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
Winter 2013 Chem 254: Introductory Thermodynamics
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
Winter 2013 Chem 254: Introductory Thermodynamics
Chapter 9: Raoult’s Law, Colligative Properties, Osmosis 97
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
Winter 2013 Chem 254: Introductory Thermodynamics
Chapter 9: Raoult’s Law, Colligative Properties, Osmosis 98
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
Winter 2013 Chem 254: Introductory Thermodynamics
Chapter 9: Raoult’s Law, Colligative Properties, Osmosis 99
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
Winter 2013 Chem 254: Introductory Thermodynamics
100
Reverse osmosis
Presses out pure water (at 27 atm)
ln 0m solventV RT x
If increase solventx must decrease pure water comes out
Winter 2013 Chem 254: Introductory Thermodynamics
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
Winter 2013 Chem 254: Introductory Thermodynamics
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]
Winter 2013 Chem 254: Introductory Thermodynamics
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
Winter 2013 Chem 254: Introductory Thermodynamics
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
Winter 2013 Chem 254: Introductory Thermodynamics
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
Winter 2013 Chem 254: Introductory Thermodynamics
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
Winter 2013 Chem 254: Introductory Thermodynamics
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
Winter 2013 Chem 254: Introductory Thermodynamics
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)