Transcript CFD of an RCM

Molecular Interactions
The most important to producing
phases and interfaces in the
materials
1
Background
Atoms and molecules with complete
valence shells can still interact with
one another even though all of their
valences are satisfied. They attract
one another over a range of several
atomic diameters and repel one
another when pressed together.
2
Molecular interactions account for:
– condensation of gases to liquids
– structures of molecular solids (surfaces)
– structural organisation of biological
macromolecules as they pin molecular building
blocks (polypeptides, polynucleotides, and
lipids) together in the arrangement essential
to their proper physiological function.
3
van der Waals interactions
• Interaction between partial charges in polar
molecules
• Electric dipole moments or charge distribution
• Interactions between dipoles
• Induced dipole moments
• Dispersion interactions
– Interaction between species with neither a net charge nor a
permanent electric dipole moment (e.g. two Xe atoms)
4
The total interaction
• Hydrogen bonding
• The hydrophobic effect
• Modelling the total interaction
• Molecules in motion
5
van der Waals interactions
Interactions between molecules include
the attractive and repulsive
interactions between the partial
electric charges of polar molecules and
the repulsive interactions that prevent
the complete collapse of matter to
densities as high as those
characteristic of atomic nuclei.
6
van der Waals interactions (contd.)
Repulsive interactions arise from the
exclusion of electrons from regions of
space where the orbitals of closed-shell
species overlap.
Those interactions proportional to the
inverse sixth power of the separation
are called van der Waals interactions.
7
van der Waals interactions
• Typically one discusses the potential
energy arising from the interaction.
If the potential energy is denoted V, then
the force is –dV/dr. If V = -C/r6
the magnitude of the force is:
d  C  6C
 6   7
dr  r  r
8
Interactions between partial charges
Atoms in molecules generally have partial
charges.
Atom
Partial charge/e
C(O)
0.45
C(–CO)
0.06
H(–C)
0.02
H(–N)
0.18
H(–O)
0.42
N
0.36
O
0.38
9
Interactions between partial charges
• If these charges were separated by a
vacuum, they would attract or repel one
another according to Coulomb’s Law:
q1q2
V
4 0 r
where q1 and q2 are the partial charges
and r is their separation
10
Charges Interactions
• Coulombic Inteaction between q1 and
q2
q1q2
V
40 r
• Partial atomic Charges
Approximated distribution of
electron in molecule
0.387 -0.387
11
Interactions between partial charges
However, other parts of the molecule, or
other molecules, lie between the charges, and
decrease the strength of the interaction.
Thus, we view the medium as a uniform
continuum and we write:
q1q2
V
4r
Where  is the permittivity of the medium
lying between the charges.
12
• The permittivity is usually expressed as a
multiple of the vacuum permittivity by
writing  = r0, where r is the relative
permittivity (dielectric constant). The
effect of the medium can be very large,
for water at 250C, r = 78.
The PE of two charges separated by bulk
water is reduced by nearly two orders of
magnitude compared to that if the charges
were separated by a vacuum.
13
Coulomb potential for two charges
vacuum
fluid
14
Ion-Ion interaction/Lattice Enthalpy
Consider two ions in a lattice
15
Ion-Ion interaction/Lattice Enthalpy
two ions in a lattice of charge numbers z1 and
z2 with centres separated by a distance r12:
( z1e) x ( z2e)
V12 
4 0 r12
where 0 is the vacuum permittivity.
16
Ion-Ion interaction/Lattice Enthalpy
To calculate the total potential energy
of all the ions in the crystal, we have to
sum this expression over all the ions.
Nearest neighbours attract, while
second-nearest repel and contribute a
slightly weaker negative term to the
overall energy. Overall, there is a net
attraction resulting in a negative
contribution to the energy of the solid.
17
For instance, for a uniformly spaced line of
alternating cations and anions for which z1
= +z and z2 = -z, with d the distance
between the centres of adjacent ions, we
find:
z 2e 2
V 
x 2 ln 2
4 0 r
18
 z 2e 2 z 2e 2 z 2e 2 z 2e 2

V
x



 ..... 

