Chapter 7 : Slide 1

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Chapter 7 Changes of State - Mixtures
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OUTLINE SECTION 7.1 - Partial molar
quantities SECTION 7.2 - Thermodynamics of
mixing SECTION 7.3 - The chemical potentials of
liquids
HOMEWORK EXERCISES 4, 5, 12-14 PROBLEMS none
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PARTIAL MOLAR QUANTITIES
Recall our use of partial pressures
p xA p xB p
pA xA p is the partial pressure
We can define other partial quantities
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PARTIAL MOLAR QUANTITIES
The contribution of one mole of a substance to
the volume of a mixture is called the partial
molar volume of that component.
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PARTIAL MOLAR VOLUME
The total change in volume is nAVA nBVB .
(Composition is essentially unchanged).
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PARTIAL MOLAR VOLUME
Illustration What is the change in volume of
adding 1 mol of water to a large volume of water?
The change in volume is 18cm3
A different answer is obtained if we add 1 mol of
water to a large volume of ethanol.
The change in volume is 14cm3
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PARTIAL MOLAR QUANTITIES
VA is not generally a constant it is a function
of composition
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Self-test 7.1 At 25o C the density of a 50 by
mass ethanol/water solution is 0.914 g cm-3.
Given that the partial molar volume of water in
the solution is 17.4 cm3 mol-1, what is the
partial molar volume of the ethanol?
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The Partial Molar Gibbs Energy (recap from
Chapter 6)
The partial molar Gibbs energy is called the
chemical potential
At constant T and p
G nAµA nBµB
(At equilibrium dG 0)
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The Wider Significance of the Chemical Potential
mi
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The Gibbs-Duhem Equation
A useful expression may be obtained by
differentiating
And then imposing equilibrium i.e. dG 0 (and
recall dG m1dn1 m2dn2).
The Gibbs-Duhem Equation. A similar expression
may be deduced for all partial molar quantities
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The Gibbs-Duhem Equation
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Example 7.1 Using the Gibbs-Duhem Equation The
experimental values of the partial molar volume
of K2SO4(aq) at 298 K are given by the
expression V(K2SO4)/(cm3 mol-1) 32.280
18.216 b1/2 where b is the numerical value of
the molality of K2SO4. Use the Gibbs-Duhem
equation to derive an expression for the partial
molar volume of water in the solution. The molar
volume of pure water at 298 K is 18.079 cm3 mol-1.
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THE THERMODYNAMICS OF MIXING Imagine a system of
two perfect gases in amounts nA and nB at equal T
and p are separated by a barrier. The initial
total Gibbs energy of the system Gi is given by
Gi nAµA nBµB
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THE THERMODYNAMICS OF MIXING
After mixing each gas exerts a partial pressure
pJ, where pA pB p. The final G is given by
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THE THERMODYNAMICS OF MIXING
Writing xJ for the mole fraction of component J
nJ xJ n and pJ /p xJ, so DmixG nRT
(xA ln xA xB ln xB) which is negative.
The Entropy of Mixing
which is positive.
What About the Enthalpy of Mixing?
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THE CHEMICAL POTENTIALS OF LIQUIDS So far we know
how to describe µ for a (perfect) gas. When a
liquid and its vapor are in equilibrium then both
phases must have equal µ, otherwise transfer of
material from one phase to the other would yield
a non-zero DG. Ideal Solutions µA(l) µA?
RT ln pA/p? where the refers to the pure
substance so e.g. pA is the vapor pressure of
pure A. If another substance is dissolved in the
liquid, µA(l) µAq
RT ln pA/pq µA(l)
RT ln pA/pA . For SOME pairs of liquids,
RAOULT'S LAW that pA xA pA is obeyed, so
µA(l) µA(l) RT ln xA
. (NOTE this defines an IDEAL SOLUTION)
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The Measurement of Vapor Pressure of Solutions
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The Measurement of Vapor Pressure of Solutions
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An example of an ideal solution obeying Raoults
Law
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An example of a positive deviation from Raoults
Law, CS2 (CH3)2CO.
(A negative deviation arises from (CH3)2CO
CHCl3.)
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Ideal-Dilute Solutions
For real solutions at low concentrations i.e. xB
ltlt 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 Henrys Law
pB
0
xB
1
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Liquid Mixtures
Ideal Solutions
DmixG nRT (xA ln xA xB ln xB)
DmixH 0
Exactly the same as gases!
Excess Functions We discuss properties of real
solutions in terms excess functions XE. The
excess entropy for example is defined as S E
Dmix S- Dmix S ideal.
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Colligative Properties

