The Shapes And Structures Of Molecules Pt. 2

• no they have different energies • we can use photoelectron spectroscopy • this bombarded a sample with x-ray photons and measuring the KE of the removed electron to tell the difference

2 of opposite spins

Degenerate

• which ‘shell’ it is in • whether it’s an s,p,d,f orbital • which of the orbitals in each subshell it’s in • each of these is expressed by a quantum number

The principal quantum number, n, specifies which shell is being referred to n takes an integer value 1,2,3… For a one electron system, n alone determines the energy of the electron

The angular momentum number, l, specifies which type of subshell orbital we are referring to (s,p,d,f…) • l takes integer values from 0 to (n-1) Each value of l has a different letter associated with it Value of l: 0,1,2,3,4,5 Letter used: s,p,d,f,g,h E.g. where n = 3, l = 0,1,2, these are s,p,d

The value of l determines the orbital angular momentum of the electron Angular momentum = (h/2(pi)) sqrt(l(l+1)) Orbitals with different l numbers have different spatial arrangements

The magnetic quantum number, ml, is the number that tells us which orbital in the subshell it is ml takes integer steps from +l to -l E.g. for a P orbital, l = 1 So ml = -1, 0, 1 Hence there are 3 p orbitals

s = 1/2

• the orientation of the electron’s intrinsic angular momentum, ms ms = +1/2 or -1/2 for spin up (up arrow) or spin down (down arrow) respectively

• we can think of the orbital angular momentum as the angular momentum the electron has due to its movement within an orbital • we can think of the intrinsic angular momentum as the momentum an electron has due to its spin on its own axis

• n specifies the electron energy • l specifies the magnitude of the orbital angular momentum (s,p,d etc.) • ml specifies the orientation of the orbital angular momentum (which s,p,d orbital) • ms specifies the orientation of the spin angular momentum (spin up or spin down)

• in an orbital if there are two electrons they MUST have opposite spin • any electron in an atom MUST have a unique set of 4 quantum numbers

a wavefunction represented by psi

a wavefunction, its a function of coordinates that describes the probability of finding the electron e.g. (xyz)

ψ[n,l,ml] (x,y,z) ψ represents the wavefunction [n,l,ml] shows that the specific function changes depending on the orbital, where our orbital is given by n,l,ml (x,y,z) are examples of coordinates where our wavefunction is a function of coordinates

• the wavefunction, ψ, gives numbers associated with different coordinates • ψ ^2 or (ψ)(ψ*) gives the probability density

using the Schrodinger equation

• we can solve Schrodinger’s equation exactly for any, one-electron system • the solutions give the atomic orbitals • there are a variety of solutions depending on the quantum numbers of the orbital

En = (- (mee^4)/(8Eo*h^2)) (z^2/n^2) where me = mass of electron e = charge on an electron h = Planck’s constant Eo = permittivity of free space z = nuclear charge of atom n = principal quantum number this can be simplified to En = -Rh (z^2/n^2) where Rh = a constant

• the energy of an orbital depends on n only • this means that 2s and 2p have the same energy, and 3s, 3p and 3d all have the same energy unless in a multi-electron system • the predicted energies are also negative because zero energy represents an electron totally free from the nucleus • this makes sense because as n –> infinity En –> 0

Rh = ionisation energy for 1H atom from 1s shell

• spherical polar coordinates (r,θ,φ) r = radius θ = polar angle φ = azimuthal angle • this simplifies our equation from psi [n,l,ml] (r,θ,φ) = R [n,l] (r) x Y [l,ml] (θ, φ) where R [n,l] (r) is the radial part of the wavefunction, dependent on n and l and defined in terms of r and Y is the angular part of the wavefunction dependent on l and ml and defined in terms of θ and φ

radius, note: this is only the distance from the nucleus, it does not take into account displacement