Physics 249 Lecture 26, Nov 5th 2012
Reading: Chapter 9
HW 8 available on web site, due Nov 9th
1) Ionic bonding (in more detail)
Bonding between atoms with low ionization energy, which donate an election, and atoms
with high electron affinity, which accept electrons and can lower their potential energy by
binding the extra electron.
However the net ionization energy, which is equal to the ionization energy minus the
electron affinity energy is greater than zero. There must be additional contributions to the
energy, negative to make the full potential energy negative and as a function of radius
rising to either side of a minimum position to create a bound state. The overall potential
energy is:
𝑈(𝑟) = 𝐸𝐸𝑀 + 𝐸𝑒𝑥 + 𝐸𝑖𝑜𝑛 − 𝐸𝑎𝑓𝑓
𝑘𝑒 2 𝐴
=−
+ 𝑛 + 𝐸𝑖𝑜𝑛 − 𝐸𝑎𝑓𝑓
𝑟
𝑟
A and n depend on the ionic bond being considered.
You should include unshielded proton repulsion at low separation distance, but this is
typically small in ionic bonding.
The equilibrium position occurs when the forces due to the electrostatic attraction and the
exclusion principle repulsion balance. Moving to lower or higher radius will cause a
restoring force toward the equilibrium position. As noted previously the system acts
approximately like a harmonic oscillator.
The dissociation energy is the energy to go from the sum of electrostatic and exclusion
principle energies to zero potential, which is less that the difference between the
ionization minus electron affinity energies.
A note on exclusion principle repulsion. Compare the equilibrium separations of:
NaCl 0.236nm
NaBr 0.250nm
Bromine is a atom with an additional complete shell with 4s, 3d and 4p orbital subshells
compared to Cl. You would expect the electron probability distribution to be
substantially larger. However, the increase in size is not that large since Bromine has
more protons which bind the inner electron more tightly. In fact Br- is 15% larger than
Cl-. This is not consistent with the 6% difference in the equilibrium separations.
However, even though Bromine has the extra electrons they do not overlap Na+ until the
corresponding outer 1p orbital of Na+ overlaps the 1p orbitals of Cl and Br. The
observed separation is used to measure the size of Na+, 0.116nm which is consistent
with the size of 1p orbitals for atoms in the B – F series.
On the size of 1p orbitals. The Bohr radius, a, goes as the inverse of the charge so we can
estimate the adjusted Borh radius using the Zeff*q for the 1p orbitals. Zeff ~ 3. Hydrogen
p orbitals have substantial probability to order 6a so with a Zeff of 3 we would expect
orbital repulsion to start around 6a/3 = 2a = 0.106 times 2.
2) Covalent bonding
There many cases where atoms bond into diatomic molecules where ionic bonding is not
possible. This means that there is no separation distance for which the total energy
including ionization, the negative electron affinity, the negative coulomb attraction, the
positive exclusion principle repulsion, and for low separations the proton repulsion is
negative.
However, many atoms can still bond via covalent bonding. Covalent bonding exists
when atoms share electrons.
The simplest example of covalent bonding is H2. Solving this would involve
understanding a 4 body kinematic, and quantum system. However, we can learn a
substantial amount by examining the qualitative features of the system.
You expect a bound state should should exist because in the limiting case of completely
overlapping the atoms you they it should behave approximately like a helium atom.
For two hydrogen atoms in the ground state far apart the electron radial wave functions
should be exponentially falling. As you move them together they should start to overlap.
They would not feel a exclusion repulsion if they had spins oriented in opposite
directions since they would not be in the same quantum state. The overlapping region
would have a substantial probability for negative charge to exist there and would cause a
force that attracts the two protons. The attractive potential energy lowers as you move to
lower separation differences since there is more negative charge in between the protons.
Eventually the protons become un-screened and repulse each other raising the potential
energy and forming a minimum. Thus you have a minimum and a bound state.