Mixture Behavior, Stability, and Azeotropy

Transcription

1 7 Mixture Behavior, Stability, and Azeotropy Copyrihted Material CRC Press/Taylor & Francis 6 BASIC RELATIONS As compounds mix to some deree in the liquid phase, the Gibbs enery o the system decreases toward a minimum at equilibrium. The Gibbs enery chane obtained by ormin a mixture startin rom all the pure compounds can be written as = x µ m k k k (7.) The terms o Equation 7. can be written more explicitly as c c c k( k k) k m = x = x µ k µ k = RT xk ln k k (7.) For a binary system, m RT = x ln + x ln (7.3) Thus, expressin the uacities in terms o the pure compound standard state, that is, deinin the ideal behavior in Lewis s sense, rom Equation 5. we write m RT = x ln x + x ln x + x ln γ + x ln γ,, (7.4a) The irst bracket o the riht-hand side ives the contribution o the ideal mixture, Equation 5.9, and as both mole ractions are less than unity, its value is always neative. This suices to prove that an ideal mixture is always a stable mixture. The second bracket ives the contribution o nonideality o the mixture, and it can be identiied with the excess Gibbs enery in Lewis s sense, Equation 6.9. Thus, =, m m id E + (7.4b) RT Rt RT 7

2 8 Classical Thermodynamics o Fluid Systems POSITIVE AND NEGATIVE DEVIATIONS FROM IDEAL BEHAVIOR IN LEWIS S SENSE For neative deviations rom ideality in Lewis s sense, γ i, <, the second bracket o the riht-hand side o Equation 7.4a is also neative and makes the mixture still more stable than an ideal solution. For positive deviations rom ideality in Lewis s sense, γ i, >, the second bracket o the riht-hand side o Equation 7.4a is positive. I this contribution is lare enouh to overcome the contribution o the ideal mixture term, it produces instability, that is, liquid liquid phase separation. A ood solvent or a iven compound should ive neative deviations, ideal mixin, or at most weak positive deviations. Althouh molecular eects are complex and resist eneralizations, it is possible to establish some rules o thumb and be prepared to ind exceptions. For the case o oranic compounds, there are two main characteristics o their molecules that help in anticipatin the behavior o their mixtures:. Lenth and shape o the hydrocarbon skeleton. Functional roups present Copyrihted Material CRC Press/Taylor & Francis 6 The eect o the unctional roup is so dominant that we classiy the oranic molecules in homoloous amilies. In this respect, water can be considered the shortest alcohol, that is, an alcohol without a hydrocarbon chain. When mixin compounds, the roup roup interactions overn the physical and chemical behavior o the mixture. Some roup roup interactions are so stron as to cause a chemical reaction and the ormation o new compounds. Good examples are the cases o roups COOH and OH, COO and H O, and CO and NH. We restrict our discussion here to nonreactive cases. Lare dierences in the lenths o the hydrocarbon chains tend to ive neative deviations. These are important in hydrocarbon hydrocarbon systems, but they tend to be overshadowed by the larer eect o unctional roups when these are present. The eect o the shape o the hydrocarbon skeleton is minor, and in most practical situations, it can be inored. However, the eect o unsaturation in hydrocarbon structures may be important since unsaturated aliphatic compounds and also aromatic compounds can participate in electron donor acceptor complexes and thus avor mixin. BERG CLASSIFICATION OF LIQUIDS AND MIXTURE BEHAVIOR Ber [] proposed a classiication o liquids in ive roups that we extend here to six by dierentiatin between saturated and unsaturated hydrocarbons. The classiication is based mostly on the capacity o the molecules to orm hydroen bonds and electron donor acceptor complexes, but it also rouhly orders the compounds on a scale o decreasin polarity. The modiied Ber s classiication is as ollows: Group : Molecules ormin tridimensional networks o H bonds. These are hihly associated liquids, such as water, polyalcohols, polyphenols, hydroxyacids, amino alcohols, and amides.