4 0 
r
2r
3r
4r

1
z 2e 2 
1 1 1

V 
x 1     ..... 
4 0 r 
2 3 4

z 2e 2
V 
. 2 ln 2
4 0 r
19
Born-Haber cycle for lattice enthalpy
20
Lattice Enthalpies, DHL0 / (kJ mol-1)
LiF
1 037
LiCl
852
LiBr
815
LiI
761
NaF
926
NaCl
786
NaBr
752
NaI
705
KF
821
KCl
717
KBr
689
KI
649
MgO
3 850
CaO
3461
SrO
3283
BaO
3114
MgS
3 406
CaS
3119
SrS
2974
BaS
2832
Al2O3
15 900
Lattice Enthalpy ( H L) is the standard enthalpy
change accompanying the separation of the species
that compose the solid per mole of formula units.
e.g. MX (s) = M+(g) + X- (g)
21
Calculate the lattice enthalpy of KCl (s)
using a Born-Haber cycle and the following
information at 25oC:
Process
Sublimation of K (s)
Ionization of K (g)
Dissociation of Cl2 (g)
Electron attachment to Cl (g)
Formation of KCl (s)
DH0
(kJ mol-1)
+89
+418
+244
-349
-437
22
Calculation of lattice enthalpy
Process
KCl (s)
K (s)
K (g)
½ Cl2 (g)
Cl (g) + e- (g)
KCl (s)
DH0 (kJ mol-1)
K (s) + ½ Cl2 (g)
K (g)
K+ (g) + e- (g)
Cl (g)
Cl- (g)
+437
+89
+418
+122
-349
K+ (g) + Cl- (g)
+717 kJ mol-1
23
Electric dipole moments
When molecules are widely separated it is
simpler to express the principal features of
their interaction in terms of the dipole
moments associated with the charge
distributions rather than with each individual
partial charge. An electric dipole consists of
two charges q and –q separated by a distance
l. The product ql is called the electric dipole
moment, m.
24
Electric dipole moments
We represent dipole moments by an arrow
with a length proportional to m and pointing
from the negative charge to the positive
charge:
d
m
d
Because a dipole moment is the product of a
charge and a length the SI unit of dipole
moment is the coulomb-metre (C m)
25
Electric dipole moments
It is often much more convenient to report a
dipole moment in debye, D, where:
1D = 3.335 64 x 10-30 C m
because the experimental values for
molecules are close to 1 D. The dipole moment
of charges e and –e separated by 100 pm is
1.6 x 10-29 C m, corresponding to 4.8 D.
26
Electric dipole moments: diatomic molecules
A polar molecule has a permanent
electric dipole moment arising from
the partial charges on its atoms. All
hetero-nuclear diatomic molecules are
polar because the difference in
electronegativities of their two
atoms results in non-zero partial
charges.
27
Electric dipole moments

m/D
/(1030 m3)
Ar
0
1.66
CCl4
0
10.5
C6H6
0
10.4
H2
0
0.819
H2O
1.85
1.48
NH3
1.47
2.22
HCl
1.08
2.63
HBr
0.80
3.61
HI
0.42
5.45
28
Electric dipole moments: diatomic molecules
More electronegative atom is usually the negative end of the
dipole. There are exceptions, particularly when anti-bonding
orbitals are occupied.
– CO dipole moment is small (0.12 D) but negative end is on C atom.
Anti-bonding orbitals are occupied in CO and electrons in anti-bonding
orbitals are closer to the less electronegative atom, contributing a
negative partial charge to that atom. If this contribution is larger
than the opposite contribution from the electrons in bonding orbitals,
there is a small negative charge on the less electronegative atom.
29
Electric dipole moments: polyatomic molecules
Molecular symmetry is of the greatest
importance in deciding whether a polyatomic
molecule is polar or not. Homo-nuclear
polyatomic molecules may be polar if they
have low symmetry
– in ozone, dipole moments associated with each
bond make an angle with one another and do not
cancel.
d+ d+
m
m
d
d
Ozone, O3
30
Electric dipole moments: polyatomic molecules
Molecular symmetry is of the greatest
importance in deciding whether a
polyatomic molecule is polar or not.
– in carbon dioxide, dipole moments
associated with each bond oppose one
another and the two cancel.
d-
m
d+ d+
m
d-
Carbon dioxide, CO2
31
Electric dipole moments: polyatomic molecules
It is possible to resolve the dipole
moment of a polyatomic molecule into
contributions from various groups of
atoms in the molecule and the direction
in which each of these contributions lie.
32
Electric dipole moments: polyatomic molecules
1,2-dichlorobenzene: two chlorobenzene
dipole moments arranged at 60o to each
other. Using vector addition the resultant
dipole moment (mres) of two dipole moments
m1 and m2 that make an angle  with one
another is approximately:
mres
m res  m  m  2m1m 2 cos  
2
1
2
2
1/ 2
m1

