Transcript 6065

CHAPTER 4
INTERMOLECULAR FORCES AND
CORRESPONDING STATES AND
OSMOTIC SYSTEMS
Tables of Contents
4.1 Potential-Energy Function
4.2 Electrostatic Forces
4.3 Polarizability and Induced
Dipoles
4.4 Intermolecular Forces between
Nonpolar Molecules
4.5 Mie's Potential-Energy Function
for Nonpolar Molecules
4.6 Structural Effects
4.7 Specific (Chemical) Forces
4.8 Hydrogen Bonds
4.9 Electron Donor-Electron Acceptor
Complexes
4.10 Hydrophobic Interactions
4.11 Molecular Interactions in Dense
Fluid Media: Osmotic Pressure
and Donnan Equilibria
4.12 Molecular Theory of
Corresponding States
4.13 Extension of CorrespondingStates Theory to More
Complicated Molecules
4.14 Summary
Introduction
Thermodynamic properties are determined by
intermolecular forces
 The objective of this chapter is to give a brief
introduction to the nature and variety of forces
between molecules


In the classical approximation
Qtrans  Qkin  N , T  Z N  N , T ,V 
Qkin
 2mkT 


 h2 
3 2 N
1
N!
ZN
 t  r1    r N  
 dr1    dr N
      exp 

V
kT


 the
kinetic energy (Qkin) depends only on the
temperature.
 ZN, the configurational partition function




t(r1rN) is the potential energy of the entire system of N
molecules whose positions are described by r1rN.
depends on intermolecular forces.
Z Nid  V N
For an ideal gas (t = 0)
The complete canonical partitional function
Q Qint N , T Qkin N , T Z N N , T ,V 
Intermolecular forces considered

Electrostatic forces: charged particles (ions); permanent dipoles,
quadrupoles, and higher multipoles.

Induction forces : a permanent dipole (or quadrupole) and an
induced dipole.

Attraction (dispersion forces) and repulsion between nonpolar
molecules.

Specific (chemical) forces leading to association and solvation,
i.e., to the formation of loose chemical bonds; hydrogen bonds and
charge-transfer complexes.
4.1 Potential-Energy Functions

Force F between molecules (spherically symmetric molecules)
d
F
dr

r
Potential energy

(r)   Fdr
r
A general form for complicated
molecules

Force should be as a function of distance, angle of
orientation of molecules
F  r , ,  ,...   r , ,  ,....
(4-2)
4.2 Electrostatic Forces

Coulomb's relation (Inverse-square law)
F
 qi
qi q j
4o r 2
, point electric charge
 o , the dielectric permittivity of a vacuum
o = 8.85419 10-12 C2 J-1 m-1
(4-3)

Potential energy
ij   
r

qi q j

4o r
2
dr 
Let
r  , ij  0
ij 
qi q j
qi q j
4o r
 constant of integratio n
4o r
For qi = zi e, e = unit charge = 1.60218 × 10-19 C
ij 
zi z j e 2
4o r
C2
[] 2 1 1  J
CJ m m
(4-5)

For a medium other than vacuum
ij 


zi z j e 2
4r

zi z j e 2
40 r r
, the absolute permittivity, =o r
r=dielectric constant (permittivity relative to that of a
vacuum)
Comparison to physical intermolecular
energies

Coulomb energy is large and long range
isolated ion (Cl- and Na+) in contact, the sum
of the two ionic radii = 0.276 nm = 2.76 A
2
zi z j e 2

 1 11.60218  1019 
C2
ij 


4o r 4 8.8542  1012 0.276  10 9  C 2  J 1  m 1  m
 Consider
 8.36  10 19 J
(N  m)
r = 2.76A
 The
same order of magnitude as typical covalent bond
(200kT at room temperature)


kT  1.38  1023 J K 1  300K  0.0414  1019 J
200 kT  8.28  1019 J
 When
two ions are 560 A apart, Coulomb energy = kT
Electrostatic forces-long range
Have a much longer range than most other
intermolecular forces that depend on higher
powers of the reciprocal distance
 Salt crystal, very high melting points of salt
 Long range nature of ionic forces is
responsible for the difficult in constructing a
theory of electrolyte solutions

Dipole Moment

A particle has two electric charges of the same
magnitude e but of opposite sign, held a distance d
apart.
  ed
e+
d
(4-6)
e-
Units and constants
used in this chapter





1 D(ebye) =3.33569×10-30 C m
ε0 = 8.85419×10-12 C2 J-1 m-1
k = 1.38066×10-23 J K-1
e = 1.60218×10-19C
The dipole moment of a pair +e and –e separated
by 0.1 nm (1 A)
  1.60218  1019 C1  1010 m   1.60218  1029 C m
 4.8 Debye
Table 4-1 Permanent dipole moments
Molecules
(Debye)
Molecules
(Debye)
CO
0.10
H2O
1.84
HBr
0.80
HF
1.91
NH3
1.47
CH3CN
3.94
SO2
1.61
KBr
9.07
Potential energy of two permanent
dipoles

