ACER 9/23 Lecture #7: Many-Electron Atoms

+ Continuation of Lecture #6: Atomic Orbitals (QM of Hydrogen)

Why are orbitals with l > 0 not symmetrical?

the angular wave functions Y(θ,ϕ) have lobes with a positive and negative phase separated by a nodal plane

What are the two types of nodes? How do you determine how many of each is in an orbital?

Angular nodes (aka nodal Planes) and radial nodes

The total number of nodes is n - 1.

The number of angular nodes in an orbital is the same number as l, the angular momentum number.

The number of radial nodes is n - l - 1.

Why do the radial wavefunction graphs for 2s and 3p look the same?

Because both orbitals have 1 radial node. 3p also has an angular node but that isn’t shown in radial wavefunction graphs.

How are Nodal Planes shown in radial wavefunction graphs and radial distribution graphs?

Nodal Planes are shown in angular distribution graphs but not in radial wavefunction graphs or radial distribution graphs, since radial wavefunction graphs measure wavefunction from the nucleus outward.

How many p-orbitals are there? Describe them and what they mean.

There are three p-orbitals (l = 3). They each align along a different Cartesian axis: P_x, P_y, P_z.

For the P_z orbital, the orbital points along the ±z axis so the electron has zero probability to be found in xy plane. The same applies for the other orbitals: for the P_x orbital, the orbital points along the ±x axis so the electron has zero probability of being found in the yz plane.

Draw a 2p orbital

Draw a 3p orbital

Draw a 3d orbital

Stern-Gerlach Experiment

sent silver ions (which have unpaired electrons) through a non-uniform magnetic field and the beam split, showing the opposite spins

proved the Pauli Exclusion Principle; proved that the spatial orientation of angular momentum is quantized

Compare Hydrogen atoms and many-electron atoms

Hydrogen atoms only experience Coulombic attraction between the nucleus and electrons, while many-elecron atoms also experience repulsive Coulombic interactions between electrons.

For hydrogen atoms, electrons in the same shell are degenerate, but the same does not apply to many-electron atoms due to electron-electron repulsion. For many-electron atoms, only electrons in the same subshell of an energy shell are degenerate.

The energy levels for either atom is also different.

How do you approximate the Schrödinger equation solution for many-electron atoms?

Due to electron-electron repulsion, the Schrödinger equation cannot be solved exactly for many-electron atoms.

Instead, this equation is used.

• Z_eff is effective nuclear charge which approximates the number of valence electrons by calculating Z_eff = Z - S. It influences energies and locations of electrons in the valence shells.

  • S is the number of shielding electrons

This equation is the fourth solution, modifying the third solution (for one-electron atoms) which modified the second solution (hydrogen atoms; Rydberg equation), which modified the particle in a box solution.

What is shielding?

when the outer electrons are prevented from getting close to the nucleus due to repulsion against inner electrons

• Electrons in inner subshells (lower l) shield more effectively, even in the same shell.

• Electrons in the same subshell screen each other very little.

Since orbitals in subshells are less effective in shielding further away from the nucleus, shielding does not increase proportionally across a period.

Aufbau Principle

dictates the order in which electrons are added to orbitals from lowest to highest energy (as a result of shielding was breaks degeneracy)

Hund’s Rule

for orbitals of the same energy, the orbitals are filled with one electron each first, with spins parallel (this is the most stable)

When removing electrons to make an ion, follow Hund’s rule in reverse. First, from the outermost subshell orbitals that have two electrons, remove an electron from each of them. Clear the subshell before moving to the next, more inner, subshell.