Winter 2013 Chem 254: Introductory Thermodynamics Chapter 9: Raoult’s Law, Colligative Properties, Osmosis ............................................................ 95 Ideal Solutions ........................................................................................................................... 95 Raoult’s Law .............................................................................................................................. 95 Colligative Properties ................................................................................................................ 97 Osmosis ..................................................................................................................................... 98 Final Review ................................................................................................................................ 101 Chapter 9: Raoult’s Law, Colligative Properties, Osmosis Ideal Solutions Ideal solutions include: - Very dilute solutions (no electrolyte/ions) - Mixtures of similar compounds (benzene + toluene) * Pure substance: Vapour Pressure P at particular T For a mixture in liquid phase * Pi x i P i Raoult’s Law, where xi is the mole fraction in the liquid phase This gives the partial vapour pressure by a volatile substance in a mixture In contrast to mole fraction in the gas phase yi Pi y i P total This gives the partial pressure of the gas xi , yi can have different values. Raoult’s Law Rationalization of Raoult’s Law Pure substance: Rvap AK evap where Rvap = rate of vaporization, A =area, Kevap = rate constant * Rcondense AK condense P 95 Winter 2013 Chem 254: Introductory Thermodynamics RRvap condense * AKvap AK condense P i * Kvap Pi (this is for a pure substance) Kcondense In solution/mixture: Rvap AK vap x i * Rcondense AK condense P i * AKvap x i AKPcondense i K vap * Pxii Pi x i P i Kcondense Raoult’s Law holds: - A-B mixture (even 50-50), A-A, A-B, B-B, interaction similar - dilute solution A >>> B (~99.9%), law holds for solvent, essentially only A-A interactions For solution in equilibrium with vapours solution vapour ii solution o Pi iiT RT ln Po * o PP Pi i T RTln RT ln* xi * Po P Pi solution * i iRTln x i * i = at vapour pressure of pure substance xi = mole fraction in liquid phase Like in Chapter 6 Gmix n total RT x iln x i i Smix n total R x iln x i i HGTSmix 0 Chapter 9: Raoult’s Law, Colligative Properties, Osmosis 96 Winter 2013 Chem 254: Introductory Thermodynamics Colligative Properties - Consider volatile solvent: add a bit of solute (salt or sucrose), liquid phase becomes a mixture - liquid freezes solid is pure solvent - liquid evaporates vapour is also pure solvent For solvent: melting point reduced (ice +salt) boiling point increased The increase/decrease depends only on molar concentration NOT nature of solute solid gas pure phase , liquid * iiRTln x = mole fraction in liquid phase * 2 RMsolvent T melt Tm meltH solute xi fus * 2 RMsolvent T boil Tmboil solute Hvap Where M solvent is the molar mass of the solvent in kg/mol, T * is the normal boiling/freezing temperature of the pure solvent Chapter 9: Raoult’s Law, Colligative Properties, Osmosis 97 Winter 2013 Chem 254: Introductory Thermodynamics msolute is the molality (mol/kg) of the solute nsolute msolute masssolvent Application: Dissolve 5 g of protein find difference in Tvap 5g mass ~ mass soluteM solute solute Osmosis I : Pure solvent (water) II : solute dissolved in solvent Interface: membrane permeable to solvent not solute Pressure in solution II is significantly higher Thermodynamics Explanation I II solvent solvent ** solventT,,, P solvent T P x solvent : extra pressure (in Bar) xsolvent : mole fraction of solvent 1 xxsolvent1 solute **T, P T , P RT ln x solvent solvent (Raoult’s) RTln xsolvent 0 **TPTP,, ; 0 **P T, P T , P RT ln Po for gases only not liquids (this is NOT the correct formula!) Pressure dependence of or liquids dG SdT VdP Chapter 9: Raoult’s Law, Colligative Properties, Osmosis 98 Winter 2013 Chem 254: Introductory Thermodynamics d Smm dT V dP * d Smm dT V dP Vm for liquid is constant with P PP d* V dP V P , P V PPm m m ** T, P T , Po V m RT ln x solvent Vm RTln 1 x solute 0 (exact) Vm RTx solute 0 nnsolute solute Vm RT RT nsolute n solvent n solvent Vm n solvent RTn solute V RTnsolute (osmotic pressure) nsolute nsolute RT is molarity V V Concentration of solute 0.5 mol/L RTL Bar / K mol K mol / L Bar 12 Bar Huge Pressures Flowers keep them pretty Osmotic pressure pushes