Untitled - Technische Universiteit Eindhoven

Appendix BDerivation of partitioning equationIn this appendix the Gibbs free energies of mixing for both phases are defined from whichthe expressions for the chemical potentials of TMOS in either phase are derived. Thedifferences in molecular sizes are neglected. The oil consists of a single representativehydrocarbon. The free energy of mixing of the oleic phase G o is given byβG o = n o lnn om o+ m o ln ,n o + m o n o + m o(B.1)where β = (R g T ) −1 , n o is the number of TMOS molecules in the oleic phase and m o isthe number of hydrocarbon molecules. The chemical potential of TMOS in oil µ o followsfrom a derivation of the free energy, i.e.βµ o =( ) ∂βGo∂n oT,p,m o= lnn on o + m o.(B.2)In the aqueous phase there is an extra contribution to the free energy G ∗ due to theinteraction of TMOS with water, which favors or hinders the mixing. The free energy ofmixing of the aqueous phase G w is given byβG w = n w lnn wm w+ m w ln + βG ∗ ,n w + m w n w + m w(B.3)where n w is the number of TMOS molecules in the aqueous phase and m w is the combinednumber of water, methanol and silicic acid molecules. Hence, the chemical potential ofTMOS in water µ w becomesβµ w =( ) ∂βGw∂n wThe interaction term is defined asT,p,m w= lnn wn w + m w+ β ∂G∗∂n w.(B.4)G ∗ = n w ε(n m ),(B.5)where ε is an interaction parameter that depends on the methanol concentration n m .Equating the chemical potentials µ o and µ w leads to the partitioning expression[n w n o= exp − ε(n ]m). (B.6)n w + m w n o + m o R g T147