Bonding
Table 2.2 Common Types of Interactions and their Pair-Potentials w(r) between Two
Atoms, Ions, or Small Molecules in a Vacuum (3 ¼ 1)a
Type of interaction
Interaction energy w(r)
Covalent, metallic
Complicated, short range
Charge–charge
þQ1Q2/4p30r (Coulomb energy)
Charge–dipole
#Qu cos q/4p30r2
#Q2u2/6(4p30)2kTr4
Dipole–dipole
#u1u2[2 cos q1 cos q2 # sin q1 sin q2 cos f]/4p30r3
#u12 u22 =3ð4p30 Þ2 kTr 6 ðKeesom energyÞ
Charge–non-polar
#Q2a/2(4p30)2r4
Dipole–non-polar
#u2a(1 þ 3 cos2 q)/2(4p30)2r6
#u2a/(4p30)2r6 (Debye energy)
Two non-polar molecules
Hydrogen bond
a
3 hva2
#
ðLondon dispersion energyÞ
4 ð4p30 Þ2 r 6
Complicated, short range, energy roughly
proportional to #1/r2
w(r) is the interaction free energy or pair-potential (in J); Q, electric charge (C); u, electric dipole moment (C m); a, electric polarizibility
(C2 m2 J#1); r, distance between the centers of the interacting atoms or molecules (m); k, Boltzmann constant (1.381 & 10#23 J K#1);
T, absolute temperature (K); h, Planck’s constant (6.626 & 10#34 J s); n, electronic absorption (ionization) frequency (s#1); 30, dielectric
permittivity of free space (8.854 & 10#12 C2 J#1 m#1). The force F(r) is obtained by differentiating the energy w(r) with respect to
distance r: F ¼ #dw/dr. The stabilizing repulsive “Pauli Exclusion” interactions (not shown) usually follow an exponential function w(r)f
exp(#r/r0), but for simplicity they are usually modeled as power laws: w(r)fþ1/rn (where n ¼ 9–12).
Dipole Moment
u =ql
Where l is the distance between two charges of value q
Example
Two electronic charges q = ± e
separated by l =0.1 nm
Dipole moment is (1.602 x 10-19)(10-10)=1.6 X 10 -29 Cm = 4.8D
1D = 3.336 X10-30 Cm
Interaction between ions
Interactions between
dipoles and ions
74 INTERMOLECULAR AND SURFACE FORCES
EQ
(a)
C
EQ
Charge
Q=ze
1
Dipole
moment
u =q
2
r
1
A
B
EQ
Eu
(b)
(c)
2
74 INTERMOLECULAR AND SURFACE FORCES
EQ
(a)
C
EQ
Charge
Q=ze
1
Dipole
moment
u =q
2
r
1
A
B
EQ
Eu
(b)
(c)
2
Ion Dipole
76 INTERMOLECULAR AND SURFACE FORCES
5 ´ 10–19
0
0.1 nm
0.02 nm
r
1D
u q
q
w (r) (J)
e r
90
q
0
0.1
r (nm)
0.2
0.3
kT at 300 K
r
0.02 nm
0
0.1 nm
−5 ´ 10–19
FIGURE 4.2 Charge-dipole interaction energy in vacuum (3 ¼ 1) between a unit charge e and a dipole of moment u ¼
Dipole-Dipole
Interactions
θ
ɸ
θ
Dipole-Dipole
NTERMOLECULAR AND SURFACE FORCES
0
Parallel
r
q
kT at 300 K
q
0.1 nm
0
w (r) (J)
0.1 nm
q
In line
–19
Parallel
–10
0
r
In line
u q
0.1
q
0.2
r (nm)
1D
0.3
0.4
Not quite as strong as ion-ion or dipole ion
rotating Dipoles
If you allow the dipole
to rotate freely are
large distances what
happens (ion-dipole)
e
Chapter 4 • Interactions Involving Polar
Z
1
qi ¼
4p
2
2
hsin qi ¼ ;
3
hcos2
p
0
2
cos q sin q dq
Z
2p
0
1
df ¼ ;
3
1
hsin fi ¼
fi ¼ ;
2
hsin qi ¼ hcos qi ¼ hsin q cos qi ¼ 0;
2
hcos2
hsin fi ¼ hcos fi ¼ hsin f cos fi ¼ 0:
less than kT, we can expand Eq. (4.11):
"wðrÞ=kT
!
"
#2
$
wðrÞ
wðr; UÞ 1 wðr; UÞ
þ/ ¼ 1"
þ
¼ 1"
"/
4p
2
2
hsin qi ¼ ;
3
0
3
0
1
hsin fi ¼
fi ¼ ;
2
hsin qi ¼ hcos qi ¼ hsin q cos qi ¼ 0;
2
hcos2
hsin fi ¼ hcos fi ¼ hsin f cos fi ¼ 0:
n w(r, U) is less than kT, we can expand Eq. (4.11):
e"wðrÞ=kT
!
"
#2
$
wðrÞ
wðr; UÞ 1 wðr; UÞ
þ/ ¼ 1"
þ
¼ 1"
"/ ;
kT
kT
2
kT
!
$
2
wðr; UÞ
wðrÞ ¼ wðr; UÞ "
þ/ :
2kT
angle-averaged free energy for the charge-dipole interaction is therefore,
4.5) for w(r, U),
wðrÞ ¼
!
z"
"
Qu cos q
Qu
"
"
2
4p30 3r
4p30 3r 2
Q2 u2
2
6ð4p3 3Þ kTr 4
for
#2
cos2
2kT
$
q
þ/
Qu
kT >
;
2
4p30 3r
Dipole Dipole
Take home
Ion dipole r^2
dipole-dipole r^3
Ion-rotating dipole r^4 (thermal contraints)
rotating dipole-dipole r^6 (thermal contraints)
dipole interactions weaker than ion
dipole moment = u = ql