University of Washington Department of Chemistry Chemistry 452/456 Summer Quarter 2013

Transcription

1 Lecture 8/8/3 Unversty o Washngton Departent o Chestry Chestry 45/456 Suer Quarter 3 A. The Gbbs-Duhe Equaton Fro Lecture 7 and ro the dscusson n sectons A and B o ths lecture, t s clear that the actvty o each coponent o a soluton can be related to the ugacty o the correspondng vapor coponent. Soluton actvty coecents can be obtaned or any volatle soluton coponent ro a knowledge o the ugactes o the vapor coponents. However, a gven solute s not volatle, ts vapor ugacty cannot be obtaned. How then to characterze ts soluton actvty? A relatonshp between the checal potentals o the varous soluton coponents can be obtaned ro the Gbbs-Duhe equaton. The G.-D. equaton relates the actvty o volatle soluton coponents to the actvtes o non-volatle coponents. The G.-D. equaton s the oundaton or the theory o so-called collgatve propertes. The Gbbs-Duhe equaton s ounded on a atheatcal property o state unctons called hoogenety. Suppose s a uncton o N ndependent varables x, x, x 3, etc. such that ( x, x xn. The uncton s rst order hoogeneous ( λx, λx λxn λ ( x, x xn where λ s a constant. Functons that are rst order hoogeneous have useul relatonshps wth ther partal dervatves. Because ( λx, λx λxn λ ( x, x xn t ollows that d ( λx, λx λxn d ( λ ( x, x xn ( x, x xn dλ dλ But by the chan rule o calculus d ( λx, λx λxn d d ( λx d x d d( x. λ λ dλ d( λx The precedng equaton ust be vald or any value o λ. Thereore settng λ we ust conclude d ( λx, λx λxn d ( x, x, xn x dλ d( x State uncton are hoogenous rst order unctons. To see how to use ths property consder the state uncton U. The natural varables or U are S, V, and {n }. Thereore, ro the dscusson above U U U U S + V + n S V,{ n} V S,{ n} k n SV,,{ nk } TS PV + n k

2 Now we take the derental o U du TdS + SdT PdV VdP + dn + n d ( k However, by denton we also know that du TdS PdV + dn k These two equatons can only be reconcled SdT VdP + n d Ths expresson s called the Gbbs-Duhe equaton. At constant T and P and or a bnary soluton the Gbbs-Duhe equaton has the or n nd + nd d d n n The equaton d d eans a change n checal potental o n the solvent s related to a change n the checal potental o the solute by a sple rato o the oles present. Ths equaton s portant n evaluatng actvty coecents o non-volatle solutes. For practcal calculatons the Gbbs-Duhe eqn s coonly expressed n terso actvtes and olalty s used nstead o ole racton. Usng the denton + RTln a t ollows that d RTdln a The G-D eqn now has the equvalent ors n χ dln a dln a dln a dln a n χ B. Applcatons: Measurng Solute Actvty by Measurng Solvent Actvty Let us assue we can easure the actvty coecent or water by easurng ts vapor pressure as a uncton o solute concentraton. Assung the water vapor behaves deally: P a γχ P I the solute s non-volatle, we can use the G-D equaton to deterne the solute actvty and ts actvty coecent at soe solute concentraton. Frst we ntegrate the G.-D. equaton: χ dln a dln γ + dln χ dln a χ χdln γ + dχ χdln γ dχ Now because χ+ χ t s true that dχ+ dχ. Then the G.-D. equaton ntegrates to: χ dln γ dlnγ χ

3 State s pure solvent. In ths lt γ ( becoes: d ( ( (. Thereore the G.-D. equaton χ ln γ ln γ ln γ ln γ dln γ χ χ Ths equaton eans we plot as a uncton o ln γ (whch we obtan or χ P each χ ro the equaton γ χ, the area under the curve s ln γ (. P Note when you plot the data, the ntegral wll dverge as χ. Thereore state s dened as χ c where c s a sall enough nuber that ln γ (. C. Collgatve Propertes A collgatve property s a physcal property o the solvent that vares as the actvty o the solvent. Vapor pressure, reezng pont, bolng pont, and osotc pressure are all collgatve propertes. At the noral reezng pont, ce s n equlbru wth pure water: sold lqud. At Pat, ths teperature s T 73.5K. The presence o a solute wll aect the reezng pont. The condton or equlbru s now + RT ln a sold lqud sold lqud lqud sold Guson, ln a RT RT RT Now derentate wth respect to T: ln a G uson, Guson, G uson, T T RT RT T RT S G T S + T S + RT RT RT RT ln a uson, T RT uson, uson, uson, uson, uson, uson, The ost useul or o ths equaton s obtaned by ntegratng wth respect to T ro state (pure water to state (soluton T uson, dln a dt RT T The actvty o pure water (state s so lna or state. Thereore:

