Chapter 7 Mixtures Colligative properties...

Chemical Thermodynamics : Georg Duesberg

Chapter 7

Mixtures

Colligative properties

Debye-Hückel

Deviations from Raoult’s Law

Methanol, ethanol, propanol

mixed with water. Which one

is which? (All show positive

deviations from ideal behavior)

*

AAA PxP Similar liquids can form an ideal solution

obeying Raoult’s Law

Raoult’s law generally describes well solvent vapour pressure when solution is dilute, but not the solute vapour pressure

Experimentally found (by Henry) that vp of solute is proportional to its mole fraction, but proportionality constant is not the vp of pure solute.

Duesberg, Chemical Thermodynamics

Chapter 7 : Slide 3

Ideal-Dilute Solutions (Henry’s Law)

For real solutions at low concentrations i.e. xB << xA the vapor pressure of

a the solute is proportional to its mole fraction but the proportionality

constant is not pA* but some empirical constant KB

pB = xBKB Henry’s Law

In a dilute solution, the solvent molecules are in an

environment that differs only slightly from that of the

pure solvent. The solute particles, however, are in an

environment totally unlike that of the pure solute.

Raoult’s and Henry’s law

1 as * AAAA xPxP

0 as , AAHAA xkxP

Raoult’s law

Henry’s law constant: *

, AAH Pk

The Henry’s law constant reflects the intermolecular interactions between

the two components.

Solutions following both Raoult’s and Henry’s Laws are called ideal-dilute

solutions.

Henry’s behavior:

Henry’s law (Raoult’s Law)

Deviations from Raoult’s Law

CS2 and dimethoxymethane: Positive

deviation from ideal (Raoult’s Law) behavior. trichloromethane/acetone: Negative

deviation from ideal (Raoult’s Law) behavior.

*

AAA PxP

6

Gas solubility

KH/(kPa m3 mol1

)

Ammonia, NH3 5.69

Carbon dioxide, CO2 2.937

Helium, He 282.7

Hydrogen, H2 121.2

Methane, CH4 67.4

Nitrogen, N2 155

Oxygen, O2 74.68

Henry’s law constants for gases dissolved in water at 25°C

Concentration of 4 mg/L of oxygen is required to support aquatic life,

what partial pressure of oxygen is required for this?

Application-diving

Table 1

Increasing severity of nitrogen narcosis symptoms with depth in feet and

pressures in atmospheres.

Depth P Total P N2 Symptoms

100 4.0 3.0 Reasoning measurably slowed.

150 5.5 4.3 Joviality; reflexes slowed; idea

fixation.

200 7.1 5.5 Euphoria; impaired concentration;

drowsiness.

250 8.3 6.4 Mental confusion; inaccurate

observations.

300 10. 7.9 Stupefaction; loss of perceptual

faculties.

Gas narcosis caused by nitrogen in normal air dissolving into nervous

tissue during dives of more than 120 feet [35 m] Pain due to expanding or contracting trapped gases, potentially leading to

Barotrauma. Can occur either during ascent or descent, but are potentially

most severe when gases are expanding.

Decompression sickness due to evolution of inert gas bubbles.


Chapter 7 : Slide 8

Colligative Properties Colligative properties are the properties of dilute solutions that

depend only on the number of solute particles present.

They include:

1. The elevation of boiling point

2. The depression of boiling point

3. The osmotic pressure

All colligative properties stem from the reduction of the solvents m by

the presence of the solute.

µA(l) = µA*(l) + RT ln xA

9

Colligative properties

Chemical potential of a solution (but not vapour or solid) decreases by a factor (RTlnxA) in the presence of solute

Molecular interpretation is based on an enhanced molecular randomness of the solution

Get empirical relationship for FP and BP (related to enthalpies of transition)

mKT

mKT

bb

ff

Kb=ebullioscopic constant

Kf =cryoscopic constant

Boiling-point elevation

Tb = KbbB

Freezing-point depression

Tf = KfbB

where bB is the molality of the

solute B in the solution

Colligative Properties


Chapter 7 : Slide 11

The Elevation of Boiling Point

µA*(g) = µA

*(l) + RT ln xA

Want to know T at which:

Presence of a solute at xB causes an increase in the boiling temp from

T * to T * + T where:

T = KxB H

RTKb

vap

2*

T = K bb b is the molality of the solute (proportional to xB).

Kb is the ebullioscopic constant of the solvent.