• Colligative properties are the properties of
  dilute solutions that depend only on the number
  of solute particles present.
• They include
• The elevation of boiling point
• The depression of boiling point
• 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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The Elevation of Boiling Point
Want to know T at which
µA(g) µA(l) RT ln xA
Presence of a solute at xB causes an increase in
the boiling temp from T to T DT where
DT KxB
b is the molality of the solute (proportional to
xB). Kb is the ebullioscopic constant of the
solvent.
DT K bb
The Depression of Freezing Point
Identical arguments lead to DT K f b where Kf
is the cryoscopic constant.
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Examples 7.8(b) Calculate the cryoscopic and
ebullioscopic constants of naphthalene. 7.10(b)
The addition of 5.0 grams of a compound to 250
grams of naphthalene lowered the freezing point
of the solvent by 0.780 K. Calculate the molar
mass of the compound.
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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
Vant 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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Osmosis
Solution
Height Proportional to Osmotic Pressure P
Solvent A with chemical potential mA(p)
Semipermeable Membrane
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Osmosis
It is assumed that the vant Hoff equation is
only the first term of a virial-like
expression P B R T
1 BB . . .
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• Activities ? How can we adjust previous equations
  to account for deviations from ideal behavior?
• solvent
• solute

Solvent activity General form of the chemical
potential of a real OR ideal solvent
mA mA RT ln
(pA/pA) Ideal solution Raoults 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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Solvent activity
aA is the activity of A essentially an
effective mole fraction
aA pA/pA
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Solute activity ? approach ideal dilute (Henrys
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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Real solutes permit deviations from
ideal-dilute behavior mB mB RT ln aB
Where aB pB/KB and aB gB xB
Note As xB ? 0, aB ? xB and gB ? 1
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Measuring Activity
Use the following information to calculate the
activity and activity coefficient of chloroform
in acetone at 25oC, treating it first as a
solvent and then as a solute with KB 165 Torr.
xC 0 0.2 0.4 0.6 0.8 1.0
pC / Torr 0 35 82 142 200 273
Chloroform regarded as solvent
a 0 0.13 0.30 0.53 0.73 1.0
a p / p
g 0.65 0.75 0.87 0.91 1.0
g a / xC
Chloroform regarded as solute
a 0 0.21 0.50 0.86 1.21 1.65
a p / KB
g 1 1.05 1.25 1.43 1.51 1.65
g a / xC
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Activities in terms of molalities, bB
For an ideal-dilute solute we had written in
terms of mole fractions
mB mB RT ln xB
with
mB mB RT ln (KB /pB)
Molality in terms of mole fraction
bB nB / ( nA Mr(A) )
xB nB / (nAnB) ? nB / nA
bB xB / Mr(A) ? xB bB Mr(A)
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Activities in terms of molalities, bB
xB bB Mr(A)
mB mB RT ln xB
mB mB? at standard molality b 1 mol kg-1
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Activities in terms of molalities, bB
As bB ? 0, mB ? -8 In other words, as a solution
becomes increasingly diluted, the solution
becomes more stabilized It becomes difficult to
remove the last little bit of solute.
To allow for deviations from ideality we
introduce (in the normal way) aB gB bB
(assuming unit-less) Then
mB mB? RT ln aB
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• SUMMARY
• Partial molar quantities and the Gibbs-Duhem
  equation. Tells us how chemical potentials vary
  with composition of a mixture.
• Chemical potentials µ of liquids are accessed
  via µ for the vapor in equilibrium.
• Raoult's Law, Henrys Law
• Real and ideal gases ? activity
• In general µ µs RT ln a.