4 T T T ln a R T T R TT TT R T uson, T ln γ+ ln χ R uson, uson, uson, ( ( T ( There are two lts or ths equaton. In ts present or, the actvty coecent o the solvent can be deterned as a uncton o χ ro the reezng pont depresson T. Then the actvty coecent or the solute at soe value o χ ay be deterned as descrbed above ro the G.-D. equaton. But n the dlute lt where γ we get: ln χ ln ( χ χ R uson, ( T ( T uson, + uson, MR T n n n M T χ M n n n n M R T T ( D. Other Applcatons o Gbbs-Duhe: The Gbbs Isother Equaton Returnng to the therodynacs o suraces, we consder the surace tenson o a soluton: dg SdT + VdP + γd + dn We need an equaton whch descrbes the relatonshp between the olar ree energy o the surace, and propertes o the surace that we can easure (e.g. surace tenson. Consder an ar-lqud nterace where the lqud s a soluton. A coponent o the soluton s dstrbuted between the bulk soluton phase, the vapor phase β, and the surace. I s n equlbru between the three phases we have β. The total ree energy s the su o the ree energes or a coponet o the syste n the vapor, bulk lqud, and n the surace G ST + γ + n+ n β where n n + n + n For the bulk soluton phase at constant teperature the Gbbs-Duhe equaton s: n nd + nd d d n

5 Derentatng G ST + γ + n+ nand applyng the Gbbs- Duhe equaton we get a useul equaton or the surace tenson and how t depends on soluton coposton: β β dγ + n d + n d dγ + n + n + n d + n + n + n d ( ( dγ + n d + nd d n dγ ( n d + n d n n n n d Γ Γ n n where Γ s the surace adsorpton o speces and has unts o oles per. n For dlute solutons << and the equaton can be rearranged to: n dγ Γ d Use the denton o the checal potental or a dlute soluton d RTd ln a RTd ln γ C RTd ln C The relatonshp between surace tenson and surace adsorpton (.e. surace concentraton s gven by γ C γ Γ RT ln C RT C T T Ths eans that a graph o the surace tenson o a soluton versus logarth o the bulk concentraton o a gven coponent gves the surace adsorpton. Ths s called the Gbbs Isother Equaton. E. Suractants The surace tenson o a pure lqud can be rased or lowered by the presence o certan solutes. The graph below shows typcal behavor or soe solutes.

6 Inorganc salts generally rase the surace tenson when they are added to water. Certan olecules drastcally lower the surace tenson when they are added to water. Such olecules requently aphphlc n the sense that they are lnear olecules coposed o a charged end that s attracted to water and a hydrophobc end that s repelled by water. Exaples nclude organc acds lke butanoc acd. CH 3 CH CH COO - adsorbs to the surace o the water wth the COO - group solvated and the alphatc chan drected away ro the water surace (see dagra above, rght. Such olecules are called surace-actve olecules or suractants. Other exaples nclude lpds and detergents. The total concentraton o the suractant c and the change n surace tenson o the solvent γ γ where γ s the surace tenson o the pure solvent. For alphatc acds ths relatonshp s γ γ alogb + bcg. The constants a and b are deterned by the acd. Acd a (N/ b (L/ole Propanoc acd Butanoc acd Caproc acd Exaple: What s the surace tenson o a.m soluton o caproc acd? Soluton: γ γ alogb+ bcg γ γ alogb+ bcg γ. 775N / b. 98N / glog b g. 775N / b. 98N / gb385. g. 347N / The relatonshp between the concentraton o suractant olecules at the surace, called the surace adsorpton Γ, and the change n surace tenson per unt change n total concentraton o suractant c s gven by the Gbbs Isother Equaton c F dγ Γ H G I RT dckj PT, Exaple: Calculate the surace adsorpton Γ or a.m soluton o caproc acd. a γ γ alogb + bcg γ lnb + bcg. 33 c dγ Γ RT dc c ab RT P, T ( + bc oles / c RT d dc ( + bc (.M (.98J / ( 3.7M ole K 98K ( + 3.7M.M ( 8.3J / ( ( (.5 a γ ln.33 5 oles /