Chapter 9: Raoult’s Law, Colligative Properties, Osmosisscienide2.uwaterloo.ca/~nooijen/website_new_20_10_2011/Chem254... · Winter 2013 Chem 254: Introductory Thermodynamics Chapter

Winter 2013 Chem 254: Introductory Thermodynamics

95

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 *

i i iP x P Raoult’s Law, where ix 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 iy

i i totalP y P

This gives the partial pressure of the gas

ix ,

iy can have different values.

Raoult’s Law

Rationalization of Raoult’s Law

Pure substance:

vap evapR AK

where vapR = rate of vaporization, A =area, evapK = rate constant

*

condense condenseR AK P


Chapter 9: Raoult’s Law, Colligative Properties, Osmosis 96

vap condenseR R

*

vap condense iAK AK P

* vap

i

condense

KP

K (this is for a pure substance)

In solution/mixture:

vap vap iR AK x

*

condense condense iR AK P

A vap iK x A *

condense iK P

vap

i i

condense

KP x

K *

i i iP x P

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

i i

lnsolution o i

i i

o

PT RT

P

*

*ln lno

i

o

P PT RT RT

P P

*

i

i

i

Px

P

* lnsolution

i i iRT x

*

i = at vapour pressure of pure substance

ix = mole fraction in liquid phase

Like in Chapter 6

lnmix total i i

i

G n RT x x

lnmix total i i

i

S n R x x

0mixH G T S



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

* lnliquid

i iRT x ix = mole fraction in liquid phase

2*

solvent melt

melt solute

fus

RM TT m

H

2*

solvent boil

boil solute

vap

RM TT m

H

Where solventM is the molar mass of the solvent in kg/mol,

*T is the normal boiling/freezing temperature of the pure solvent



solutem is the molality (mol/kg) of the solute

mass

solute

solute

solvent

nm

Application:

Dissolve 5 g of protein find difference in vapT

5 g

mass ~ masssolute solute

soluteM

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 solvent solventT P T P x

: extra pressure (in Bar)

solventx : mole fraction of solvent 1

1solvent solutex x

* *, , ln solventsolventT P T P RT x (Raoult’s)

ln 0solventRT x

* *, ,T P T P ; 0

* *, , lno

PT P T P RT

P

for gases only not liquids (this is NOT the correct formula!)

Pressure dependence of or liquids

dG SdT VdP



m md S dT V dP

*

m md S dT V dP

mV for liquid is constant with P

* ,P P

m m mP P

d V dP V P P V

* *, , lno m solventT P T P V RT x

ln 1 0m soluteV RT x (exact)

0m soluteV RTx

solute solute

m

solute solvent solvent

n nV RT RT

n n n

m solvent soluteV n RTn

soluteV RTn (osmotic pressure)

solutenRT

V is molaritysoluten

V

Concentration of solute 0.5 mol/L

L Bar / K mol K mol / LR T

Bar 12 Bar Huge Pressures

Flowers keep them pretty

Osmotic pressure pushes outwards giving flower rigidity


100

Reverse osmosis

Presses out pure water (at 27 atm)

ln 0m solventV RT x

If increase solventx must decrease pure water comes out


Final Review 101

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

- Calculate ,p xK K from o

r TG , r H

- Calculate temperature and pressure dependence - Phrase

xK as a function of + solve for eq (or

xK from eq )

- Relation between , ,p x fK K K

2) Chapter 8 : Phase equilibrium

- Phase diagrams P vs T or vs T

- Use Clapeyron Equation to o Calculate

fP at fT (or conversely)

o Calculate ,H S from ,P T data

o mV (s-l) from densities + use s-l clapeyron

3) Binary Mixtures: general use of relations between

, , , , *, ii i i tot i i

total

nx y z P P x

n

4) Question that tests knowledge from Chapter 1 to 5

- Calculate ,H S for reaction/process

- Ideal gas cycle - S system – bath


Final Review 102

Chapter 6:

U

H U PV

A U TS

G H TS U PV TS

Gibbs free energy most important for chemistry: at constant ,T P , no non- PV work

