Properties of Solutions
Concentration Terms



Dilute - not a lot of solute.
Concentrated - a large amount of
solute.
Concentration can be expressed
quantitatively is many ways:
• Molarity
• Molality
• Percentage
• Mole fraction
Molarity and Molality

The molarity is the number of moles of
solute in 1 litre of solution.
•

The molality is the number of moles of
solute in 1 kg of solvent.
•

M = moles of solute / V sol’n (litres)
M = moles of solute / kg solvent
Conversion between the two requires
the solutions density.
Partial Molar Thermodynamic
Properties

Define a partial molar
thermodynamic property
as

Euler’s Theorem
 Y
Y J  
 n J


T ,P ,n'
Y   n JY J
J
The Chemical Potential

We define the chemical
potential of a substance as
 G
 J  
 n J


T ,P ,n'
The Wider Significance of 

Shows how all
the extensive
thermodynamic
properties
depend on
system
composition
 U 

 J  
 n J S ,V ,n'
 H 

 J  
 n J S ,P ,n'
 A
 J  
 n J


T ,V ,n'
Thermodynamics of Mixing


Spontaneous mixing of two or
more substances to form
solutions
Gibbs energy of the solution
must be less than G(pure
components)
The Gibbs Energy of Mixing


mix G  nRT   X J ln X J 
 J

The Enthalpy and Entropy
 mix G 



  nR   X J lnX J 
 J

 T P
 mix S
 mix G

T

T



mix H

  T2

P
The Ideal Solution
TmixS/n
kJ/mol
0
TmixH/n
TmixG/n
XA
The Volume and Internal Energy
of Mixing
 mix G 

  mixV
 P T
mix U  mix H  PmixV
Ideal Solution Def’n

For an ideal solution
mixV  0 ; mix H  0
mix U  mix H  PmixV
0
Raoult’s Law

Consider the following system
Raoult’s Law #2

The chemical potential
expressions
 A liq    liq   RT ln X A 
*
A
 A vap    vap   RT ln p A 
O
A
Raoult’s Law: Depression of
Vapour pressure


VP of solution relates to VP of pure
solvent
PA = XAP*A
Solutions that obey Raoult’s law are
called ideal solutions.
Raoult’s Law Example


The total vapour
pressure and partial
vapour pressures of an
ideal binary mixture
Dependence of the vp
on mole fractions of
the components.
An Ideal Solution


Benzene and toluene
behave almost ideally
Follow Raoult’s Law
over the entire
composition range.
Henry’s Law

Henry’s law relates
the vapour pressure
of the solute above
an ideally dilute
solution to
composition.
The Ideal Dilute Solution

Ideal Dilute
Solution
• Solvent obeys
•
Raoult’s Law
Solute obeys
Henry’s Law
Henry’s Law #2

The chemical potential
expressions
O (H )
J  sol ' n   J  liq   RT ln  X J
J vap    vap   RT ln  p J 
O
J


JO(H) is the Henry’s law standard
state.
It is the chemical potential of J in
the vapour when PJ = kJ.

Henry’s Law #3



The Standard State
Chemical potential for
Henry’s Law
When the system is in
equilibrium
The chemical potential
expressions reduce to
Henry’s Law

o
J ,H
  vap  
o
J
RT ln k J 
 J sol' n    J vap 
PJ  k J X J
Henry’s Law in terms of
molalities

The Standard State
Chemical potential for
Henry’s Law
Jo ,m  Jo ,H  RT lnM J m o 

When the system is in
equilibrium
 J sol' n    J vap 

The chemical potential
expressions reduce to
Henry’s Law in terms of
molalities
PJ  k J
m 
mJ
Chemical Potentials in terms of
the Molality

The chemical potential expressions
 J sol ' n   

J ,m
 mJ 
 RT ln o 
m 
oJ,m = chemical potential of the solute in an
ideal 1 molal solution
The Gibbs-Duhem Equation

The Gibbs-Duhem gives us an
interrelationship amongst all partial
molar quantities in a mixture
0   n J dY J
J
Colligative Properties
Colligative Properties

All colligative properties
• Depend on the number and not the nature
of the solute molecules

Due to reduction in chemical potential
in solution vs. that of the pure solvent
• Freezing point depression
• Boiling Point Elevation
• Osmotic Pressure
Boiling Point Elevation

Examine the chemical potential
expressions involved
J liq   J vap 
 J vap   *J liq   RT ln X J
 J vap    J* liq   RT ln X J
 v apG
Boiling Point Elevation #2

The boiling point elevation
 RT b* J  
X B
T b  
 v apH J  


2
 RT b M J 
m J  K b mB
T b  
 v apH J  


*2
Freezing Point Depression

Examine the chemical potential
expressions involved
 J liq    s 
*
J
 J liq   *J liq   RT ln X J
 s    liq   RT ln X J
 fus G
*
J
*
J
Freezing Point Depression #2