m2
33
Electric dipole moments: polyatomic molecules
34
Electric dipole moments: polyatomic molecules
Better to consider the locations and
magnitudes of the partial charges on all the
atoms. These partial charges are included in
the output of many molecular structure
software packages. Dipole moments are
calculated considering a vector, m, with three
components, mx, my, and mz. The direction of m
shows the orientation of the dipole in the
molecule and the length of the vector is
the magnitude, m, of the dipole moment.
m  m  m  m
2
x
2
y

2 1/ 2
z
35
Electric dipole moments: polyatomic molecules
To calculate the x-component we need to
know the partial charge on each atom and the
atom’s x-coordinate relative to a point in the
molecule and from the sum:
m x   qJ xJ
J
where qJ is the partial
charge of atom J, xJ is
the x coordinate of atom
J, and the sum is over all
atoms in molecule
mz
my
m
mx
36
Partial charges in polypeptides
Atom
Partial charge/e
C(O)
0.45
C(–CO)
0.06
H(–C)
0.02
H(–N)
0.18
H(–O)
0.42
N
0.36
O
0.38
37
Calculating a Molecular dipole moment
(182,-87,0)
H
+0.18
-0.36
(132,0,0)
N
m
(0,0,0)
C
+0.45
O
-0.38
(-62,107,0)
mx = (-0.36e) x (132 pm) + (0.45e) x (0 pm)
+(0.18e) x (182 pm) + (-0.38e) x (-62 pm)
= 8.8e pm
= 8.8 x (1.602 x 10-19 C) x (10-12 m)
= 1.4 x 10-30 C m = 0.42 D
38
Calculating a Molecular dipole moment
my = (-0.36e) x (0 pm) + (0.45e) x (0 pm)
+(0.18e) x (-86.6 pm) + (-0.38e) x (107 pm)
= -56e pm = -9.1 x 10-30 C m = -2.7 D
mz = 0
m  m  m  m
2
x
2
y

2 1/ 2
z
m =[(0.42 D)2 + (-2.7 D)2]1/2 = 2.7 D
Thus, we can find the orientation of the dipole
moment by arranging an arrow 2.7 units of
length (magnitude) to have x, y, and z
components of 0.42, -2.7, 0 units
(Exercise: calculate m for formaldehyde)
39
Interactions between dipoles
The potential energy of a dipole m1 in the
presence of a charge q2 is calculated
taking into account the interaction of the
charge with the two partial charges of
the dipole, one a repulsion the other an
attraction.
m1q2
V 
2
4 0 r
l
q1
q2
-q1
r
40
 The
potential energy between a point dipole
and the point charge q (l>>r)
-q1
+q1
l
q2
r
1  q1q2
q1q2 


V 



1
1 
40  r  2 l r  2 l 
qq  1
1 
  1 2 


40 r  1  x 1  x 

q1q 2 l
40
m1q2
V 
2
40 r
41
Interactions between dipoles
A similar calculation for the more general
orientation is given as:
q
2
m1q2 cos
V 
4 0 r 2
r
l
q1