Considering the Coulombic forces between the
four charges.
The potential energy of two dipoles
ij  

i j
4o r
2 cos  cos 
i
j

 sin i sin  j cosi   j 
θi  180, θ j  180
Maximum potential
+

3
-
-
+
Minimum potential
+
-
i   j  0
θi  180, θ j  0
+
-
i   j  0
Average potential energy

The average potential energy ij between tow dipoles i and j
in vacuum at a fixed separation r is found by averaging over
all orientations with each orientation weighted according to
Boltzmann factor (Hirschfelder et al., 1964)
 ij 
 ije
e
 ij / kT
 ij / kT
d
i  j
2

d

 ije
e
 ij / kT
 ij / kT
sin 1 sin  2 d1d2 d
sin 1 sin  2 d1d2 d
2
2

2
6
3 4o  kTr
(4-8)
Potential Energy for Dipole Moment
ij  (distance)-6
 For pure polar substance
ij  (dipole moment)4

 dipole
moment < 1 debye, small contribution
 dipole moment > 1 debye, significant contribution
Quadrupole Moments

Molecules have quadrupole moments due to the
concentration of electric charge at four separate
points in the molecules
Quadrupole Moments

For a linear molecule, quadrupole moment Q is defined by the sum of the
second moments of the charges
Q   ei d i2
(4-9)
i


where the charge ei is located at a point at a distance di away from some
arbitrary origin and where all charges are on the same straight line.
For nonlinear quadrupole or for molecules having permanent dipole, the
definition of the quadrupole moment is more complicated.
Table 4-2 Quadrupole moments for
selected molecules
Molecule
Q1040(C m2)
Molecule
Q1040(C m2)
H2
+2.2
C6H6
+12
C2H2
+10
N2
-5.0
C2H4
+5.0
O2
-1.3
C2H6
-2.2
N 2O
-10
Potential energy between dipole and
quadrupole or quadrupole and quadrupole

The average potential energy is found by averaging over all orientations; each
orientation is weighted according to its Boltzmann factor (Hirschfelder et al., 1964).
Upon expanding in powers of 1/kT,

For dipole i-quadrupole j
i 2Q 2j
ij  
 ...
2
8
4o  kTr
(4-10)
 For quadrupole i-quadrupole j
Qi2Q 2j
7
ij  
 ...
2
10
40 4o  kTr
(4-11)
Dependence of Potential Energy on
Separation Distance

Charged molecules (ions, Coulomb’s relation)
 ij

 (distance)-1, long range effect
(4-5)
Dipole moment,
 ij

(distance)-6,
short range effect
(4-8)

Dipole moment-Quadrupole moment
(4-10)

Quadrupolemoment-Quadrupole moment
(4-11)
• ij  (distance)-8 , very short range effect
• ij  (distance)-10 , extremely short range effect
Remarks on
Dipole(2)/Quadrupole(4)/Octopole(8)/
Hexadecapole(16) Moments

Literature study


Dipole(extensive)>Quadrupole(less)>Octopole(little)>Hexadecapole
(much less)
Effect on thermodynamic properties

Dipole(large)>Quadrupole(less)>Octopole(negligible)>Hexadecapole
(negligible) due to short ranges
4.3 Polarizability and Induced Dipoles


A nonpolar molecule has no permanent moment but when
such a molecule is subjected to an electric field, the electrons
are displaced from their ordinary positions and a dipole is
induced.
The induced dipole moments is defined as
 i  E

Where E is the field strength and  is polarizability, a fundamental
property of the substance.
Polarizability
Indicates that how easily the molecules
electrons can be displaced by an electric field.
 Polarizability can be calculated in several
ways, most notably from dielectric properties
and from index-of-refraction data.
 For asymmetric molecules, polarizability is not
a constant but a function of the molecule’s
orientation relative to direction of field.

Polarizability volume

Polarizability has the units C2J-1 m2, however, it is
common practice to present polarizabilities in units of
volume as

C2 J -1m 2
 '
[] 2 -1 -1  m3
4o
CJ m

’ is called polarizability volume.
Table 4-3 Average Polarizabilities
Molecule
’1024(cm3)
Molecule
’1024(cm3)
H2
0.81
SO2
3.89
N2
1.74
Cl2
4.61
CH4
2.60
CHCl3
8.50
HBr
3.61
Anthracene
35.2
Mean Potential Energy nonpolar-polar molecules


A nonpolar molecule i is situated in an electric field set up by
the presence of a nearby polar molecule j, the resultant force
between the permanent dipole and the induced dipole is
always attractive.
j
The mean potential energy was first calculated
by Debye
i  j 2
ij  
 f T 
2 6
4o  r
i
(4-13)
+
-
+
Nonpolar
Polar
Mean Potential Energy polar-polar


Polar as well as nonpolar can have dipole induced
in an electric field.
The mean potential energy due to induction by
permanent dipoles is
ij



i
2
j
  j
4o  r
2
6
2
i
  f T 
i
j
-
+
+
-
Polar
Polar
(4-14)
Mean Potential Energy Quadrupole-Quadrupole