Identical arguments lead to: T = K f b where Kf is the cryoscopic constant.

The Depression of Freezing Point

Raoult’s law generally describes well solvent vapour pressure when solution is dilute, but not the solute vapour pressure

Experimentally found (by Henry) that vp of solute is proportional to its mole fraction, but proportionality constant is not the vp of pure solute.

ion vaporisatofheat vap H



Osmosis – for Greek word “push” Spontaneous passage of a pure solvent into a solution separated from it

by a semi-permeable membrane (membrane permeable to the solvent,

but not to the solute)

Osmotic pressure – P – the pressure that must be applied to the solution

to stop the influx of the solvent

Van’t Hoff equation:

P = [B] R T

where [B] = nB/V

Osmometry - determination of molar mass by measurement of

osmotic pressure – macromolecules (proteins and polymers)

13

Osmotic Pressure

P

Pp

pmAAA dpVplplIII *

,

** ),(),(: mm

),,(),(: * P AAA xplplI mm

*

Am ),( PplAm

At equilibrium:

aAaA xRTplxplII ln),(),,(: * PP mm

aAaAA xRTplxplpl ln),(),,(),( ** PP mmm

Effect of mole fraction

Effect of pressure

P

PPp

pmAAAA dpVxRTplpl *

,

** ln),(),( mm

Combine I+II and plug in III

14

Osmotic Pressure

RTnV bP

x b =1-x a … ln (1-x b)~ - xb for dilute solution

P

p

pmAA dpVxRT *

,ln

P *

,ln mAB VxRT

Assume molar volume Va constant in dilute solution and … nb /(na+nb)~ nb /na

Van’t Hoff equation

P = [B] R T where [B] = nB/V

*

Am ),( PplAm

15

Osmotic Pressure and Cells

In the figure, red blood cells are placed into saline solutions.

1. In which case (hypertonic, isotonic, or hypotonic) does the

concentration of the saline solution match that of the blood cells?

2. In which case is the saline solution more concentrated than the

blood cells?

Hemolysis Crenation

16

Osmotic Pressure and Molecular Weight

Exercise:

It is found that 2.20 g of polymer dissolved in enough water to

make 300 mL of solution has an osmotic pressure of 7.45 torr at

20 °C. Determine the molecular mass of the polymer.

Why do we use osmotic pressure to find molecular

weight and not one of the other colligative properties?

Vapor Pressure of Ideal Two-Component Solutions

* * * * * * *

1 2 1 1 2 2 1 1 1 2 1 1 2 2 1(1 ) ( ) (linear in )P P P x P x P x P x P x P P P x

The mole fraction of component 1

in the liquid phase

*

21 * *

1 2

P Px

P P

P *

1P

*

2P

P

1P

2P

1x 10

The variation of the total vapour

pressure of a binary mixture with the

mole fraction of A in the liquid when

Raoult's law is obeyed.

*

AAA PxP

Thus

11

* * * * *

1 1 2 1 1 2

* * * * * *

1 1 2 2 1 2 1 2

(Dalton's law)

1

Py

P

x P P P P P P

x P x P P P P P P P

Calculate the mole fraction y1 of component 1 in the vapor phase at a given

value P of the vapor pressure using Dalton’s law of partial pressures:

1

1

1

depends linearly on .

depends nonlinearly

(hyperbolically) on (and on )!

x P

y

P x

*

1P

*

2P

1 vs. P x

1 1 or x y10

1 vs. P y

liquid-vapor coexistence

*

21 * *

1 2

P Px

P P

Vapor Pressure of Ideal Two-Component Solutions

Gas phase liquid phase

19

Temperature-Composition Diagrams

*

22

*

11760 PxPx

*

1

*

2

*

21

760

PP

Px

760760

*

1111

PxPy

Point a: On solution line …

Point b: On vapor line …

1-propanol and 2-propanol at ambient pressure (i.e., 760 torr)

Dalton’s Law

How does this

relate to fractional

distillation?

20

Fractional Distillation-volatile liquids

Important in oil refining

*

1T

*

2T 1 vs. bT x

10

1 vs. bT y

liquid

vapor

Substance labeled 2 is assumed to

have a lower boiling point. The vapor

is richer than the solution in the more

volatile substance 2, thus y1 < x1..

http://science.howstuffworks.com/oil-refining4.htm

21

Exception:azeotropes Azeotrope: boiling without changing

High-boiling and Low-boiling

Favourable interactions between components

reduce vp of mixture

Trichloromethane/propanone

HCl/water (max at 80% water, 108.6°C)

Unfavourable interactions between

components increase vp of mixture

Ethanol/water (min at 4% water, 78°C)



Activities : How can we adjust previous equations to account for deviations from

ideal behavior of liquids?