0dG : direction of spontaneous process

0dG : equilibrium

Temperature and Pressure dependence

, , lnf

f i

i

PG T P G T P RT

P

1 1

, ,f i i

f i

G T P G T P H TT T

fT

iT

2

f

i

T

T

H TdT

T

: chemical potential = molar Gibbs free energy

Two phases are in equilibrium if every species has the same chemical potential

I II

i i

From i i totP x P (for ideal gas)

ix : mole fraction (

iy in Chapter 9)

lnmixing tot i i

i

G n RT x x 0

lnmixing tot i i

i

G n RT x x 0

totn : total # of moles

[mixing of ideal gases]


Final Review 103

Chemical Equilibrium

A B C Dm A m B m C m D

0i i

i

A i im products (right)

i im reactants (left)

r i i

i

G

lno

r T pG RT Q

i

ip

i o

PQ

P

o o

r T i f T

i

G G i

G of formation at ,oP T

o

r TG : each species at oP

Equilibrium: 0rG pK

ln o

p r TRT K G

1 1

ln lnr i

p f p i

f i

H TK T K T

R T T

Related : i

x i

i

Q x

xK

p x

o

PK K

P

Extent of Reaction:

o

i i in n

: extent of reaction

i : stoichiometry

o

in : initial number of moles


Final Review 104

tot i

i

n n , ii

tot

nx

n

i

x i

i

Q x

p x

o

PQ Q

P

All are functions of . If I know one of the ix at equation. i

i

tot

Px

P

o

eq x p r TK K G

Example: 2 2 22 2O H H O

Initially : 1 mole of 2O

2 moles of 2H

No 2H O

o

in in ix

2O 1 (1 ) 1

(3 )

2H 2 (2 2 ) 2 2

(3 )

2H O 0 2 2

(3 )

3totn

2

2

2 2

1 2 2

1 2 23 3

2 32

3

xQ


Final Review 105

Chapter 8

- Generic T phase diagram

- Generic P T phase diagram

Curves: indicate phase-coexistence lines at particular ,P T 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

1 1

lnphase transition

f m

i f i

P H

P R T T

sub fus evapH H H

Solid – liquid:

m

m

SP

T V

where

mS is some constant

3 310 10l s

m m m l s

m mV V V


Final Review 106

m : the molar mass in g

: density in kg m-3

fusion

m

fusion

HS

T

Chapter 7

Compression factor

m m

actual

m

V V PZ

V RT

Every substance has ,, ,c c m cP T V at critical point

define reduced dimensionless variables

r

c

PP

P ,

r

c

TT

T , ,

,

mm r

m c

VV

V

,r rZ P T “Universal function”

Fugacity Coefficients:

0

1,

fP

f

ZT P dP

P

,f T P P

lno

o

fT RT

P

lno

o

PT RT

P

i i i

f i i p i x

i i i o

PK f K K

P

p x

o

PK K

P

ln 0o

r T fG RT K

Or

ln 0o

r T pG RT K 1


Final Review 107

Chapter 9 : Binary Solutions

totn : tot # of moles

ln : # of moles of liquid

vn : # of moles of vapour

ix : mole fraction in liquid

iy : mole fraction in vapour

iz : total mole fraction

totP : total vapour pressure

*iP : Vapour pressure of pure i at T

lx : fraction of liquid

vx : fraction of vapour

Raoult: *i i iP x P (ideal solution)

Ideal gas: i i totP y P

1 1 1 2* 1 *totP x P x P

2

1

1 2

*

* *

totP Px

P P

21 1 11

1 2

** *

* *

tot

tot tot

P Px P Py

P P P P

1 1

1 1

l y zx

y x

l l

totn x n

1 1

1 1

1v l z x

x xy x

v v

totn x n

Colligative Properties:

fusion f soluteT k m


Final Review 108

kg solute

mole of solutem

of solvent

2

solvent fusion

f

fusion m

R M Tk

H

boiling b soluteT k m

2

solvent fusion

b

vap m

R M Tk

H

Osmotic Pressure :

solutenRT

V

Or

* ln 0m solventV RT x (more precise)

Chapter 9: Raoult’s Law, Colligative Properties, Osmosisscienide2.uwaterloo.ca/~nooijen/website_new_20_10_2011/Chem254... · Winter 2013 Chem 254: Introductory Thermodynamics Chapter

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