The average potential energy of induction
between two quadrupole molecules


2
2
3  iQ j   j Qi
ij  
 f T 
2 8
2 4o  r
(4-15)
Mean potential energy due to
moments
Due to Induced dipole moment is smaller
than that to permanent dipole-dipole
interactions
 Due to Induced quadrupole moment is
smaller than that to permanent quadrupolequadrupole interactions

4.4 Intermolecular Forces between Nonpolar
Molecules



In 1930 it was shown by London that nonpolar molecules are,
in fact, nonpolar only when viewed over a period of time; if an
instantaneous photograph of such a molecule were taken, it
would show that, at a given instant, the oscillations of the
electrons about the nucleus has resulted in a distortion of
electron arrangement sufficient to cause a temporary dipole
moment.
This dipole moment, rapidly changing its magnitude and
direction, averages zero over a short period of time; however,
these quickly varying dipoles produce an electric field which
then induces dipoles in the surrounding molecules.
The result of this induction is an attractive force called the
induced dipole-induced dipole force.
Potential energy for nonpolar
molecules

Using quantum mechanics, London showed that subject to
certain simplifying assumptions, the potential energy
between two simple, spherically symmetric molecules i and j
at large distances is given by
3  i  j  h 0i h 0 j
ij  
2 4o 2 r 6  h 0i  h 0 j





(4-16)
Where h is Planck’s constant, and vo is a characteristic electronic frequency
for each molecule in its unexcited state.
First ionization potential I for hv0

For a molecule i, the product hv0 is very nearly equal
to its ionization potential, Ii
hv0  I
3  i  j  h 0i h 0 j
ij  
2 4o 2 r 6  h 0i  h 0 j
3  i  j  I i I j

2 4o 2 r 6  I i  I j










For molecules i and j
3  i  j  I i I j
ij  
2 4o 2 r 6  I i  I j




(4-18)
For the same molecules, i = j
3 i I i
ii  
4 4o 2 r 6
2

Potential energy f(T)

Potential energy  r-6
(4-19)
Table 4-4 First ionization potentials
Molecule
I (eV)
Molecule
I (eV)
C6H5CH3
8.9
CCl4
11.0
C6H6
9.2
C2H2
11.4
N-C7H16
10.4
H2O
12.6
C2H5OH
10.7
CO
14.1
1 eV  1.60218  10
19
J
Polarizability dominate over ionization
potential
London’s formula is more sensitive to the
polarizability () than it is to the ionization
potential (I)
 For typical molecules, polarizability () is
roughly proportional to molecular size while
the ionization potential (I) does not change
much form one molecule to another

ij  k '


i  j
r
6
;
2
(4-20)
Where k’ is a constant that is approximately the same for the three
types of interaction, i-i, i-j, and j-j.
The attractive potential between two dissimilar molecules is
approximately given by the geometric mean of the potentials
between the like molecules at the same separation
ij  ii jj 
1/ 2

j
i
ii  k ' 6 ; jj  k ' 6
r
r
2
(4-21)
The above equation gives some theoretical basis
for the"geometric-mean rule" .
Comparison of dipole, induction, and
dispersion forces

London has presented calculated potential energies:
Two Identical Molecules
i
2
ii  
2
3 4o  kTr6
4
ii  
2  
2
i i
2 6
o
4  r
3 i I i
ii  
4 4o 2 r 6
4-8
B
ii   6
r
2
4-19
4-13
Table 4-5 Relative magnitudes
Molecule
CCl4
Dipole,
debye
0
CO
Bx1079Jm6, Bx1079Jm6, Bx1079Jm6,
dipole
Induction Dispersion
0
0
1460
0.10
0.0018
0.0390
64.3
HBr
0.80
7.24
4.62
188
HCl
1.08
24.1
6.14
107
H2O
1.84
203
10.8
38.1
(CH3)CO
2.87
1200
104
486
Remarks on Table 4-5
The contribution of induction forces is small;
even for strongly polar substances such as
ammonia, water, or acetone
 the contribution of dispersion forces is far from
negligible.
 the contribution of dipolar moment is large for
dipole moment > 1.0 debyte.

Table 4-6 Relative magnitudes
Molecule Molecule
(1)
(2)
Dipole
(1)
Dipole
(2)
Bx1079Jm6,
dipole
Bx1079Jm6,
Bx1079Jm6,
Induction Dispersion
CCl4
c-C6H12
0
0
0
0
1510
CCl4
NH3
0
1.47
0
22.7
320
CO
HCl
0.10
1.08
0.206
2.30
82.7
H2O
HCl
1.84
1.08
69.8
10.8
63.7
(CH3)2CO
NH3
2.87
1.47
315
32.3
185
(CH3)2CO
H2O
2.87
1.84
493
34.5
135
Remarks on Table 4-6
Polar forces are not important when the dipole
moment is less than about 1 debye
 induction forces always tend to be much
smaller than dispersion forces.

