Shapes and Structures 2 - Orbital Theory

Question 1

What is n?

Answer 1

Principal quantum number, which shell is being referred to

Takes integer values 1 up to n

Question 2

What is l?

Answer 2

Angular momentum quantum number
Whether we are referring to an s, p, d or f sub-shell
Takes integer values 0 up to (n-1)

Question 3

What are the values of l for each sub-shell?

Answer 3

s: l=0
p: l=1
d: l=2
f: l=3

Question 4

What is mₗ?

Answer 4

Magnetic quantum number

Takes integer values from +l to -l

Question 5

What are the mₗ values for each sub-shell?

Answer 5

s: mₗ = 0 only -> so one s orbital for each n value
p: mₗ = +1, 0, -1 -> so 3 p orbitals for each n value
d: mₗ = +2, +1, 0, -1, -2 -> so 5 d orbitals for each n value

Question 6

What is s?

Answer 6

The magnitude of the spin angular momentum

Question 7

What is mₛ?

Answer 7

The orientation of the spin angular momentum

Takes integer steps from +s to -s

Question 8

What is the value of s for an electron?

What is the conclusion?

Answer 8

s = 1/2

All electrons have the same intrinsic angular momentum

Question 9

What are the values of mₛ for an electron?

Answer 9

mₛ = +1/2 (spin up) or -1/2 (spin down)

Question 10

What do the four quantum numbers specify?

Answer 10

n specifies the energy
l specifies the magnitude of the orbital angular momentum
mₗ specifies the orientation of the orbital angular momentum
mₛ specifies the orientation of the spin angular momentum

Question 11

What is the wavefunction used to describe an orbital?

Answer 11

Ψₙ,ₗ,ₘₗ(x, y, z)

Question 12

What does the plot of Ψ^2 show?

Answer 12

A measure of the probability of finding the electron around a given position

Question 13

What is the equation for the energy of a described wavefunction?

Answer 13

Eₙ = ((-mₑ x e^4)/(8ε₀^2 x h^2)) x (z^2)/(n^2)
mₑ = mass of an electron
ε₀ = vacuum permittivity
h = Planck's constant
e = elementary charge
z = nuclear charge of atom/ion

Question 14

How is the equation for the energy of a described wavefunction written more conveniently?

Answer 14

Eₙ = -Rₕ x (z^2)/(n^2)

Rₕ is called the Rydberg constant (units energy)

Question 15

What does the energy of an orbital depend on?

Answer 15

Only n

2s and 2p orbitals have the same energy (degenerate)

Question 16

What is the main feature of the energy of a wavefunction equation?

Answer 16

The negative sign
Means that Eₙ tends towards zero
Zero Eₙ is where the electron is separated from the nucleus

Question 17

What is the form of a wavefunction in polar coordinates?

Answer 17

Ψₙ,ₗ,ₘₗ(r, θ, Φ)

Question 18

How can the polar form of a wavefunction be expressed as the product of two functions? And what does each represent?

Answer 18

Ψₙ,ₗ,ₘₗ(r, θ, Φ) = Rₙ,ₗ(r) x Yₗ,ₘₗ(θ, Φ)
Rₙ,ₗ(r) is the radial part of the wavefunction
Yₗ,ₘₗ(θ, Φ) is the angular part of the wavefunction

Question 19

What is the wavefunction for the 1s orbital proportional to?
What does the graph look like?
What does a 3D isoplot look like?

Answer 19

Proportional to e^(-r/a₀)
Exponential decay from a maximum at r = 0
Spherical isoplot for any given wavefunction

Question 20

What does the RDF (Radial Distribution Function) against r graph show?

Answer 20

Probability of finding an electron at a given radius, summed over all angles

Question 21

What does RDF give?

Answer 21

Probability of finding an electron in a thin shell of thickness δr at radius r from the nucleus

Question 22

What does Ψ^2 give?

Answer 22

Probability of finding the electron in a tiny volume element δv at position x, y, z from the nucleus

Question 23

How is RDF found?

Answer 23

Ψ^2 x r^2

Question 24

What is a radial node?

Answer 24

A value of r at which Ψ = 0

Visible in the density plot as a white ring

Question 25

What kind of nodes do p orbitals have one of?

What are they for the 2p orbital?

Answer 25

Angular node
2pₓ: Φ = 90°, yz nodal plane
2pᵧ: Φ = 0°, xz nodal plane
2pz: θ = 90°, xy nodal plane

Question 26

How does the total number of nodes relate to the principal quantum number n?

Answer 26

Total number of nodes = angular nodes + radial nodes

= n - 1

Question 27

How many angular nodes are there for an orbital?

Answer 27

l angular nodes

Where: for s = 0, p = 1, d = 2, f = 3

Question 28

How many radial nodes are there for an orbital?

Answer 28

n - 1 - l radial nodes

Question 29

For the 2s e- in Li, how could orbital approximation be used to remove the effects of the 1s e-?

Answer 29

Averaging out the effects of the two 1s e- on the 3+ nucleus and 2s e- is like the 2s e- being attracted to a roughly 1+ nucleus (“hydrogen-like” system)

Question 30

How much does each 1s e- shield each other by?

Answer 30

Roughly 30% of one proton’s charge

Question 31

What is the term for the nuclear charge each electron experiences?

Answer 31

Effective nuclear charge

Question 32

For multi-electron systems such as Li, why do the 2s and 2p orbitals have different energy?

Answer 32

2s orbital penetrates 1s more than 2p does
So 2s experiences greater nuclear charge than 2p
So 2s has slightly lower energy than 2p (using Eₙ equation)

Question 33

Why is the 4s lower than the 3d orbital for atoms like K or Ca but not for H?

Answer 33

For K/Ca, greater nuclear charge contracts orbitals and penetration of lower orbitals by 4s is sufficient for it to be lower energy
Not the case in H as 4s e- is further away on average

Question 34

What does the graph of orbital energies against atomic number show?

Answer 34

Orbital energy decreases across a period due to increased Z(eff)
Core electrons (not in valence shell) have very low energy and take little part in reactions