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Dr. Henry Curran
Physical Chemistry (CH313)
Atkins and de Paula, Chapter 17
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.
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.
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.
If the potential energy is denoted V, then the force is –dV/dr. If V = -C/r6
the magnitude of the force is:
Atoms in molecules generally have partial
where q1 and q2 are the partial charges
and r is their separation
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:
Where e is the permittivity of the medium lying between the charges.
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.
Consider two ions in a lattice
two ions in a lattice of charge numbers z1 and
z2 with centres separated by a distance r12:
where e0 is the vacuum permittivity.
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.
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:
Lattice Enthalpy ( ) 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)
ProcessDH0 (kJ mol-1)
Sublimation of K (s) +89
Ionization of K (g) +418
Dissociation of Cl2 (g) +244
Electron attachment to Cl (g) -349
Formation of KCl (s) -437
ProcessDH0 (kJ mol-1)
KCl (s) K (s) + ½ Cl2 (g) +437
K (s) K (g) +89
K (g) K+ (g) + e- (g) +418
½ Cl2 (g) Cl (g) +122
Cl (g) + e- (g) Cl- (g) -349
KCl (s) K+ (g) + Cl- (g) +717 kJ mol-1
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.
We represent dipole moments by an arrow with a length proportional to m and pointing from the negative charge to the positive charge:
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)
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.
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.
More electronegative atom is usually the negative end of the dipole. There are exceptions, particularly when anti-bonding orbitals are occupied.
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
Molecular symmetry is of the greatest importance in deciding whether a polyatomic molecule is polar or not.
Carbon dioxide, CO2
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.
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 q with one another is approximately:
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 mshows the orientation of the dipole in the molecule and the length of the vector is the magnitude, m, of the dipole moment.
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:
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
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
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 =[(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)
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.
A similar calculation for the more general orientation is given as:
If q2 is positive, the energy is lowest when q = 0 (and cos q = 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.
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.
Interaction energy between two dipoles m1 and m2:
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.
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.
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 q = 180o, m1 = m2 = 2.7 D, corresponding to 9.1 x 10-30 C m
The average energy of interaction between polar molecules that are freely rotating in a fluid (gas or liquid) is zero (attractions and repulsions cancel). However, because the potential energy for dipole-dipole interaction depends on their relative orientations, the molecules exert forces on one another, and do not rotate completely freely, even in a gas. Thus, the lower energy orientations are marginally favoured so there is a non-zero interaction between rotating polar molecules.
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.
E a 1/r6 => van der Waals interaction
E a 1/T => greater thermal motion overcomes the mutual orientating effects of the dipoles at higher T
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* = aE
where a is the polarizability of the molecule.
Polarizability also depends on the orientation of the molecule wrt the electric field unless the molecule is tetrahedral (CCl4), octahedral (SF6), or icosahedral (C60).
The polarizability volume has the dimensions of volume and is comparable in magnitude to the volume of the molecule
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?
A polar molecule with dipole
moment m1 can induce a dipole
moment in a polarizable
the induced dipole interacts with the permanent dipole of the first molecule and the two are attracted together
the induced dipole (light arrows) follows the changing orientation of the permanent dipole (yellow arrows)
For a molecule with m = 1 D (HCl) near a molecule of polarizability volume a’ = 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 a 1/r6 => van der Waals interaction
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.
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)
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, as in: d-X—Hd+ ……Yd-
(DS < 0).
CH4 (in CCl4) = CH4 (aq)
has DH = - 10 kJ mol-1, DS = - 75 J K-1 mol-1, and DG = + 12 kJ mol-1 at 298 K.
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.
The total attractive interaction energy between rotating molecules that cannot participate in hydrogen bonding is the sum of the contributions from the dipole-dipole, dipole-induced-dipole, and dispersion interactions.
Only the dispersion interaction contributes if both molecules are non-polar.
All three interactions vary as the inverse sixth power of the separation. Thus the total van der Waals interaction energy is:
where C is a coefficient that depends on the identity of the molecules and the type of interaction between them.
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 dominate the attractive forces when molecules are squeezed together.
graph of the potential energy of two closed-shell species as the distance between them is changed
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
This very simple assumption is surprisingly useful in assessing a number of properties.
Another approximation is to express the short-range repulsive potential energy as inversely proportional to a high power of 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
The sum of the repulsive interaction with n = 12 and the attractive interaction given by:
is called the Lennard-Jones (12,6)-potential.It is normally written in the form:
The two parameters are e (epsilon), 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