Intermolecular Forces between Nonpolar
Molecules
London's formula does not hold at very
small separations
 Repulsive forces between nonpolar
molecules at small distances are not
understood
 Theoretical considerations suggest that the
repulsive potential should be an exponential
function of intermolecular separation

Total potential energy for nonpolar
molecules

Attractive potential
(London, 1937)
B
 6
r

Repulsive potential
(Amdur et al., 1954)


Total potential energy
(Mie, 1903)
A
rn
A B
total  repulsive  attractive  n  m
r
r
4.5 Mie's Potential-Energy Function
for Nonpolar Molecules

Mie's potential (1903)
 n / m
n



m 1/  n  m
nm
   n    m 
     
r 
 r
(4-25)
Lennard-Jones potential
  12    6 
  4      
 r  
 r 
(4-26)
Parameters in potential function


Parameters: , , m, n
For a Mie (n, 6) potential
1 /  n 6 
6
 
n


rmin
Parameters can be computed (with simplifying assumptions) from the
compressibility of solids at low temperatures or from specific heat
data of solid or liquids.
Parameters can also be obtained from the variation of viscosity or
self-diffusivity with temperature at low pressures, and from gas phase
volumetric properties (second virial coefficients).
Application of Mie’s potential



Mie’s potential applies to two nonpolar, spherically
symmetric molecules that are completely isolated.
In nondilute systems, and especially in condensed
phases, two molecules are not isolated but have
many other molecules in their vicinity.
By introducing appropriate simplifying assumptions,
it is possible to construct a simple theory of dense
media using a form of Mie’s two-body potential.
Simple theory of dense media using
Mie’s potential




Consider a condensed system near the triple point.
Assume total potential energy is due to primarily to interactions between nearest
neighbors.
Let the number of nearest neighbors is in a molecular arrangement be designated
by z.
In a system containing N molecules, the total potential energy t is then
approximately given by
1
t  Nz
2

Where  is the potential energy of an isolated pair; ½ is needed to avoid counting each
pair twice.

Substituting Mie’s equation
1
1 A B
t  Nz  Nz n  m 
2
2 r
r 

To account for additional potential energy resulting from
interaction of a molecule with all of those outside its nearestneighbor shell, numerical constant sn and sm are introduced by
1  sn A sm B 
t  Nz n  m 
2  r
r 
Determine of sn and sm


When the condensed system is considered as a lattice such as that existing in a
regularly spaced crystal, the constants sn and sm can be accurately determined from
the lattice geometry.
For example, a molecule in a crystal of the simple-cubic type has 6 nearest
neighbors (z = 6) at a distance r, 12 at a distance r(2)1/2, 8 at a distance r(3)1/2.
The attractive energy of one molecule with respect to all of the others is given
6

12
8
t, attractive  B  m 

 
m
m
r


2r
3r
 zBs
6B 
2
4
 m 1 

   m m
m
m2
r 
r

2
3


2
4
sm  1 

 
m
m


2
3
   
   
   
Table 4-7 Summation constant sn and
sm (Moelwyn-Hughes, 1961)
n or m
m=6
Simple cubic Body-centered Face-centered
cubic, z=8
cubic, z=12
z=6
sm =1.4003 sm = 1.5317 sm = 1.2045
n=9
sn =1.1048
sn = 1.2368
sn = 1.0410
n= 12
sn = 1.0337
sn = 1.1394
sn = 1.0110
n= 15
sn = 1.0115
sn = 1.0854
sn = 1.0033
Relation of rmin(isolated pair)
and rmint(pair in a condensed system)

At equilibrium, the potential energy of the condensed system
is a minimum
 dt 
0


 dr  r  rmin t
rmin t 
nm
sn nA

sm mB
rmin  rmin t
1  sn A sm B 
t  Nz n  m 
2  r
r 
 rmin

 rmin t



nm
sm

sn
4.6 Structural Effects

Intermolecular forces of nonspherical molecules depend not only on
the center-to-center distance but also on the relative orientation of
the molecules.

Branching lower the boiling point; the surface area per
molecule decreases

Mixing liquids of different
degrees of order usually
brings about a net decrease
of order, and hence positive
contributions to the
enthalpy h and entropy s
of mixing

At mole fraction x = 0.5,
mixh for the binary
containing n-decane is
nearly twice that for the
binary containing isodecane
4.7 Specific (Chemical) Forces

Chemical forces:specific forces of attraction
which lead to the formation of new
molecular species
 Association:

acetic acid consists primarily of dimers due to
hydrogen bonding
 Solvation:

to form polymers, dimers, trimers
to form complexes,
a solution of sulfur trioxide in water by formation of
sulfuric acid
4.8 Hydrogen Bonds

Hydrogen fluoride

Crystal structure of ice

The bond strength
 hydrogen
bonds
8 to 40 kJ /mol
 covalent bond
200 to 400 kJ /mol

