chem 241 lec 2
What does ℓ = 0 mean for an orbital?
It’s an s orbital with no angular dependence, so it is spherically symmetric.
Why is an s orbital spherical?
The wavefunction does not depend on θ or φ → probability is the same in all directions.
What does ∣Ψ∣ 2 represent?
Probability density (probability per unit volume).
What is the radial probability distribution for an electron?
P(r)=4πr2∣R(r)∣2
Why is there a 4πr 2 term in radial probability?
Because the area/volume of a spherical shell increases with r.
what does "orbital shape"usually represent (like 90%)
A boundary enclosing ~90% probability of finding the electron.
Why are electron energy levels negative in atoms?
Because E=0 is defined for a free electron at infinite separation; bound electrons have E<0 since energy must be added to remove them.
What does the principal quantum number
n control?
The energy level and average size of the orbital.
What does ℓ represent?
Orbital shape (ℓ=0 s, ℓ=1 p, ℓ=2 d, ℓ=3 f).
What does mℓ represent?
Orbital orientation in space, values from −ℓ to +ℓ.
How many radial nodes does an s orbital have?
n−1
How many radial nodes do 1s, 2s, 3s have?
1s: 0, 2s: 1, 3s: 2.
What is a node?
A region where Ψ = 0 so ∣Ψ∣ 2 =0 → zero probability of finding the electron.
Why are electron energies quantized in atoms?
Orbitals are standing wave solutions; only wavefunctions that satisfy boundary conditions are allowed.
What quantum number ℓ corresponds to a p orbital?
ℓ= 1
What values can mℓ take for p orbitals?
−1,0,+1
How many p orbitals exist in one shell?
3 (because 2ℓ+1=3)
Why does the p subshell start at n = 2?
Because ℓ can only be 0 to n−1; to have
ℓ=1, n must be ≥ 2.
What is the node in a p orbital?
A nodal plane through the nucleus where Ψ = 0
What do the + and – signs on p orbital lobes represent?
The phase/sign of the wavefunction Ψ, not electrical charge.
What are px, py, pz?
Three p orbitals oriented along the x, y, and z axes (same energy/shape, different orientation).
What is the difference between Ψ and Ψ²?
Ψ can be +/– (phase); Ψ² is always positive and gives probability density.
What do the +/– lobes in a p orbital represent?
Wavefunction phase/sign, not charge.
How many angular nodes does a p orbital have?
1 (since angular nodes = ℓ and for p, ℓ=1).
How many radial nodes does 2p have?
0 (since n−ℓ−1=0).
Why do p orbitals form two lobes?
A nodal plane through the nucleus splits the wavefunction into opposite phases → two lobes.
Why does orbital phase matter in bonding?
Same-phase overlap is constructive → bonding; opposite-phase overlap is destructive → antibonding.
How many angular nodes does a 3p orbital have?
1 (because ℓ=1).
How many radial nodes does a 3p orbital have?
1 (because n−ℓ−1=3−1−1=1).
Why does 3p have inner + outer lobes?
Because it has a radial node that splits the orbital into inner and outer regions.
What happens to Ψ at a radial node?
Ψ=0 and the wavefunction changes sign (phase flips).
Do +/– lobes represent charge?
No — they represent the phase of Ψ.
Why are outer lobes larger than inner lobes in 3p?
Higher n orbitals are more spread out → more probability at larger r.
What causes a p orbital to have two lobes?
A nodal plane through the nucleus splits Ψ into two opposite phases → two lobes.
What does the + / – sign on orbital lobes represent?
Phase (sign) of Ψ, not electrical charge.
Why does orbital shape matter for bonding?
Bond strength depends on how well orbitals overlap between nuclei (constructive overlap).
What is constructive orbital overlap?
Overlap of orbitals with the same phase → builds electron density between nuclei → bonding.
Why does inner orbital structure (nodes/inner lobes) affect energy in multi-electron atoms?
It affects penetration and shielding, which control how strongly the nucleus stabilizes the electron.
Why are orbitals described as standing waves?
Only certain wave patterns fit boundary conditions, producing stable wavefunctions with nodes.
What is ℓ for d orbitals?
ℓ = 2
How many d orbitals exist in a d subshell and why?
