Shapes and Structures 2 - Orbital Theory Flashcards
What is n?
Principal quantum number, which shell is being referred to
Takes integer values 1 up to n
What is l?
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)
What are the values of l for each sub-shell?
s: l=0
p: l=1
d: l=2
f: l=3
What is mₗ?
Magnetic quantum number
Takes integer values from +l to -l
What are the mₗ values for each sub-shell?
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
What is s?
The magnitude of the spin angular momentum
What is mₛ?
The orientation of the spin angular momentum
Takes integer steps from +s to -s
What is the value of s for an electron?
What is the conclusion?
s = 1/2
All electrons have the same intrinsic angular momentum
What are the values of mₛ for an electron?
mₛ = +1/2 (spin up) or -1/2 (spin down)
What do the four quantum numbers specify?
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
What is the wavefunction used to describe an orbital?
Ψₙ,ₗ,ₘₗ(x, y, z)
What does the plot of Ψ^2 show?
A measure of the probability of finding the electron around a given position
What is the equation for the energy of a described wavefunction?
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
How is the equation for the energy of a described wavefunction written more conveniently?
Eₙ = -Rₕ x (z^2)/(n^2)
Rₕ is called the Rydberg constant (units energy)
What does the energy of an orbital depend on?
Only n
2s and 2p orbitals have the same energy (degenerate)
What is the main feature of the energy of a wavefunction equation?
The negative sign
Means that Eₙ tends towards zero
Zero Eₙ is where the electron is separated from the nucleus
What is the form of a wavefunction in polar coordinates?
Ψₙ,ₗ,ₘₗ(r, θ, Φ)
How can the polar form of a wavefunction be expressed as the product of two functions? And what does each represent?
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
What is the wavefunction for the 1s orbital proportional to?
What does the graph look like?
What does a 3D isoplot look like?
Proportional to e^(-r/a₀)
Exponential decay from a maximum at r = 0
Spherical isoplot for any given wavefunction
What does the RDF (Radial Distribution Function) against r graph show?
Probability of finding an electron at a given radius, summed over all angles
What does RDF give?
Probability of finding an electron in a thin shell of thickness δr at radius r from the nucleus
What does Ψ^2 give?
Probability of finding the electron in a tiny volume element δv at position x, y, z from the nucleus
How is RDF found?
Ψ^2 x r^2
What is a radial node?
A value of r at which Ψ = 0
Visible in the density plot as a white ring
What kind of nodes do p orbitals have one of?
What are they for the 2p orbital?
Angular node
2pₓ: Φ = 90°, yz nodal plane
2pᵧ: Φ = 0°, xz nodal plane
2pz: θ = 90°, xy nodal plane
How does the total number of nodes relate to the principal quantum number n?
Total number of nodes = angular nodes + radial nodes
= n - 1
How many angular nodes are there for an orbital?
l angular nodes
Where: for s = 0, p = 1, d = 2, f = 3
How many radial nodes are there for an orbital?
n - 1 - l radial nodes
For the 2s e- in Li, how could orbital approximation be used to remove the effects of the 1s e-?
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)
How much does each 1s e- shield each other by?
Roughly 30% of one proton’s charge
What is the term for the nuclear charge each electron experiences?
Effective nuclear charge
For multi-electron systems such as Li, why do the 2s and 2p orbitals have different energy?
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)
Why is the 4s lower than the 3d orbital for atoms like K or Ca but not for H?
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
What does the graph of orbital energies against atomic number show?
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