Hydrogen bond is broken rather easily
Characteristic properties of hydrogen
bonds (see Figure 4-5)




Distances between the neighboring atoms of the two functional
groups (X-H- - -Y) in hydrogen bonds are substantially smaller than
the sum of their van der Waals radii.
X--H stretching modes are shifted toward lower frequencies (lower
wave numbers) upon hydrogen-bond formation.
Polarities of X-H bonds increase upon hydrogen-bond formation, often
leading to complexes whose dipole moments are larger than those
expected from vectorial addition.
Nuclear-magnetic-resonance (NMR) chemical shifts of protons in
hydrogen bonds are substantially smaller than those observed in the
corresponding isolated molecules. The deshielding effect observed is
a result of reduced electron densities at protons participating in
hydrogen bonding.
Solvent effect on hydrogen bonding

The thermodynamic constants for hydrogen-bonding

reactions are generally dependent on the medium in which
they occur.
1: 1 hydrogen-bonded complex of trifluoroethanol (TFE) with
acetone in the vapor phase and in CCl4 solution (inert
solvent).

Vertical transfer reaction
Horizonal complex-formation reaction
Transfer energy for complex into CCl4/(separated isomers into CCl4)
(-8.7)/(-5.7-4.75)=0.83;
Gibbs energy of transfer for complex into CCl4/(separated isomers into CCl4)
(-4.1)/(-3.2-2.0)=0.79
The transfer energy and Gibbs energy of the complex are not even
approximately canceled by the transfer energies and Gibbs energies of the
constituent molecules.

For most hydrogen-bonded complexes,
stabilities decrease as the solvent changes from
aliphatic hydrocarbon to chlorinated (or aromatic)
hydrocarbon, to highly polar liquid.
Strong effect of hydrogen bonding on
physical properties of pure fluids

Dimethyl ether and ethyl alcohol (hydrogen
bonding), both are C2H6O
Strong dependence of the extent of
polymerization on solute concentration

When a strongly hydrogen-bonded substance such as ethanol is
dissolved in an excess of a nonpolar solvent (such as hexane or
cyclohexane), hydrogen bonds are broken until, in the limit of
infinite dilution, all the alcohol molecules exist as monomers
rather than as dimers, trimers, or higher aggregates.
Hydrogen-bond formation between
dissimilar molecules


Acetone and
chloroform(with
hydrogen bonding)
Acetone with carbon
tetrachloride (no hydrogen
bonding )

Freezing-point data
Enthalpy of mixing data

The enthalpy of mixing of
acetone with carbon tetrachloride
is positive (heat is absorbed),
whereas the enthalpy of mixing
of acetone with chloroform is
negative (heat is evolved), and it
is almost one order of magnitude
larger.

These data provide strong
support for a hydrogen
bond formed between
acetone and chloroform.
4.9 Electron Donor-Electron Acceptor
Complexes
Table 4-9 Sources of experimental
data for donor-acceptor complexs
(Gutman, 1978)
Data
Type
1
Frequencies of charge-transfer absorption bonds
primary
2
Geometry of solid complexes
primary
3
NMR studies of motion in solid complexes
primary
4
Association constants
secondary
5
Molar absorptivity or other measurement of
absorption intensity
secondary
6
Enthalpy changes upon association
secondary
7
Dipole moments
secondary
8
Infrared frequency shifts
secondary
Primary and secondary data
“Primary” indicates that the data can be
interpreted using well-established theoretical
principles.
 “Secondary” indicates that data reduction
requires simplifying assumptions that may be
doubtful.

Charge-transfer complexes (loose complex formation)
 If a complex is formed between A and B, light absorption is larger.
 At different temperatures, it is possible to calculate also the
enthalpy and entropy of complex formation

Table 4-10 Spectroscopic equilibrium constants and
enthalpies of formation for s-trinitrobenzene(electron
acceptor)/ aromatic complexs (electron donor) dissolved in
cyclohexane at 20 oC
Equilibrium
constant
0.88
-h(kJ mol-1)
Mesitylene
3.51
9.63
Durene
6.02
11.39
Pentamethylbenzene
10.45
14.86
Hexamethylbenzene
17.50
18.30
Aromatic
Benzene
6.16
Remarks on Table 4-10

Complex stability rises with the number of
methyl groups on the benzene ring, in
agreement with various other measurements
indicating that -electrons on the aromatic ring
become more easily displaced when methyl
groups are added.
Table 4-11 Spectroscopic equilibrium constants
for formation for polar solvent/p-xylene
complexes dissolved in n-hexane at 25 oC
Polar solvent
Equilibrium constant
Acetone
0.25
Cyclohexanone
0.15
Propionitrile
0.07
Nitropropane
0.05
2-Nitro-2-methylpropane
0.03
Remarks on Table 4-11