outwards giving flower rigidity Chapter 9: Raoult’s Law, Colligative Properties, Osmosis 99 Winter 2013 Chem 254: Introductory Thermodynamics Reverse osmosis Presses out pure water (at 27 atm) Vm RTln x solvent 0 If increase xsolvent must decrease pure water comes out 100 Winter 2013 Chem 254: Introductory Thermodynamics Final Review Chapters to be covered: 1, 2, 3.1-3.6, 4.1-4.5, 5.1-5.11, 6, 7, 8.1-8.5, 9.1-9.4 4 Questions 1) Chapter 6 : Chemical Equilibrium o - Calculate KKpx, from rTG , r H - Calculate temperature and pressure dependence - Phrase K x as a function of + solve for eq (or K x from eq ) - Relation between KKKp,, x f 2) Chapter 8 : Phase equilibrium - Phase diagrams P vs T or vs - Use Clapeyron Equation to o Calculate Pf at Tf (or conversely) o Calculate HS, from PT, data o Vm (s-l) from densities + use s-l clapeyron 3) Binary Mixtures: general use of relations between ni xi, y i , z i , P tot , P i *, x i ntotal 4) Question that tests knowledge from Chapter 1 to 5 - Calculate HS, for reaction/process - Ideal gas cycle - S system – bath Final Review 101 Winter 2013 Chem 254: Introductory Thermodynamics Chapter 6: U H U PV A U TS G H TS U PV TS Gibbs free energy most important for chemistry: at constant TP, , no non- PV work dG 0 : direction of spontaneous process dG 0 : equilibrium Temperature and Pressure dependence Pf G T, Pfi G T , P RT ln Pi 11 GTPGTPHT f,, i i TTfi Tf Ti Tf HT 2 dT Ti T : chemical potential = molar Gibbs free energy Two phases are in equilibrium if every species has the same chemical potential I II ii From Pi x i P tot (for ideal gas) xi : mole fraction ( yi in Chapter 9) Gmixing n tot RT x iln x i 0 i Gmixing n tot RT x iln x i 0 i ntot : total # of moles [mixing of ideal gases] Final Review 102 Winter 2013 Chem 254: Introductory Thermodynamics Chemical Equilibrium mABCD A m B m C m D iiA 0 ii m products (right) i iim reactants (left) rG i i i o rG T RTln Q p i Pi Qp i Po oo rG T i f G T i i G of formation at PTo , o rTG : each species at Po Equilibrium: rG 0 K p o RTln Kp r G T HT 11 lnKTKT ln ri p f p i RTTfi i Related : Qxxi K x i P KKpx Po Extent of Reaction: o nni i i : extent of reaction i : stoichiometry o ni : initial number of moles Final Review 103 Winter 2013 Chem 254: Introductory Thermodynamics ni nntot i , xi i ntot i P Qxxi QQpx i Po Pi All are functions of . If I know one of the xi at equation. xi Ptot o eq KKG x p r T Example: OHHO222 2 2 Initially : 1 mole of O2 2 moles of H 2 No HO2 o ni ni xi 1 O 1 (1 ) 2 (3 ) 22 H 2 (2 2 ) 2 (3 ) 2 HO 0 2 2 (3 ) ntot 3 2 1 2 2 2 33 1 2 2 Qx 22 2 23 3 Final Review 104 Winter 2013 Chem 254: Introductory Thermodynamics Chapter 8 - Generic T phase diagram - Generic PT phase diagram Curves: indicate phase-coexistence lines at particular PT, for pure substance Eg. Vapour pressure as a function of T for liquid - Paths through phase diagrams Clapeyron indicates pressure – temperature dependence of gas/liquid vs solid coexistence curve P H phase transition 11 ln f m PRTTi f i HHHsub fus evap Solid – liquid: P Sm where Sm is some constant TVm mm1033 10 VVV ls m m m ls Final Review 105 Winter 2013 Chem 254: Introductory Thermodynamics m : the molar mass in g -3 : density in kg m H fusion Sm Tfusion Chapter 7 Compression factor VVPmm Z actual Vm RT Every substance has PTVc,, c m, c at critical point define reduced dimensionless variables P T Vm Pr , Tr , Vmr, Pc Tc Vmc, ZPT rr, “Universal function” Fugacity Coefficients: Pf Z 1 T, Pf dP 0 P f T, P P o f T RT ln Po o P T RT ln Po i i i P Kf f i i K p i K x i i i Po P KKpx Po o rG T RTln K f 0 Or o rG T RTln K p 0 1 Final Review 106 Winter 2013 Chem 254: Introductory Thermodynamics Chapter 9 : Binary Solutions ntot : tot # of moles l n : # of moles of liquid v n : # of moles of vapour xi : mole fraction in liquid yi : mole fraction in vapour zi : total mole fraction Ptot : total vapour pressure Pi * : Vapour pressure of pure i at T l x : fraction of liquid v x : fraction of vapour Raoult: Pi x i P i * (ideal solution) Ideal gas: Pi y i P tot Ptot x1 P 1* 1 x 1 P 2 * PPtot 2 * x1 PP12** x1 P 1**PPtot 2 * P 1 y1 PPPPtot12** tot l yz11 x yx11 ll n x ntot vl zx11 xx1 yx11 vv n x ntot Colligative Properties: Tfusion k f m solute Final Review 107 Winter 2013 Chem 254: Introductory Thermodynamics mole of solute m solute kg of solvent 2 RMT solvent fusion k f fusionH m Tboiling k b m solute 2 RMT solvent fusion kb vapH m Osmotic Pressure : n solute RT V Or Vm* RT ln x solvent 0 (more precise) Final Review 108 .