5, because m ℓ =−2,−1, 0, +1, +2 (so 2ℓ+1=5).
Name the five d orbitals.
d xy, d xz, dyz, dx2-y2, dz2
which d orbital points directly along the x and y axes
dx2-y2
which d orbital points mainly along the z axis and has a donut ring
d z2
for dxy where are the lobes located
between the x and y axes in the xy plane
what do + and - signs in d orbitals represent
wavefunction phase/sign not charge
Where are the lobes of dxy
located?
in the xy plane between the x and y axes
what are the angular nodes of dxy
the xz plane and yz plane
how many radial nodes does 4d have
4-2-1= 1
total nodes in 4d orbital
n-1= 3
What ℓ value corresponds to f orbitals?
ℓ =3
how many f obritals exist in an f subshell
7 (since 2ℓ+1=7)
What are the possible mℓ values for f orbitals?
−3,−2,−1,0,+1,+2,+3
What is the lowest n that can contain f orbitals?
n=4 (4f is the first).
Why are 4f orbitals weakly involved in bonding in lanthanides?
They’re core-like and shielded → poor overlap with ligands → mostly ionic bonding.
Why do lanthanides often have strong magnetic properties?
Many unpaired f electrons → high magnetic moments.
What does a radial probability distribution show?
The probability of finding an electron at a distance r from the nucleus.
Why do orbitals with higher n extend further from the nucleus?
Higher n means higher energy and larger average radius.
In hydrogen, what determines orbital energy?
Only n. So orbitals with same n are degenerate (3s = 3p = 3d).
In multi-electron atoms, what determines orbital energy?
Both n and ℓ, due to shielding and penetration.
What is penetration and why does it matter?
Penetration is how much electron density is close to the nucleus; more penetration → higher Zeff → lower energy.
Energy order for n=3 orbitals in multi-electron atoms?
3s<3p<3d
How many angular (planar) nodes do s orbitals have?
0
How many angular nodes do p orbitals have and what is it?
1 nodal plane through nucleus
How many angular nodes do d orbitals have?
2
What rule gives the number of orbitals in a subshell?
2ℓ+1.
How many orbitals are in s, p, d subshells?
s = 1, p = 3, d = 5.
Which d orbital points along x and y axes?
dx2-y2
What changes when moving from H atom to many-electron atoms?
: Electron–electron repulsion causes shielding/penetration → orbital energy depends on n and ℓ, not just n.
State the Aufbau principle.
Electrons fill the lowest energy orbitals first (ground state configuration).
State the Pauli exclusion principle (practical form).
Max 2 electrons per orbital with opposite spins.
State Hund’s rule.
In degenerate orbitals, electrons occupy singly first with parallel spins, then pair.
In hydrogen, are 3s, 3p, 3d the same energy?
Yes (degenerate, energy depends only on n).
In many-electron atoms, are 3s, 3p, 3d the same energy?
No. Usually 3s<3p<3d due to penetration/shielding.
How many orbitals exist in s, p, d, f subshells?
s:1, p:3, d:5, f:7
What is effective nuclear charge Zeff?
The net positive charge an electron feels after shielding by other electrons
Give the equation for effective nuclear charge.
Zeff= Z - S
What does S represent?
Shielding/screening by other electrons (depends on electron configuration/orbitals).
Why is Zeff <Z?
Because electron–electron repulsion shields the nucleus.
why does Zeff generally increase across a period
Z increases while shielding changes only slightly (electrons added to same shell)
Why is Li reactive based on Zeff?
its 2s valence electron is shielded by 1s^2 --> low Zeff--> easy to remove
why is Zeff needed
In multi-electron atoms, electrons repel each other and shield nuclear charge → electrons feel less than +Z.
Define effective nuclear charge.
Zeff = Z-S
why is H solvable but He isnt exactly
H is a 2body problem; he is a 3 body problem with e- - e- repulsion
Electron configuration of H and He?
H: 1s^1, He: 1s^2
Why does He put its second electron into 1s instead of n=2?
Even with pairing repulsion, 1s is far lower energy than 2s.
What is pairing energy?
Energy cost of placing two electrons in the same orbital (mainly from e⁻–e⁻ repulsion).
what is the first orbital in n = 2?
2s