Chapter 5 Solubility and Activity Coefficients in Water First let s get a feeling for the driving force of mixing in terms of the partial molar free energy G or chemical potential i of a compound in a phase which is i i RT ln f i f i if we choose f i as the fugacity of the pure liquid and f i i X i f i pure liquid i i RT ln i X i i w i RT ln i w X i w the organic phase the water phase The difference in chemical potentials for our compound in interest i in the two phases is i w i RT ln i w X i w RT ln i X i 1 where i w i solG i the molar free energy of solution or the driving force for phase transfer Initially if we start with pure organic and water and just consider the X i w organic in water X i w is zero solG i RT ln i w X i w RT ln i X i at some small time dt after the phase transfer starts let say i 999 and X i 0 999 ln i Xi 0 002 At this same time dt let s say two organic molecules have gone into the water phase X iw will be very small eg 2 divided by 6 02 x 10 23 molecules for one mole of water or X iw 10 24 and say iw 1 for a very dilute solution of the organic in water remember toluene 2 iw in water 1x10 4 the product of Xiw and iw is still than 1 so ln iw Xiw ln 1x10 24 x 10 4 46 multiplying by RT gives RT ln iw Xiw 114 kJ mole 1 for solG This makes G negative it s chemical potential difference between the two phases the sign tells the direction of the desired transfer This process continues until iw X iw i X i or where and f iw f i f iw iw X iw f io f i i X i f io going back to solGi RT ln iw X iw RT ln i Xi 3 For the majority of compounds Xi the mole fraction of the organic in the organic layer is essentially one i e there is almost no water in the organic phase we will also assume that the activity coef in the organic phase is essentially ideal and is close to one From Chapter 3 RT ln i hx Xi hx RT ln iH2OXi H2O xiH2O RT ln x xiH2O KiH 2O hx xihx ihx RT ln iH2O RT ln ihx Ki12 1 2G RT ln KH2O hx Free energy of transfer 12Gi RT ln iw RT ln where Xiw G RT ln iw E 4 RT ln Xiw is called the entropy of ideal mixing the RT ln iw term is the molar excess free energy GiE of the liquid compound in water due to the non ideally of solution of the organic in water 5 Chapter 5 compares saturated and infinitely dilute activity coefficients 6 Table 5 2 page 80 old book see page 141 new book 7 Solubility of solids and gases in liquids for liquid liquid interactions organic water 12Gi RT ln iw RT ln Xiw and we have shown over and over again that 1 Xiw iw In dissolving a solid into a liquid we need to also account for melting From the Gibbs Duhem equation for gases in equilibrium with a liquid vapGi vap i RT ln p iL po if po is one bar or atm vapGi vap i RT ln p iL for gas solids by analogy subGi RT ln p iS and fusGi subGi vapGi RTln p iL p iS to account for melting 8 12Gi RT lnXiw RT ln iw ideal mixing nonideal effects p RT ln iL pis melting so 1 iw sat 1 iw sat Xiwsat Xiw sat liquids pis piL solids Note that Xsatiw Vmix Xsatiw L Vmix pis piL pis Csatiw Csatiw L piL Xiw sat 1 iw sat pig piL gases sat As far as computing from Ciw values sat the new book gives log Cw corrected for solid liquid interactions the old book gives both new book example page 140 9 Estimate Csatiw L satiwand GEiw for di n butyl phthalate 1st Csatiw L Csatiw on page 1206 log Csatiw 4 36 Csatiw 4 37x10 5 Csatiw Xi Vmix 1 i Vmix So i 1 4 37x10 5 x 0 018 1 27x106 GEiw RTln I 3483 J molK What about solid hexa chlorocyclohexane If we can estimate p is p iw sat sat pis 1 X iw p solids and plug into iwsat 1 Csatiw Vmix pp solids p a poor man s estimate of lnp is iw iL is iL is iL 10 ln pis piL pis piL 56 5 R Tm Tamb 1 for Tm 1130C 0 076 iwsat Csatiw 2 5x10 5 moles L 1 69 x105 and GEiw RTln I 11 Heats of Solution relationships book Figure 5 3 page 83 old Hcav H1 H2 H3 H1used to break orgainc organic bonds H2 used to break H2O H2O bonds and forming a cavity H3 heat released from organic H2O bonds H1 H2 1 H3 1 Hice water molecules around organic are attracted to outside water molecules and solidified in place HsE Hcav Hice molecular size 12 Enthalpies of solution appear to be related to surface area of the molecule 13 14 Is there a relationship between solubility and molar volumes with in a compound class 15 16 This suggests a generalized relationship ln iw a size b or ln Csatiw a size b 17 Entropy of Dissolution Gs RT ln w RT ln Xw entropy term It is difficult to derive an exact analog between excess enthalpies of solution Hes and excess entropies of solution Ses Since entropy is an indicator of randomness for an ideal solution Sidealmix R nsoluteln Xsolute nsolvent ln Xsolvent here it is assumed that each molecule has approximately the same size and shape The non ideal mixing of large organic molecules results in the displacement of many water molecules It is suggested old book that a better description of the displacement of water molecules is the volume fraction Srealmix R norgln X org nH2O ln X H2O where X is the volume fraction 18 Srealmix R norgln X org nH2O ln X H2O the volume fraction of X H2O is almost 1 so Srealmix R norgln X org if we represent X org as vol In the organic phase totalvol S re a l m ix Rn o rg ln n n o rg o rg V o rg V o rg n V H2O H 2O because nH2O norg S re a l m ix Rn o rg ln n n o rg V o rg H2O V H2O separating the ln term and norg nH2O Xorg S re a l m ix Rn o rg ln x o rg Sidealmix per mole T S e m ix Rn o rg …