-q1
If q2 is positive, the energy is lowest
when  = 0 (and cos  = 1), as the partial
negative charge of the dipole lies closer
than the partial positive charge to the
point charge and the attraction outweighs
the repulsion.
42
Interactions between dipoles
The interaction energy decreases
more rapidly with distance than that
between two point charges (as 1/r2
rather than 1/r), because from the
viewpoint of the point charge, the
partial charges on the dipole seem to
merge and cancel as the distance r
increases.
43
-q1
+q1
q2
l
r
q2
l
r
q2
l
• Increasing the distance,
the potentials of the
charges decrease and the
two charges appear to
merge.
• These combined effect
approaches zero more
rapidly than by the
distance effect alone.
r
44
Interactions between dipoles
Interaction energy between two dipoles m1
and m2:
l2
m1m 2 (1  3 cos2  )
V
4 0 r 3
q2
l1
-q2
r

q1
-q1
For dipole-dipole interaction the potential
energy decreases as 1/r3 (instead of 1/r2
for point-dipole) because the charges of
both dipoles seem to merge as the
separation of the dipoles increases.
45
• The potential energy between two parallel dipoles
-q1
+q1
l
l
r
1  q1q2 q1q2 q1q2 q1q2 
V





40  r  l
r
r
r l 
q1q2  1
1



 2

40 r  1  x 1  x


2q1q 2  l 
 
40 r  r 
m1m 2

20 r 3
2
+q2
-q2
+q2
+q1
l
l

-q2
r
-q1
m1m 2 f  
V
40 r 3
2


f   1 3 cos 
This applies to polar molecules in a fixed,
parallel, orientation in a solid.
46
Interactions between dipoles
The angular factor takes into account how the like or
opposite charges come closer to one another as the
relative orientations of the dipoles is changed.
– The energy is lowest when  = 0 or 180o (when 1 – 3 cos2 = -2), because
opposite partial charges then lie closer together than like partial
charges.
– The energy is negative (attractive) when  < 54.7o (the angle when 1 – 3
cos2 = 0) because opposite charges are closer than like charges.
– The energy is positive (repulsive) when  > 54.7o because like charges are
then closer than opposite charges.
– The energy is zero on the lines at 54.7o and (180 – 54.7) = 123.3o
because at those angles the two attractions and repulsions cancel.
47
Interactions between dipoles
Calculate the molar potential energy of the
dipolar interaction between two peptide links
separated by 3.0 nm in different regions of a
polypeptide chain with  = 180o, m1 = m2 = 2.7 D,
corresponding to 9.1 x 10-30 C m
m1m 2 (1  3 cos2  )
V
4 0 r 3
30
(9.110 C m)  (2)
V
4 (8.8541012 J 1 C 2 m1 ) (3.0 109 m)3
23
V   5.610
2
J   34J mol
1
48
• Freely rotating dipoles: Liquid, Gas
– The interaction energy of two freely rotating
dipoles is zero.
– Real molecules do not rotate completely freely
due to the fact that their orientations are
controlled partially by their mutual interaction.
49
Interactions between dipoles
When a pair of molecules can adopt all relative
orientations with equal probability, the favourable
orientations (a) and the unfavourable ones (b) cancel,
and the average interaction is zero.
In an actual fluid (a) predominates slightly.
50
Interactions between dipoles
2m m
V 
2
6
3(4 0 ) kTr
2
1
2
2
2m12 m22
V 
3(4 0 ) 2 kTr 6
E  1/r6 => van der Waals interaction
E  1/T => greater thermal motion overcomes the
mutual orientating effects of the dipoles at higher T
51
• The average interaction energy of two
polar molecules rotating at a fixed
separation r
m1m 2 f
V 
3
40 r
– Probability that a particular orientation is
given by Boltzmann distribution
pe
V / kT
– Keesom Interaction
m1m 2 f
V
3
40 r
C
2m12 m 22
V  6 ; C
2
r


