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
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
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:
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
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.
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.98 6.7 Butanoc acd.98 9.64 Caproc acd.98 3.7 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 + 3. 7. 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.33.693 6. 4 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 /