No complex formation with saturated hydrocarbons
(such as 2-nitro-2-methylpropane, 0.03, and 2nitropropane, 0.05) and as a result we may expect
the thermodynamic properties of solutions of these
polar solvents with aromatics to be significantly
different from those of solutions of the same solvents
with paraffins and naphthalenes.
Evidence for complex
formation from
thermodynamic
measurements
Electron-donating
Power of the hydrocarbon
Evidence for
the existence of
a donor-acceptor
Interaction between
Tricholrobenzene
And aromatic
hydrocarbons
4.10 Hydrophobic Interaction

Some molecules have a dual nature





One part of molecule is soluble in water, hydrophilic, water-loving
part
While another part is not water-soluble, hydrophobic, water-fearing
part
Have a unique orientation in an aqueous medium; to
form suitably organized structures.
Such molecules called “amphiphiles”.
The organized structure called “micelles”
Hydrophobic part (a long-chain hydrocarbon)
is kept away from water
 Hydrophilic terminal groups at the surface of
the aggregates are water solvated and keep
the aggregations in solution.
 Reverse miscelles, by addition of a small
amount of water to a surfactant containing
organic nonpolar phase

hydrophobic effect 斥水性






The hydrophobic effect arises mainly from the strong attractive forces
(hydrogen bond) between water molecules in highly structured liquid water.
These attractive forces must be disrupted (使分裂) or distorted (扭曲)
when a solute is dissolved in water.
Upon solubilization of a solute, hydrogen bonds in water are often not
broken but they are maintained in distorted form.
Water molecules reorient, or rearrange, themselves such that they can
participate in hydrogen-bond formation, more or less as in bulk pure liquid
water.
In doing so, they create a higher degree of local order than that in pure
liquid water, thereby producing a decrease in entropy.
It is this loss of entropy (rather than enthalpy) that leads to an unfavorable
Gibbs energy change for solubilization of nonpolar solutes in water.
Table 4-12 Change in standard molar Gibbs energy (go), enthalpy (ho), and entropy
(Tso) for the transfer of hydrocarbons from their pure liquids into water at 25 oC
Hydrocarbon
go
ho
Tso
Ethane
16.3
-10.5
-26.8
Propane
20.5
-7.1
-27.6
N-Butane
24.7
-3.3
-28.0
N-Hexane
32.4
0
-32.4
Benzene
19.2
+2.1
-17.1
Toluene
22.6
+1.7
-20.9
Remarks on Table 4-12



The standard entropy of transfer is strongly negative,
due to the reorientation of the water molecules
around the hydrocarbon.
The poor solubility of hydrocarbons in water is not
due to a large positive enthalpy of solution but rather
to a large entropy decrease caused by what is called
the hydrophobic effect.
This effect is, in part, also responsible for the
immiscibility of nonpolar substances (hydrocarbons,
fluorocarbons, etc) with water.



Closely related to the hydrophobic effect is the hydrophobic
interaction. This interaction is mainly entropic and refers to the
unusually strong attraction between hydrophobic molecules in
water, in many cases, this attraction is stronger than in vacuo.
Energy of interaction of two contacting methane in vacuo is 2.5 x 10-21 J.
Energy of interaction of two contacting methane in water is 14 x 10-21 J.
4.11 Molecular interactions in dense
fluid media

Intermolecular forces
In the low pressure gas phase, interact in a “free” medium
(i.e., in a vacuum), described by a potential function (e.g.
Lennard-Jones)
 In the liquid solvent, interact in a solvent medium,
described by the potential of mean force
The essential difference is that the interaction between two
molecules in a solvent is influenced by the molecular nature
of the solvent but there is no corresponding influence on the
interaction of two molecules in (nearly) free space.


In the low pressure gas phase
Two solute molecules in a solvent
10/31/2006





For two solute molecules in a solvent, their intermolecular pair potential
includes not only the direct solute-solute interaction energy, but also any
changes in the solute-solvent and solvent-solvent interaction energies as
the two solute molecules approach each other.
A solute molecule can approach another solute molecule only by
displacing solvent molecules from its path.
Thus, at some fixed separation, while two molecules may attract each
other in free space, they may repel each other in a solvent medium if the
work that must be done to displace the solvent molecules exceeds that
gained by the approaching solute molecules.
Further, solute molecules often perturb (擾亂) the local ordering of solvent
molecules.
If the energy associated with this perturbation depends on the distance
between the two dissolved molecules, it produces an additional solvation
force between them.