3 40 kT
Average interaction
52
is attractive.
Interactions between dipoles
• At 25oC the average interaction energy for
pairs of molecules with m = 1 D is about -1.4 kJ
mol-1 when the separation is 0.3 nm.
• This energy is comparable to average molar
kinetic energy of 3/2RT = 3.7 kJ mol-1 at 25oC.
• These are similar but much less than the
energies involved in the making and breaking of
chemical bonds.
53
Induced dipole moments
A non-polar molecule may acquire a temporary
induced dipole moment m* as a result of the
influence of an electric field generated by a
nearby ion or polar molecule. The field distorts
the electron distribution of the molecule and
gives rise to an electric dipole. The molecule is
said to be polarizable.
The magnitude of the induced dipole moment is
proportional to the strength of the electric
field, E, giving:
m* =  E
where  is the polarizability of the molecule.
54
Induced dipole moments
• The larger the polarizability of the molecule
the greater is the distortion caused by a given
strength of electric field.
• If a molecule has few electrons (N2) they are
tightly controlled by the nuclear charges and
the polarizability is low.
• If the molecule contains large atoms with
electrons some distance from the nucleus (I2)
nuclear control is low and polarizability is high.
55
Induced dipole moments
Polarizability also depends on the orientation
of the molecule toward the electric field
unless the molecule is tetrahedral (CCl4),
octahedral (SF6), or icosahedral (C60).
– Atoms and tetrahedral, octahedral, and
icosahedral molecules have isotropic
(orientation-independent) polarizabilities
– All other molecules have anisotropic
(orientation-dependent) polarizabilities
56
Polarizability volume
The polarizability volume has the dimensions
of volume and is comparable in magnitude to
the volume of the molecule
The same magnetude as the
actual molecular volumes (Å3)

'
 
4 0
C2 m2 J-1
C2 J-1 m-1
57
Polarizability volumes

m/D
/(1030 m3)
Ar
0
1.66
CCl4
0
10.5
C6H6
0
10.4
H2
0
0.819
H2O
1.85
1.48
NH3
1.47
2.22
HCl
1.08
2.63
HBr
0.80
3.61
HI
0.42
5.45
58
Polarizability volume
What strength of electric field is required to induce
an electric dipole moment of 1 mD in a molecule of
polarizability volume 1.1 x 10-31 m3?