1. solvent

2. solute Solvent activity:

General form of the chemical potential of a real or ideal solvent:

mA = mA* + RT ln (pA/pA*)

Ideal solution – Raoult’s law is obeyed:

mA = mA* + RT ln xA i.e xA = pA/pA*

Real solution – we can write:

mA = mA* + RT ln aA

23

Activity of solvent

jj

sol

j aRTl ln)(* mm

*

j

j

jP

Pa ja 1jx

j

j

jx

a

For non-ideal solutions:

Activity

Activity defined as: as

Activity coefficient (a measure of deviation from ideality):

jjjj RTxRT mm lnln* Similar to real gas, fugacity…

All non- ideality in γ


Solute activity approach ideal dilute (Henry’s law)

behavior as xB 0

Ideal-dilute: pB = KB xB

mB = mB* + RT ln (pB/pB*)

= mB* + RT ln (KB /pB*) + RT ln xB

The second term on the rhs of the above equation is

composition independent, so we may define a new reference

state:

mB+ = mB* + RT ln (KB /pB*)

So that: mB = mB+ + RT ln xB



Real solutes permit deviations from ideal-dilute behavior

mB = mB+ + RT ln aB

Where aB = pB/KB and aB = B xB

Note: As xB 0, aB xB and B 1

26

Ionic Activity, Molality, & Activity Coefficients

ma ma

v

vm

We can define single-ion activity coefficients…

Mean ionic molality Mean ionic activity coefficient

mmm vv2 )ln()ln(2 aRTvaRTv mmm

222 ln aRT mm

or

From this reaction…

Also know…

Therefore…

vv

aaa2 or

)()()()(2 aqAvaqCvsAC zzlOH

vv

27

Electrolyte Solutions

Electrolyte solutions deviate from ideal behavior more strongly and at lower

concentrations than non-electrolyte solutions.

Activities/activity coefficients are essential when

working with electrolytes! Examples of electrolytes…

NaCl, MgSO4, MgCl2, Na2SO4

Coulomb interactions imply oppositely charged ions attract

each other

In solutions, near an ion counter ions are found (ionic atmosphere)

Coulomb potential(f) drops as 1/r: fi = Zi/r (Zi a ionic charge)

G (& µ) of ion lowered by electrostatic interactions

– Since µi = µi ideal + RTln(+-), lowering is associated with

RTln(+-)

• ln(+-) can be calculated by modeling these interactions

– Debye-Hückel Limiting Law

28

Debye-Hückel Limiting Law

The variation of the shielded Coulomb

potential with distance for different values

of the Debye length, r/rD. The smaller the

Debye length, the more sharply the

potential decays to zero. In each case, a is

an arbitrary unit of length.

Dr

r

r

i er

ez

04

29

Debye-Hückel Theory

2/1ln cAIzz

s

j

jjc czI1

2

2

1

Debye-Hückel Theory: Assumes ions are point ions (no radii) with

purely Coulombic interactions and activity coefficients depend

only on the ion charges and the solvent properties.

Ionic Strength

For Aqueous Solutions…

2/3

0

2/1

42

Tk

eNA

Br

A

2/1509.0ln cIzz

30

Validity of Debye-Hückel Theory

Electrode-saline interface illustrating

water molecules, sodium (Na+), and

chloride (Cl-) ions. Cl- ions are largely

excluded from the region defined

approximately by the Debye length.

Ionic Strength and Molality

I = k (b/bø)

k X- X2- X3- X4-

M+ 1 3 6 10

M2+ 3 4 15 12

M3+ 6 15 9 42

M4+ 10 12 42 16

31

Table 25.3: Activity and electrolytes

32

Validity of Debye-Hückel Theory

2/1

2/1

1ln

c

c

BI

IzzA

Extended Debye-Hückel:

An experimental test of the Debye-Hückel limiting law. Although there

are marked deviations for moderate ionic strengths, the limiting slopes

as NaCl are in good agreement with the theory, so the law can be used

for extrapolating data to very low molalities. Ar high strenght

Chapter 7 Mixtures Colligative properties...

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