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

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Winter 2013 Chem 254: Introductory Thermodynamics

RR vap 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 iiT 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 xi = mole fraction in liquid phase

* 2 RMsolvent T melt  Tmmelt   solute H 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 ** solventT,,, P   solvent T P  x solvent   : extra pressure (in Bar)

xsolvent : mole fraction of solvent  1

xxsolvent1 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  RTL 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

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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

iim 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 lnKTKT ln  ri  p f  p i   RTTfi

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: OHHO222 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 mm1033 10 VVV ls     m m m ls  

Final Review 105

Winter 2013 Chem 254: Introductory Thermodynamics

m : the molar mass in g  : density in kg m-3

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 nl : # of moles of liquid nv : # 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 xl : fraction of liquid xv : 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  PPPPtot12** tot

l  yz  x   11  yx11  ll   n x ntot zx xxvl 1    11 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