3 Mixture Behavior, Stability, and Azeotropy 9 Group : Molecules havin both active hydroen atoms and electron donor atoms, such as O, N, and F. These are associated liquids, such as nitrocompounds and nitriles with hydroen atoms in the α position (i.e., in the adjacent C atom) with respect to the N atom, nitromethane and acetonitrile, phenols, acids (includin HCN, HF, HCl, and HNO 3 ), alcohols, and primary and secondary amines. Group 3: Molecules havin electron donor atoms (O, N, and F) but not havin an active hydroen atom. Nitriles and nitrocompounds without α-hydroen, esters, ketones, aldehydes, esters, tertiary amines, and so orth, are in this roup. Group 4: Molecules havin an active hydroen atom but not an electron donor atom, such as O, N, or F. These are polychlorinated compounds with Cl and H in the same or in adjacent carbon atoms. Group 5: Unsaturated hydrocarbons, aliphatic or aromatic. Group 6: Saturated hydrocarbons. Copyrihted Material CRC Press/Taylor & Francis 6 From roup 3 down, mixtures o compounds rom the same roup are close to ideal with sliht positive deviations. Within a same roup, positive deviations increase as the dierence between the molecules increases. For example, a mixture o two ketones will be more ideal than a mixture o one ketone and one nitrile. Lon hydrocarbon chains in both molecules o the same roup tend to reduce the nonideality. I the molecules have similar vapor pressures, their mixtures can exhibit maximumpressure azeotropes. When mixin compounds o dierent roups rom roup 3 down, there are two cases deservin special mention. Members o roup 3 mixed with members o roup 4 will always ive neative deviations rom ideality. I their vapor pressures are similar, they can exhibit minimum-pressure azeotropes. Second, polar solvents rom roup 3 (and also rom roup ), when mixed with members o roup 5, ive electron donor acceptor complexes and thus avor solubility. Groups and contain the most diicult personalities amon the chemical species. When members o roups or are mixed with members o roups 4, 5, or 6 they always exhibit positive deviations that can be stron enouh to cause liquid phase separation. When members o roups and are mixed with members o roup 3, there can be either weak positive or weak neative deviations. And this is also the situation when mixin members o roup amon themselves, members o roup amon themselves, and members o roup with members o roup. Neative deviations that lead to azeotropic behavior are ound in mixtures o water with stron acids or other associatin liquids o roup and also in mixtures o oranic acids or phenols with amines or alcohols. While neative deviations rom ideality in Lewis s sense produce stable mixtures, stron positive deviations may cause instability, leadin to the separation o the mixture into two liquid phases. Typical examples o a binary system ormed by two partially miscible compounds are the cases o butanol and water and the more extreme case o petroleum and water. However, even in the case o stron positive deviations, two liquid compounds are miscible to a minor extent. Startin rom one o the liquid compounds pure (solvent), it is always possible to dissolve in it a small concentration

4 3 Classical Thermodynamics o Fluid Systems o the second liquid compound (solute). However, i the concentration o the solute is increased above a certain limit, the liquid mixture may split into two separate liquid phases. Each liquid phase will be rich in one o the components and lean in the other. In ternary or hiher systems, the additional components will distribute between the two liquid phases ormed. Copyrihted Material CRC Press/Taylor & Francis 6 PARTIAL MISCIBILITY In the mixin process o two completely miscible liquid compounds, or in the miscibility reions o two partially miscible liquid compounds, that is, in the rane where the mixture is stable, the Gibbs enery decreases toward equilibrium; thus, or the ormation o a stable binary mixture the Gibbs enery chane is necessarily neative, m < Fiure 7. presents a plot (Δ m /RT) versus x showin that or a system in which the two compounds are totally miscible in all o the composition rane, the slope o the curve Δ m /RT versus x increases continuously with an increase in x. Thus, d d m / RT T So, the criterion or the stability o a binary mixture can be written as ( m / RT ) x (7.5) T Δ m /RT (+) Stable ( ) x FIGURE 7. Stable system.

5 Mixture Behavior, Stability, and Azeotropy 3 where x reers to x or to x. When Equation 7.5 is satisied over all o the concentration rane, the compounds are miscible in all proportions. Binary systems havin stron positive deviations rom ideality in Lewis s sense may exhibit immiscibility i Equation 7.5 ails to be satisied. I this happens, the mixture will be unstable and split into two phases. This behavior is depicted in Fiure 7., where between points and, the slope o the curve o (Δ m /RT) versus x decreases with composition. In this case, the Gibbs enery o the system is less when there are two phases present instead o one. For a binary liquid mixture, Equation 7. takes the orm µ m x xµ (7.a) Takin the second derivative with respect to either x or x, or the condition o stability we obtain Copyrihted Material CRC Press/Taylor & Francis 6 m = (7.6) T This result shows a very important conclusion; that is, both the plots o Δ m versus x and versus x convey the same inormation reardin a possible instability o the system. I at any concentration rane the second derivative o the unction becomes neative, that is, the curve presents an inlection point, the mixture becomes unstable and separates into two phases. In Chapter 6, we discussed the relations between molar properties and partial molar properties, takin the molar volume as an example, but observin that the conclusions were also valid or all other molar and partial molar quantities. Hence, by analoy to what was shown in Fiure 6., Fiure 7.3 shows that a tanent line to T Δ m /RT (+) Partial miscibility ( ) x FIGURE 7. System showin instability ap k.