The molecular nature of the solvent can produce potentials of
mean force that are much different for the corresponding twobody potential in vacuo.
The potential of mean force is a measure of the
intermolecular interaction of solute molecules in liquid solution.
Solution theories, such as McMillan-Mayer theory (1945),
provide a direct quantitative relation between the potential of
mean force and macroscopic thermodynamic properties (the
osmotic virial coefficients) accessible to experiment.
Osmotic (滲透) virial coefficients are obtained through
osmotic-pressure measurements.
Osmotic Pressure
Van’t Hoff (1890)
 Fig 4-17

Fig 4-17

The semi-permeable membrane is permeable to the solvent (1) but impermeable
to the solute (2). The pressure on phase  is P, while that on phase  is P + .
1  1
(4-38)
1  pure1 T, P 
(4-38a)
1  pure1 T , P    RT ln a1
(4-39)
For a pure fluid,
    v
 P T
 pure1 P     pure1 P   vpure1
(4-40)
 pure1 (T , P )  1  1  pure1 (T , P  )  RT ln a1
 pure1 (T , P )  vpure1  RT ln a1
0   vpure1  RT ln a1
 ln a1 
vpure1
RT
(4-41)
If the solution in phase  is dilute, x1 is close to unity, and 1 is also
close to unity. a1 = 1 x1
 ln a1   ln x1 
vpure1
(4-42)
RT
When x2  1( very dilute ), ln x1   ln 1 - x2    x2
vpure1  x2 RT
(4-43)
Because x2  1, n2  n1 and x2  n2 / n1
n2
n2
vpure1  x2 RT 
RT  RT
n2  n1
n1
V  n1vpure1  n2 RT
V  n2 RT
Van’f Hoff equation for osmotic pressure 
(4-44)
Van’f Hoff equation

Assumptions
 The
 The

solution is very dilute.
solution is incompressible.
Application
, T, and mass concentration of solute, the
solute’s molecular weight can be calculated.
 A standard procedure for measuring molecular weights of
large molecules (polymer or biomacromolecules such as
proteins) whose molecular weight cannot be accurately
determined from other colligative property measurements
(boiling point elevation or freezing point depression)
 Measure
For finite concentration
Van’t Hoff’s equation is a limiting law for the
concentration of solute goes to zero.
 For finite concentration, a series expression is
used

Osmotic virial expansion

For finite concentration, it is useful to write a series
expansion in the mass concentration c2(in g/liter),
 1


2
 RT 
 B * c2  C * c2  
c2
M2




Where M2 is the molar mass of solute
B*, the osmotic second virial coefficient
C*, the osmotic third virial coefficient
(4-54)

For dilute solution, we can neglect three-body
interaction (C*). Thus, a plot of /c2 against c2
is linear, with intercept equal to RT/M2 and
slope equal to RTB*.
Table 4-13 Osmotic second virial coefficients and number-average molecular weights
for alpha-chymotrypsin, lysozyme, and ovalbumin in aqueous buffer solutions,
regressed from the data shown in Fig. 4-18
Attractive
force
Repulsive
force
Macroscopic and microscopic
Osmotic second virial coefficients ( a
macroscopic property) are related to
intermolecular forces (microscopic property)
between two solute molecules.
 B22* can provide useful information on
interaction between polymer of protein
molecules in solution.

Donnan Equilibria

The osmotic-pressure relation given by van’t Hoff was derived for solutions
for nonelectrolytes or for solutions of electrolytes where the membrane’s
permeability did not distinguish between cations and anions.

Consider a chamber divide into two parts by a membrane that exhibits ion
selectivity, i.e., some ions can flow through the membrane while others
cannot. In this case, the equilibrium conditions become more complex
because , in addition to the usual Gibbs equations for equality of chemical
potentials, it is now also necessary to satisfy an additional criterion:
electrical neutrality for each of the two phases in the chamber.
Donnan Equilibria


Consider an aqueous system containing three ionic
species: Na+, Cl- and R-, where R- is some anion
much larger than Cl-. Water is in excess; all ionic
concentrations are small.
The chamber is divided into two equisized parts,
phase αand phase β, by an ion-selective membrane.
The membrane is permeable to water, Na+ and Clbut it is impermeable to R-.
Fig. 4-19
Electroneutrality before equilibrium is attained
0
0
cNa
  c R
and
0
0
cNa
  c Cl
(4-46)
Let  represent the change in Na+ concentration in  phase. Because
R- cannot move from one side to the other, the change in ClConcentration in  phase is –.
In  :
f
0
cNa

  c
Na 
cClf-  
cRf-  cR0-
(4-47)
In  :
f
0
cNa

  c
Na 
cClf-  cCl0-  
cRf-  0
(4-48)

Na+
Cl-
Calculate δ from know initial
concentration

For the solvent, we write
 s  s
(4-49)
*s  P  vs  RT ln as   s  s  *s  P  vs  RT ln as
(4-50)
*s is pure liquid solvent at system temperatu re and at zero pressure.
*s  P  vs  RT ln as  *s  P  vs  RT ln as
P  vs  RT ln as  P  vs  RT ln as
RT as
ln   P   P   
vs
as
solvent
(4-51)
We also have


 NaCl
  NaCl
(4-52)
NaCl is total dissociate d into Na  and Cl

 Na
 Cl   Na
 Cl



*
Na 
*
Na 
-

-
(4-53)