 
4 0
'
31
1.110


 1.2241041 J-1 C2 m2
4 0
6
30
1
.
0
x
10
(
3
.
335

10
)
*
64
m  E  E 
1.22410 41
 2.72510 JC m  2.72510 Vm
5
1
1
5
 2.725102 kVm1  2.725kV cm1
1
59
Dipole-induced dipole moments
A polar molecule with dipole
moment m1 can induce a dipole
moment in a polarizable
molecule
the induced dipole interacts
with the permanent dipole of
the first molecule and the two
are attracted together
m 2
V 
6
 0 r
2
1
the induced dipole (light
arrows) follows the changing
orientation of the permanent
dipole (yellow arrows)
60
Dipole-induced dipole moments
For a molecule with m = 1 D (HCl) near a
molecule of polarizability volume ’ = 1.0 x
10-31 m3 (benzene), the average interaction
energy is about -0.8 kJ mol-1 when the
separation is 0.3 nm.
E  1/r6 => van der Waals interaction
61
Dispersion interactions
• Interactions between species with neither
a net charge nor a permanent electric
dipole moment
– uncharged non-polar species can interact
because they form condensed phases such as
benzene, liquid hydrogen and liquid xenon
• The dispersion interaction (London Force)
between non-polar species arises from
transient dipoles which result from
fluctuations in the instantaneous positions
of their electrons
62
Dispersion interactions
Electrons from one molecule may
flicker into an arrangement that
results in partial positive and
negative charges and thus gives an
instantaneous dipole moment m1.
This dipole can polarize another
molecule and induce in it an
instantaneous dipole moment m2.
Although the first dipole will go on
to change the size and direction of
its dipole (≈ 10-16 s) the second
dipole will follow it; the two dipoles
are correlated in direction, with
the positive charge on one molecule
close to a negative partial charge on
the other molecule and vice versa.
An instantaneous dipole on
one molecule induces a
dipole on another molecule,
and the two dipoles attract
thus lowering the energy.
63
Dispersion interactions
• Overall, net attractive interaction
• Polar molecules interact by:
– dispersion interactions and dipole-dipole
interactions
– dispersion interactions often dominant
• Dispersion interaction strength depends on:
– polarizability of first molecule which is decided by
nulcear control
• loose => large fluctuations in e- distribution
– polarizability of second molecule
V  1 2
64
Dispersion interactions
2 1' 2' I1 I 2
V  6 
3 r
I1  I 2
London
formula
I1, I2 are the ionization energies of the two molecules
Potential energy of interaction is proportional to
1/r6 so this too is a contribution to the van der
Waals interaction. For two CH4 molecules, V = -5 kJ
mol-1 (r = 0.3 nm)
65
Total interaction- Hydrogen bonding
The coulombic interaction between the partly
exposed positive charge of a proton bound to an
electron withdrawing X atom (in X—H) and the
negative charge of a lone pair on the second atom Y,
d-X—Hd+ ……Ydas in:
• Strongest intermolecular interaction
• Denoted X—H……Y, with X and Y being N, O, or F
– only molecules with these atoms
• ‘Contact’ interaction
– turns on when X—H group is in contact with Y atom
66
Hydrogen Bonding
– A dipole-dipole force with a
hydrogen atom bonded to
nitrogen, oxygen or fluorine.
– The energy of a hydrogen
bond is typically 5 to 30
kJ/mole.
– These bonds can occur
between molecules or within
different parts of a single
molecule.
– The hydrogen bond is a very
strong fixed dipole-dipole van
der Waals-Keesom force, but
weaker than covalent, ionic
and metallic bonds.
67
• A hydrogen atom attached to a
relatively electronegative atom
(usually fluorine, oxygen, or
nitrogen) is a hydrogen bond
donor.
• An electronegative atom such as
fluorine, oxygen, or nitrogen is a
hydrogen bond acceptor,
regardless of whether it is
bonded to a hydrogen atom or
not. Hydrogen bond Strength
F—H...F
O—H...N
O—H...O
N—H...N
N—H...O
HO—H...OH3+
40 kcal/mol
6.9 kcal/mol
5.0 kcal/mol
3.1 kcal/mol
1.9 kcal/mol
4.3 kcal/mol
 The length of hydrogen bonds depends
on bond strength, temperature, and
pressure.
 The typical length of a hydrogen bond in
water is 1.97 Å.
68
Hydrogen bonding
• Leads to:
–
–
–
–
–
–
rigidity of molecular solids (sucrose, ice)
low vapour pressure (water)
high viscosity (water)
high surface tension (water)
secondary structure of proteins (helices)
attachment of drugs to receptor sites in
proteins
69
Interaction potential energies
Interaction type
Distance dependence
Typical energy
of potential energy
(kJ mol )
Ion–ion
1/r
250
Ion–dipole
1/r
2
15
Dipole–dipole
1/r
3
2
Between stationary polar
molecules
1/r
6
0.3
Between rotating polar molecules
1/r
6
2
Between all types of molecules
and ions
London (dispersion)
Comment
1
Only between ions
The energy of a hydrogen bond X–H  Y is typically 20 kJ mol1 and occurs on contact for X, Y  N, O, or F.
70
The Hydrophobic effect
• An apparent force that influences the shape of
a macromolecule mediated by the properties of