6 3 Classical Thermodynamics o Fluid Systems = µ = µ x ( x ) x FIGURE 7.3 Stable binary system. Copyrihted Material CRC Press/Taylor & Francis 6 the curve o versus x intercepts the vertical axis located at x = at a value equal to the chemical potential o compound, and the vertical axis located at x = at a value equal to the chemical potential o compound at the particular composition where it touches the curve. I the curve o versus x does not have any inlection point, the system presents complete miscibility in all the composition rane. In this case, the slope o the curve with respect to x will continuously increase with respect to an increase in x or with respect to an increase o x. However, i Equation 7.6 is not satisied in some composition rane, as shown in Fiure 7.4, the tanent will touch the curve in two points representin two liquid phases o dierent compositions havin identical values or the chemical potential o compound and also identical values or the chemical potential o compound. Thus, i the condition or stability is not satisied, the mixture will split into two liquid phases o dierent compositions. The points o contact o the tanent line with the curve or the molar Gibbs enery, describin the compositions o the liquid phases at equilibrium, are called binodal points, and the inlection points are called spinodal points. For modelin purposes, it is o interest to relate stability to the excess Gibbs enery in the Lewis convention. From Equations 7.4b and 7.5, we conclude that to show instability in some rane o composition, the ollowin condition must be met: or, m m id E = + < E id, m <

7 Mixture Behavior, Stability, and Azeotropy 33 µ µ x FIGURE 7.4 System showin instability ap. Takin the second derivative o the contribution o the ideal mixture, Equation 5.9, with respect to either x or x, we obtain id, m RT = xx Copyrihted Material CRC Press/Taylor & Francis 6 As the riht-hand side o this expression is always positive, we conirm that the ideal mixture is stable in all o the composition rane. The most interestin conclusion, however, is obtained when we combine this later result with that obtained or the second derivative o the excess Gibbs enery and write that or instability, E RT < (7.7) xx Equation 7.7 shows that in order to represent liquid liquid separation in a binary system, that is, in order to represent instability, an excess Gibbs enery unction must have a neative value o its second derivative in at least part o the composition rane. This is equivalent to say that in some part o the composition rane, excess Gibbs enery unction must present two inlection points in which its derivative chanes sin. Notably, the two-parameter expression o Wilson [], which is one o the most successul or the representation o vapor liquid equilibrium, ails to meet this condition, and thus it is unable to represent liquid liquid equilibrium. In act, or a binary system, the Wilson equation or the excess Gibbs enery can be written as E Wilson = RT [x ln (x + x Λ ) + x ln (x Λ + x )]

8 34 Classical Thermodynamics o Fluid Systems where Λ and Λ are temperature dependent but composition independent adjustable binary parameters. Takin the second derivative o this expression, one obtains E = RT x Wilson Λ Λ + x x + x Λ x Λ + x The riht-hand side o this expression is always positive, no matter the values o the binary parameters Λ and Λ. Hence, the Wilson equation in its two-parameter orm cannot satisy Equation 7.7. CONDITION FOR STABILITY OF LIQUID MIXTURES Particularly useul orms o the stability condition are those written in terms o the uacities o the compounds. For clarity, beore ivin the inal orms, we present three alternative ormulations o the Gibbs Duhem equation or a binary liquid mixture under isothermal conditions. From Equation 4.9, or this case we write x d ln + x ln = and or an ininitesimal chane in composition o the mixture, Copyrihted Material CRC Press/Taylor & Francis 6 ln x d ln + x d = Replacin x by ( x ) and rearranin, we write = x while replacin x by ( x ) in Equation 4.9a, we obtain = x Takin the derivative o Equation 7.3 with respect to x, (4.9a) (4.9b) (4.9c) d( m/ RT) ln ln = ( ln ln ) + x d + x d ( ln ln ) We reconize here that the second bracket o the riht-hand side is null, accordin to Equation 4.9a above, and the third bracket is a constant, independent o composition. Thus, or the second derivative, the only important term is the irst bracket. Hence, d m/ RT =

9 Mixture Behavior, Stability, and Azeotropy 35 Comparison o this result with inequality (Equation 7.5) shows that the condition or stability can be written as > (7.8a) or, equivalently, as =, + > (7.8b) Combinin the inequality in Equation 7.8a with Equations 4.9b and 4.9c o the Gibbs Duhem equation written above, or stability it is necessary that and > (7.9) > (7.) Accordin to the inequalities in Equations 7.9 and 7., in a stable liquid mixture the uacity o a compound always increases with an increase in its concentration. Copyrihted Material CRC Press/Taylor & Francis 6 AZEOTROPY For the purposes o this discussion, we start with Equation 6.4, which is valid at moderate conditions o temperature and pressure. Addin the equations or both compounds in a binary mixture, we obtain For an ideal mixture, then, s s (7.) P = x γ P + x γ P, i, P = x P + x P = P + P P x id s s s i s s Thus, at constant temperature, in a plot o P versus x, the equilibrium pressure o an ideal mixture will be a straiht line. On the other hand, i the activity coeicients are larer than unity, that is, or positive deviations rom ideality, the equilibrium pressure will be a curve above the straiht line. I the activity coeicients are smaller than unity, that is, neative deviations rom ideality, the equilibrium pressure will be a curve below the straiht line. In an ideal solution, the dierent kinds o molecules do not distinuish each other and they mix without heat eects or volume chane. In the case o positive deviations, the two compounds attract each other with orces that are weaker than those existin in an ideal solution. Similarly, in the case o neative deviations, the two compounds attract each other with orces that are stroner