U− |π = (100)V,S,ni∂ Ā66and Ā is the surface area of the adsorbent (much easier to handle than n ). So, the final expression of the internal energy of the adsorbed phase is the following:

CX− −dU = T dS P dV πdĀ + µ dn (101)i ii=1

Now, we’ll manipulate this expression in order to propose a complete thermodynamic

framework of our adsorbed phase, making also some assumptions. In fact, all terms containing the variation of volume are usually neglected because of their minor contribution, since the main characteristic property of the adsorbed phase is the surface:

C CX X− − ' −dU = T dS P dV πdĀ + µ dn T dS πdĀ + µ dn (102)i i i ii=1 i=1

From U we can compute all the other thermodynamic properties:

⇒ 'enthalpy dH dU (103)C ∂GX |⇒ ' − − (104)Gibbs f ree energy dG SdT πdĀ + µ dn with π = T,ni i i∂ Āi=1

Before continuing the derivation we make a comparison of this thermodynamic framework with respect to a traditional VLE (3D), so considering a bulk phase. This can be perfomed observing Table 15 in which for each row we have the equivalent properties of the 2 systems:

in particular we observe that in the adsorbed phase, the Gibbs free energy is a function also of the spreading pressure

π which is, once again, the change of internal energy (or Gibbs free energy) associated to the spreading of molecules adsorbed on the surface (we have an additional intensive variable). Notice that we have also an additional extensive variable (area). This has an impact of the DOF of the system because now we have 3 intensive variables, so the variance of the system becomes: −v = C φ +3 (105) (C = number of components, φ = number of phases), whereas for the VLE we would have: v =−C φ + 2. In particular, considering a mono component system: (→V LE v = 1 P = f (T )sC =1 →ADS v = 2 q = f (P, T )so also for a mono component system (adsorbed phase) we have to provide 2 intensive variables to saturate the DOF. Having underlined this aspect, we’ll exploit the dG expression to compute equilibrium relations (for the adsorbed phase). In particular we consider a constant temperature: CX−πdĀdG = + µ dn (106)i ii=167 then we consider the integrated

taking the differential: C CX X−πdĀ −dG = Ādπ + µ dn + n dµi i i ii=1 i=1

In order to be consistent with the previous definition (Equation 106), this equality must hold:

which is also called Gibbs-Duhem equation for the adsorbed phase (remember that n iand µ are the number of moles and the chemical potential of the species adsorbed). This equationiis very useful because allows us to express the spreading pressure as a function of easily accessiblequantities.

Focus on a single component system at phase equilibrium (adsorbed phase -gas phase)

in which the chemical potentials refer to the single species we have in the system. The equalityobviously holds also considering the differentials, that can be expressed as:

fugacity considering an ideal gas). Taking the equality: nRTĀdπ = dP = nRT d ln P (109)P which is called Gibbs adsorption isotherm, a fundamental relationship expressing the spreading pressure as a function of measurable variables (temperature, pressure, area, adsorbed moles). At this point we divide both terms by the mass of the solid m: sĀ ndπ = RT d ln Pm ms s nĀ , q = , By introducing area and adsorbed concentration per unit mass of solid: A = m ms we have: A dπ = qd ln PRT the first term can be defined with a quantity dψ that can be integrated from 0 to a generic quantity P value: PZψ = qd ln P (110)0 so if we know q as a function of pressure we can compute through an integration ψ and consequently the spreading pressure π. The previous derivation can be also applied to a multicomponent case: NCA Xdψ = dπ = q d ln p (111)i iRT i68 obtaining: NCpZ i Xψ = q d ln p (112)i i0 i Whereas, in the case of non-ideal gaseous mixture: NC NCpZA iX X¯

¯⇒dπ =dψ = q d ln f (P, T, y ) ψ = q d ln f (P, T, y )i i i i i iRT 0i i

We can exploit the knowledge of the spreading pressure to check the reliability of our experimental data of adsorption equilibrium. So we consider a system at constant pressure ( the constant temperature assumption is always taken into account) composed of 3 species. So we measure the adsorption equilibria by changing the composition of the ternary mixture. Given these assumptions:

d(P y )i = d ln yd ln p = ii P yiplugging this expression in Equation 111, we have:

NC NC qiX X dydψ = q d ln y = ii i y ii iEven if we have a mixture of 3 components, we focus our attention on data collected for a binary− →mixture 1 2 (y + y = 1 dy + dy = 0):1 2 1 2 q q q q1 2 1 2−dψ = dy + dy = dy