Phase Equilibria

)()( gl AA )(2)(2 gl OHOH

)()( ls AA )()( gs AA Melting-Freezing

Evaporation-Condensation

Sublimation-Condensation

)()( IIsolidIsolid AA Phase transition

mp

mmm

ST

dpVdTSddG

dpnVdT

nS

nGd

VdpSdTdG

gasm

p

gas

liquidm

p

liquid

solidm

p

solid

ST

ST

ST

diagramTinslope

Sm

Sg >> Sl > Ss

The most stable phase is that with lowest chemical potential.

http://justcallmemsfrizzle.wordpress.com/2008/12/05/phase-change-graph/

Pressure Effect

liquidm

solidm

solidm

liquidm

solidm

liquidm

gasm

gasm

T

gasliquidm

T

liquidsolidm

T

solid

mmm

VVcessubsfewVVusually

VorVV

Vp

Vp

Vp

dpVdTSddG

:tan

-T curve of gases much more largely affected by

pressure change than liquids or

solids

Pressure increase: - Boiling point elevation - Freezing point elevation/depression

Phase Diagram

CO2

Clapeyron Equation)()( gl AA )()( ls AA )()( gs AA

At Equilibrium

)()( AA

tr

tr

trtr

mmmm

mmmm

mmmm

VS

dTdp

dTSdpV

dTSSdpVV

dTSSdpVV

dpVdTSdpVdTS

dd

Clapeyron Equation

Solid-Liquid Equilibrium

)()( ls AA

m

m

fus

fus

pT

pT mfus

fusp

pmfus

fus

mfus

fus

m

fusfus

sm

lm

fus

fus

tr

tr

TT

VH

pp

TdT

VH

dpdTTV

Hdp

TVH

dTdp

TH

SSS

VS

VS

dTdp

m

m

'

12 ln

2'

1

2

1

Slope of pT-curve Usually positive ~ 40 atm/K 40 atm are needed to

change the melting point by 1 K

0,

0,

fuss

ml

m

fuss

ml

ms

ml

mfus

VVVsystemssome

VVVusuallyVVV

m

m

fus

fus

TT

VH

pp'

12 ln

mmm

mfus

mmm

mfus

TTTTVif

TTTTVif

pppp

''

''

1212

0ln0

0ln0

0Upon pressure increase

Melting point elevation

Melting point depression

waterIce skating

m

m

fus

fus

m

mm

fus

fus

m

mm

fus

fus

m

mmm

fus

fus

m

m

fus

fus

TT

VH

p

TTT

VH

pp

smallveryisxifxx

TTT

VH

pp

TTTT

VH

TT

VH

pp

'

12

'

12

''

12

1ln

1ln

lnln

if pressure is changed by p, the

melting point will change by Tm

Liquid-Gas Equilibrium )()( gl AA

vap

vap

VS

dTdp

p

RTVVV

VV

VVVV

gm

lm

gm

lm

gm

lm

gmvaptr

1000b

vapvap T

HS

121

2

121

2

2

2

11ln

11ln

2

1

2

1

TTRH

TpTp

pTpTRH

pp

TdT

RH

pdp

TdT

RH

pdp

RTp

TH

dTdp

vap

v

v

bb

vap

pT

pT

vapp

p

vap

vap

b

b

Slope of pT-curve always positive ~ 0.04 atm/K Boiling point

increases by 25 K upon increasing the pressure by 1 atm.

Clausius-Clapeyron Equation

applies also to s-g equilibrium

CTR

Hp

TdT

RH

pdp

vap

vap

1ln

2

TRH

TRH

C

vap

vap

econstp

eep

1

1

atmvb ppTTat

Benzene has a normal boiling point of 353.25 K. If benzene is to be boiled at 30oC, to what value must the pressure be lowered. Hvap=30.76 kJ/mol

Determine the change in the freezing point of ice upon pressure increase from 1 atm to 2 atm. Vm(water)=18.02 cm3/mol and Vm(ice)=19.63 cm3/mol at 273.15 K. Hfus=6.009 kJ/mol.

Phase RuleF: Number of degrees of freedomNumber of independent variables that can be changed without changing the number of phases

C: number of independent componentsP: number of coexisting phases

F=1

F=2

F=1

F=0

Liquid-Gas Equilibrium

of a binary mixtureIdeal solution: 00 mixmix VH

mixtureBABA

0 mixH Energy of interaction AA,BB = A-BIntramolecular forces AA,BB = A-B

0 mixV mixturemmixturempurempurem BVAVBVAV

mixturemLBmLAmL

200100100

Ideal solutions obey Raoults Law

oiii pxp

L

V

L

V

(pA)solvent > (pA)solution

oiii pxp

1

BA

AA nn

nx

BA pppsolution

BBAABATotal PXPXPPP

AAA PXP

BBB PXP

xbay

xpppp

pxpxpp

pxpxp

pxpxp

ppp

BoA

oB

oAsolution

oBB

oAB

oAsolution

oBB

oABsolution

oBB

oAAsolution

BAsolution

1

p-x phase diagramT=const.