Define the freezing point depression
 RT f J  
X B
T f  
 fus H J  


*2
 RT f M J  
m J  K f mB
T f  
 fus H J  


*2
Osmosis
Osmosis

The movement of water through a
semi-permeable membrane from dilute
side to concentrated side
• the movement is such that the two sides
might end up with the same concentration

Osmotic pressure: the pressure
required to prevent this movement
Osmosis – The Thermodynamic
Formulation

Equilibrium is established across
membrane under isothermal
conditions
 J P     *J P 
 J P   , X J   *J P     RT ln X J
 - the osmotic pressure
The Final Equation

The osmotic pressure is related to the
solutions molarity as follows
nB
  RT  M B RT
V
Terminology



Isotonic: having the same osmotic
pressure
Hypertonic: having a higher osmotic
pressure
Hypotonic: having a lower osmotic
pressure
Terminology #2


Hemolysis: the process that ruptures a
cell placed in a solution that is
hypotonic to the cell’s fluid
Crenation: the opposite effect
The Partial Molar Volume

In a multicomponent system
V   n JV J
J
 V
V J  
 n J


T ,P ,n'
Volume Vs. Composition

The partial molar volume of
a substance
•

slope of the variation of the
total sample volume plotted
against composition.
PMV’s vary with solution
composition
The PMV-Composition Plot


The partial molar
volumes of water and
ethanol at 25C.
Note the position of
the maxima and
minima!!
Experimental Determination of
PMV’s


Obtain the densities of systems as a
function of composition
Inverse of density – specific volume
of solution
1


mL
Vs 


g

g




mL



V mL

 A  Bm  Cm
mol
2
Example with Methanol.



Plot volumes vs. mole fraction of
component A or B
Draw a tangent line to the plot of
volume vs. mole fraction.
Where the tangent line intersects the
axis – partial molar volume of the
components at that composition
The Solution Volume vs.
Composition
The Mean Molar Volume

Define the
mean mixing
molar volume
as
• V*J – the molar
•
volume of the
pure liquid
Vm = V/nT
mixV m  V   x JV J*
J
The Mean Molar Volume Plot
 mixVm / (mL/mol)
0.20
0.00
-0.20
-0.40
-0.60
-0.80
VB-VB*
*
V
V
AA
-1.00
-1.20
0.00
0.50
XMeOH
1.00
Infinite Dilution Partial Molar
Properties

The value of a partial molar
thermodynamic property in the
limit of zero volume is its infinite
dilution value
• E.g., for the volumes
V J  x J lim 0 V J

The Definition of the Activity

For any real system, the chemical
potential for the solute (or solvent)
is given by
o
 J    RT ln a J
Activities of Pure Solids/Liquids

The chemical potential is essentially
invariant with pressure for
condensed phases
 J P    J P
o
o
o
  V dp
 
 J P
o
p
J
Po
Pure Solids and Pure Liquids

For a pure solid or a pure liquid at
standard to moderately high pressures
0  RT ln a J
or aJ = 1
Activities in Gaseous Systems

The chemical potential of a real gas
is written in terms of its fugacity
o
 J    RT ln f J
Define the Activity Coefficient


The activity coefficient (J) relates the
activity to the concentration terms of
interest.
In gaseous systems, we relate the
fugacity (or activity) to the ideal
pressure of the gas via
 J PJ  f J
Activities in Solutions


Two conventions
Convention I
• Raoult’s Law is applied to both solute and
solvent

Convention II
• Raoult’s Law is applied to the solvent;
Henry’s Law is applied to the solute
Convention I

We substitute the activity of the
solute and solvent into our
expressions for Raoult’s Law
I
J
PJ  a PJ
I
J
 xJ  a
*
I
J
Convention I (cont’d)

Vapour pressure above real
solutions is related to its liquid
phase mole fraction and the activity
coefficient
PJ   x J PJ
I
J
*
Note – as XJ  1
JI  1 and PJ  PJid
Convention II


The solvent is treated in the same
manner as for Convention I
For the solute, substitute the solute
activity into our Henry’s Law
expression
PJ  a k J
II
J
 xJ  a
II
J
II
J
Convention II (cont’d)

Vapour pressure above real dilute
solutions is related to its liquid
phase mole fraction and activity
coefficient
PJ   x J k J
II
J
Note – as XJ  0
JII  1 and PJ  PJid
Convention II - Molalities

For the solute, we use the molality
as our concentration scale
J  
o m 
J
m 
m 
 RT ln a J
m 
 J mJ  aJ
Note – as mJ  0
J(m)  1 and aJ(m)  mJ