Def 3.1: Cohesive energy of a crystal is defined as the energy that must be added to the crystal to
separate its components into neutral free atoms, at rest, at infinite separation.
Note:
The inert ("noble") gases have small cohesive energies. Thermal energy easily overcomes the
cohesion and so these are gases at RT.
Alkali metals (le) have intermediate energies.
In the middle of the periodic table the cohesive energies are typically strong.
Physical properties such as melting temperature and bulk modulii vary roughly as the
cohesive energies.
The inert gases form the simplest crystals, with atoms remaining very similar to free atoms. These elements
have their outermost electron shell completely filled, and the distribution of electon charge in the free atom
is spherically symmetric.
If the electron charge distributions were rigid, the interactions between two identical inert gas atoms would
zero. There would be no cohesion and solid crystals would not form.
However, momentary asymmetries in the electron distribution of one atom can induce dipole moments in
neighbouring atoms.
Note:
This energy gets larger as R is smaller. If we only have the aractive London interaction, then
the atoms would try to get infinitely close to each other! (ie R=0). For a real crystal with typical
atomic spacing we need to balance with a repulsive interaction.
Defn 3.3: The Pauli Exclusion Principle states that two electrons cannot have all their quantum numbers
equal; ie there cannot be two electrons overlapping in exactly the same "state".
This means that electron overlap increases the total energy of the system, and therefore gives a repulsive
interaction. (Remember, the system wants to lower its energy as much as possible).
Experimental data for inert gases fits closely to a repulsive potential of the form
Defn 3.4: It is usual to write the total potential energy of two atoms separated by R as
Note:
Since the Lennard-Jones potential only considers separation distance (and not angle), the atoms
will pack as closely as they can while minimising the repulsion interaction. All inert gas crystals
are fcc structures.
Neglecting kinetic energy of the inert gas atoms, the cohesive energy of an inert gas crystal is the sum
of the Lennard-Jones potential over all pairs of atoms in the crystal. For N atoms,
For the fcc structure, there are 12 nearest neighbours. Since the terms in the sums get smaller as R increases,
We expect that the sums will not be much greater than 12. Indeed the limits have been evaluated as
Taking the
values of
obtained from gas measurements, and R obtained from crystallography, the empirical
Note:
The deviation from 1.09 for lighter elements is mostly due to "zero-point motion" (the kinetic energy
of atoms at absolute zero). This is a quantum mechanical effect that is less important for heavier
atoms.
In fact, this zero-point motion is why He never condenses at all!
From measurements of inert gases we have predicted the laice constant of inert gas crystals!
Ionic crystals are made up of positive and negative ions. The atoms form these ions by gaining or losing
electrons to obtain a full outer shell.
These ions have the electronic configuration of an inert gas atom (full outer shell). In addition to the
London interaction, these ions will experience a direct electrostatic araction. This electrostatic interaction
is much stronger than the London interaction (by a factor of ~100)