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Chemical Thermodynamics : Georg Duesberg
Chapter 7
Mixtures
Colligative properties
Debye-Hückel
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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.
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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.
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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)
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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
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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?
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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.
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Duesberg, Chemical Thermodynamics
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
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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
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Boiling-point elevation
Tb = KbbB
Freezing-point depression
Tf = KfbB
where bB is the molality of the
solute B in the solution
Colligative Properties
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Duesberg, Chemical Thermodynamics
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
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Duesberg, Chemical Thermodynamics
Chapter 7 : Slide 12
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)
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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
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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
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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
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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?
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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
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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
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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?
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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..
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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)
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Duesberg, Chemical Thermodynamics
Chapter 7 : Slide 22
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
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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
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Duesberg, Chemical Thermodynamics
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
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Duesberg, Chemical Thermodynamics
Chapter 7 : Slide 25
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
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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
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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
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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
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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
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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
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31
Table 25.3: Activity and electrolytes
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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