A+BL

V

BoA

oB

oA

oBB

B

oBB

oAA

oAA

oB

LB

oA

LA

oB

LB

tot

BB

totBBtotAA

totVBBtot

VAA

BAgastotal

xppppxy

pxpxpx

pxpxpx

ppy

pyppyp

pxppxp

ppp

BoB

oA

oB

oA

oB

total

BoA

oB

oAtotal

BoB

oA

oB

oAB

B

ypppppp

xppppinsubstitute

yppppyx

solve for xB

L

V

T const.

Ex. Benzene and Toluene

• Consider a mixture of benzene, C6H6, and toluene, C7H8, containing 1.0 mol benzene and 2.0 mol toluene. At 20 °C, the vapor pressures of the pure substances are:P°benzene = 75 torrP°toluene = 22 torr

• Assuming the mixture obeys Raoult’s law, what is the total pressure above this solution?

23

T-x phase diagramp=const.

aB

cB

VcB

aB

L xxnxxn 'Lever Rule

totBA

totVL

nnn

nnn

Distillationp=const.

Colligative Properties

Colligative Properties

Kf and Kb

31

Ex. Boiling Point ElevationA 2.00 g sample of a large biomolecule was dissolved in 15.0 g of CCl4. The boiling point of this solution was determined to be 77.85 °C. Calculate the molar mass of the biomolecule. For CCl4, the Kb = 5.07 °C/m and BPCCl4 = 76.50 °C.

b

solventbsolute

solvent

solutebb

solvent

solutesolute

solutebb

KkgwtTn

kgwtnKT

kgwtnm

mKT

/

/

/

mCKkgkgwt

CCCTTT

obsolvent

oooob

/07.5015.0/

35.150.7685.77

molnsolute310026.4

molgmol

gnmMwt

solute

solutesolute /497

10026.42

3

Ex. Freezing Point DepressionEstimate the freezing point of a permanent type of antifreeze solution made up of 100.0 g ethylene glycol, C2H6O2, (MM = 62.07) and 100.0 g H2O (MM = 18.02).

33CCCTTT

TTT

Ckgwt

nKT

kgwtnmm

mKT

ooof

off

foff

o

EG

EGff

EG

EGEGsolute

soluteff

30300

3010.0

611.186.1/

/

molmolg

gMwt

mnEG

EGEG 611.1

/07.62100

Membranes and PermeabilityMembranes – Separators – Example: Cell walls– Keep mixtures organized and

separated

Permeability– Ability to pass substances through membrane

Semipermeable Membrane– Some substances pass, others don’t.– Selective

Osmosis and Osmotic Pressure

A. Initially, Soln B separated from pure water, A, by osmotic membrane (permeable to water). No osmosis occurred yet

B. After a while, volume of fluid in tube higher. Osmosis has occurred.

35

Flow of water molecules

Net flow

Column risesPressure increases

Increase of flow from right to leftFinally:

Equilibrium established

Flow of water molecules

Net flow = 0

Osmotic pressure (p): Pressure needed to stop the flow.

Equation for Osmotic Pressure

• Assumes dilute solutions

p = i M R T– p = osmotic pressure– i = number of ions per formula unit = 1 for molecules– M = molarity of solution

• Molality, m, would be better, but M simplifies• Especially for dilute solutions, where m M

– T = Kelvin Temperature– R = Ideal Gas constant

= 0.082057 L·atm·mol1K1

37

Eye drops must be at the same osmotic pressure as the human eye to prevent water from moving into or out of the eye. A commercial eye drop solution is 0.327 M in electrolyte particles. What is the osmotic pressure in the human eye at 25°C?

atmKmolKatmLM 00.829808206.0327.0

p

p = MRT T(K) = 25°C + 273.15

Using p to determine MMThe osmotic pressure of an aqueous solution of certain protein was measured to determine its molar mass. The solution contained 3.50 mg of protein in sufficient H2O to form 5.00 mL of solution. The measured osmotic pressure of this solution was 1.54 torr at 25 °C. Calculate the molar mass of the protein.

Lmol

KmolKatmL

torratmtorr

RTM 51028.8

29808206.0

760154.1

p

molLMVMn 735 1014.41000.51028.8

molgmolg

nmassMwt /1045.8

1014.41050.3 3

7

3