*


 P  vNa   RT ln aNa
    -  P v -  RT ln a -  
Cl
Cl
Cl

  *Cl-  PvCl-  RT ln aCl - 
 PvNa   RT ln aNa
P v
P v
P v

Na 

Na 

Na 
 

 RT ln a   P v  RT ln a 
 v   RT ln a a   P v  v   RT ln a



 RT ln aNa

P
v

RT
ln
a


Cl Cl 
Na 
Cl -


Cl -
Cl -

Na 

Cl -

Na 



a
a

- 


Na
Cl
P  P vNa   vCl -   RT ln    
 a a - 
 Na Cl 



a
a

- 
RT


Na
Cl
 P P 
ln    
vNa   vCl -  aNa  aCl - 

Cl -

solute
(4-54)

Na 
aCl -

solvent
solute


 aNa
a RT  as 
RT
ln      
ln   Cl
vs  a s 
vNa   vCl -  aNa  aCl v


aNa
a

Cl 

aNa
a

Cl -
 as 
   
 as 
Na 
v
Cl-
vs
(4-55)
as  as  1
ai  ci
In a very dilute solutions,
Activity of solute




cNa
c -  c
c Cl
Na Cl
(4-56)








cNa
c

c
c

Cl Na  Cl -
c
c
0
Na 
    c
0
Na 
  c
0
Na 
    c
0
Na 
0 2
Na 
0
Na 
0
Na 
0 2
Na 
2
0 2
Na 
0
Na 
0 2
Na 
0
Na 
0
Na 
    c
0
Na 
 
2
0
2
 2cNa
  
  2c 
c  2c   c 

c 

c  2c 
c
  c

0
Cl -
(4-58)
 can be calculated and the final equilibrium concentration can be
Calculated for eqs (4.47) and (4-48)
The fraction of original sodium chloride
in β that has moved to α

c
c 
0 2
Na 
0
Na 


0
 2cNa
 
c  c
0
Na 

1
0
cNa

0
cNa

c 
0
Na 
0
Na 
 2c
0
Na 

c
 2 

c

2

0
Na 
0
Na 

1

c
c 
1
0
Na 




1
0
Na 
0
 2cNa
 
The osmotic pressure

0
0
  2RT cNa
  c -  2
Cl

The difference in electrical potential


Because the equilibrium concentration of Na+ is not the same in both sides, we
have a concentration cell (battery) with a difference in electric potential across the
membrane.
The difference in electric potential is given by the Nerst equation

aNa

RT
 
ln 
N Aez Na  aNa 

Upon setting activities equal to concentration
0
cNa

RT
 
ln 0
0
N Aez Na  cNa   cNa

4.12 Molecular Theory of Corresponding
States
 r
  
i
 i 
ii
(4-64)
Q  Qint  N , T Qtrans  N , T ,V 
(4-65)
 t r1 ... rN  
 dr1 ... drN
...  exp 


N!
kT


3 N

 2mkT  2 1
Qtrans  

h
2


(4-66)
 t r1...rN  
Z N   ... exp  
dr1...drN
kT 

(4-67)
Equation of state
  ln Q 
  ln Z N 
P  kT 
 kT 


 V  T , N
 V  T , N
 
t   ij rij
i j
ZN  
3N
 
 rij    r1 
 rN 
  d  3  ... d  3 
 ...  exp kT 
 
   
i j

ZN  
3N
 kT V

Z  , 3 , N
  

*
N
Aconf   kT ln Z N
(4-68)
(4-69)
(4-70)
(4-71)
(4-72)
Aconf  N T , v 
 kT V  
3N 
Z N   z *  , 3  
    
N
P
  ln z *



NkT
V  T , N
V ~ P 3
~ kT ~
T
,v 
3 ,P 

N

~
~
P  F * T , v~
4.13 Extension of corresponding states
theory to more complicated molecules
z  z  z
0
1
Conclusions
Physical and chemical forces determines the
properties of systems
 Intermolecular forces responsible for
molecules behavior

Homework-5, Prob 4-17

 

 ln xw   ln 1  x A  x A2  x A  x A2 
 vpure water 
RT
basis : 1000cc of water
mole of protein  (5 g/liter)/( mw of lysozyne)
specific volume of water  1/0.997 cm3/g  ...........cm3/mol
mole of water (1000 cc)  1000/speci fic volume of water in cm3/mol
mole fraction of protein  mole of protein /m ole of water
K  10 
5
a A2
a
2
A

x A2
x A2
Homework-3, Prob 4-19
 1


2
 RT 
 B * c2  C * c2  
c2
M2


RT

 RTB * c  c0   
c  c0 M 2
Plot of

c  c0
vs
c  c0 
Determine Intercept and slope
Ans: M2= 12439 g/mol, B* = 2.93x10-7 L mol g-2
Relation of  and t(rmint)
1
2
   t r  rmin t 
N
zs m
 sn 
 
 sm 
m / n  m 
9
t r  rmin t    subh0  R D
8

 c 1T
k
c
2 N
3
c1  0.77, c2  0.75, c3  7.42, zc 
A

 c2vc
3
2 c1
 0.290
3 c2 c3
NA
NB
  t 
Z N   ...  exp
 dr  drB  ...
 kT  A

ij



i

1 / 2
j
 ij
1

2

i

j