the solvent, water.
• Why don’t HC molecules dissolve appreciably in
water?
• Experiments show that the transfer of a
hydrocarbon molecule from a non-polar solvent
into water is often exothermic (H < 0)
• The fact that dissolving is not spontaneous
must mean that entropy change is negative
(S < 0).
71
The Hydrophobic effect
• For example, the process:
CH4 (in CCl4) = CH4 (aq)
has H = - 10 kJ mol-1, S = - 75 J K-1
mol-1, and G = + 12 kJ mol-1 at 298 K.
• Substances characterized by a positive
Gibbs energy of transfer from a nonpolar to a polar solvent are classified as
hydrophobic.
72
The Hydrophobic effect
When a HC molecule is
surrounded by water, the
water molecules form a
clathrate cage. As a result of
this acquisition of structure,
the entropy of the water
decreases, so the dispersal
of the HC into water is
entropy-opposed.
The coalescence of the HC
into a single large blob is
entropy-favoured.
73
The Hydrophobic effect
• The formation of the clathrate cage decreases
the entropy of the system because the water
molecules must adopt a less disordered
arrangement than in the bulk liquid.
• However, when many solute molecules cluster
together fewer (but larger) cages are required
and more solvent molecules are free to move.
• This leads to a net decrease in the organization
of the solvent and thus a net increase in the
entropy of the system.
74
The Hydrophobic effect
• This increase in entropy of the solvent is
large enough to render spontaneous the
association of hydrophobic molecules in a
polar solvent.
• The increase in entropy that results in the
decrease in structural demands on the solvent
is the origin of the hydrophobic effect.
• The presence of hydrophobic groups in
polypeptides results in an increase in
structure of the surrounding water molecules
and a decrease in entropy.
75
Modelling the total interaction
The total attractive interaction energy
between rotating molecules that cannot
participate in hydrogen bonding is the sum
of the contributions from the dipoledipole, dipole-induced-dipole, and
dispersion interactions.
Only the dispersion interaction
contributes if both molecules are nonpolar.
76
Modelling the total interaction
All three interactions vary as the inverse
sixth power of the separation. Thus the total
van der Waals interaction energy is:
C
V  6
r
where C is a coefficient that depends on the
identity of the molecules and the type of
interaction between them.
77
Modelling the total interaction
The attractive (negative)
contribution has a long
range, but the repulsive
(positive) interaction
increases more sharply
once the molecules come
into contact.
Repulsive terms become
important and begin to
graph of the potential
dominate the attractive
energy of two closed-shell forces when molecules are
species as the distance
squeezed together.
between them is changed
78
Modelling the total interaction
• These repulsive interactions arise
primarily from the Pauli exclusion
principle, which forbids pairs of
electrons being in the same region of
space.
• The repulsions increase steeply in a way
that can be deduced only by very
extensive, complicated, molecular
structure calculations.
79
Modelling the total interaction
• In many cases one may use a greatly
simplified representation of the
potential energy.
– details ignored
– general features expressed using a few
adjustable parameters
• Hard-Sphere potential (approximation)
– Assume potential energy rises abruptly to
infinity as soon as the particles come within
some separation s
80
Modelling the total interaction
• V = ∞ for r ≤ s
• V = 0 for r > s
s
This very simple assumption
is surprisingly useful in
assessing a number of
properties.
There is no
potential energy of
interaction until
the two molecules
are separated by a
distance s when
the potential
energy rises
abruptly to infinity
81
Modelling the total interaction
Another approximation is to express the
short-range repulsive potential energy as
inversely proportional to a high power of r:
*
C
V  n
r
where C* is another constant (the star
signifies repulsion). Typically, n is set to 12, in
which case the repulsion dominates the 1/r6
attractions strongly at short separations as:
C*/r12 >> C/r6
82
Modelling the total interaction
The sum of the repulsive interaction with n = 12
and the attractive interaction given by:
C
V  6
r
is called the Lennard-Jones (12,6)-potential. It
is normally written in the form:

 s 

s  
V  4      
r
r




 


12
6
83
Modelling the total interaction
The two parameters
are , the depth of
the well, and s, the
separation at which
V = 0.
The Lennard-Jones potential models the
attractive component by a contribution that is
proportional to 1/r6, and a repulsive component
by a contribution proportional to 1/r12
84
Modelling the total interaction
Species / kJ mol-1 s / pm
Ar
128
342
Br2
536
427
C6H6
454
527
Cl2
368
412
H2
34
297
He
11
258
Xe
236
406
85
THE END
86
87