Intuition The one core ideaA coloured complex is an energy ruler : the split d-orbitals sit at two heights, and light of exactly the right energy gets swallowed to lift an electron from the low shelf to the high shelf. Every symbol on this page — λ ν ˉ Δ o ε t 2 g e g
This page assumes nothing . If the parent note wrote a symbol without explaining it, we build it here from the ground up, in the order that each idea needs the one before it. Read top to bottom; by the end every piece of notation in the parent topic will be earned.
Before any symbol, picture a wave travelling to the right.
λ λ distance between two neighbouring crests of the light wave, measured in metres (or nanometres). In the figure it is the amber horizontal bar from one peak to the next.
Picture: how "stretched out" the wave is. Long λ λ Why the topic needs it: we identify a colour by its λ
ν ν how many crests pass a fixed point each second , measured in hertz (per second). Fast-arriving crests = high frequency.
Picture: stand at the cyan dot in the figure and count crests going by. More crests per second = higher ν
These two are locked together by the speed of light c c = λ ν ⟹ ν = λ c
λ ν c λ ν squeeze the wavelength and the frequency shoots up . That single trade-off is why "short wavelength" and "high frequency" always mean the same colour.
Light also arrives in tiny lumps called photons . Each lump carries a fixed amount of energy.
Intuition Why we even needh ν h h h quantised — it comes in whole packets, and h
Notice the inverse in E = h c / λ λ E λ E
ν ˉ ν ˉ = λ 1 number of wave-crests packed into one centimetre , unit cm − 1 Picture: lay a 1 cm ruler along the wave and count how many crests fit. A tightly-packed blue wave fits many → large ν ˉ
Intuition Why chemists preferν ˉ λ E = h c / λ = h c ⋅ ( 1/ λ ) = h c ν ˉ directly proportional to ν ˉ cm − 1 Δ o
Worked example The 500 nm ↔ 20000 cm⁻¹ conversion, step by stepStep 1 — WHAT: start with λ = 500 nm WHY: it's what the spectrometer reads.
Step 2 — convert to cm: 500 nm = 500 × 1 0 − 9 m = 500 × 1 0 − 7 cm WHY: ν ˉ Step 3 — flip it: ν ˉ = 500 × 1 0 − 7 cm 1 = 20000 cm − 1 WHY: wavenumber is one-over-wavelength.
Now the metal side. A transition-metal ion has five d-orbitals — five differently-shaped clouds where a d-electron can live. In a bare ion all five sit at the same energy. Put the ion inside six ligands (an octahedron) and the clouds pointing straight at the ligands get pushed up ; the ones pointing between get pushed down .
t 2 g e g
t 2 g lower set of 3 d-orbitals (they point between the ligands, feel less repulsion). Read it "tee-two-gee".e g upper set of 2 d-orbitals (they point at the ligands, feel more repulsion). Read it "ee-gee".
Picture: a low shelf holding 3 orbitals and a high shelf holding 2, in the figure.
(The letters are group-theory labels for orbital shape; the g low shelf t 2 g e g .)
Definition Crystal-field splittingΔ o Δ o = E ( e g ) − E ( t 2 g ) energy gap between the two shelves in an o ctahedral field. The subscript o Picture: the amber double-arrow spanning the gap in the figure.
Why the topic needs it: this gap is the price of the electron's jump. A photon is absorbed only if its energy matches Δ o Δ o = h ν abs = h c ν ˉ abs
size of Δ o λ Spectrochemical Series ranks, and it grows out of Crystal Field Theory .
Definition d–d transitionAn electron sitting on the low shelf t 2 g Δ o e g Picture: the cyan electron dot leaping the amber gap in the figure.
The electron count d n
d n d n n d 1 d 5 d 10 Oxidation States of Transition Metals for how the charge fixes n Why it matters here: d 0 d 10 no d–d jump → colourless .
Definition Complementary colourThe colour you see is white light minus the absorbed slice. On the colour wheel, seen and absorbed sit opposite each other.
Picture: the wheel in the figure — absorb the green wedge, and your eye sums everything left over into the red/purple on the far side.
Common mistake "Seen colour = absorbed colour"Truth: for a see-through complex you see the leftover , i.e. the complement. Absorb green → look red. This is why [ Ti ( H 2 O ) 6 ] 3 +
Colour has two questions: which wavelength (that's λ Δ o how strongly (that's ε
Definition Molar absorptivityε ε how greedily one mole of the complex swallows its wavelength — the "darkness per unit concentration". It appears in the Beer–Lambert Law :
A = ε c l
A c l ε L mol − 1 cm − 1 Picture: two beakers of equal concentration — the one with large ε ε
ε not change the wavelength; it shrinks ε ε ∼ 1 100 ε ∼ 1 0 3 1 0 4 loudness .
S S spin rule says a jump should keep the number of unpaired electrons the same (Δ S = 0 Magnetic Properties of Complexes , which count those same unpaired electrons.
Definition Charge transfer (CT)Instead of hopping within the d-shelves, the electron leaps between the metal and a ligand . LMCT = ligand→metal; MLCT = metal→ligand. Because it isn't a g → g ε MnO 4 − Electronic Spectra & Term Symbols .
Light wave: wavelength and frequency
Photon energy E equals h nu
Wavenumber nu-bar equals 1 over lambda
d-orbitals split into t2g and eg
Crystal field gap delta-o
d-d transition: photon energy equals delta-o
Colour seen equals complement of absorbed
Molar absorptivity epsilon: how strong
Selection rules: allowed vs forbidden
Spin S and unpaired electrons
d-d weak vs charge transfer strong
Topic: Colour and Spectra
Hide the right side and test yourself. If any line stumps you, reread its section above.
λ distance between two wave crests; identifies the colour
ν crests passing per second; frequency, tied to λ c = λ ν
Why E = h c / λ energy rises as wavelength shrinks — bluer light is more energetic
ν ˉ 1/ λ
Convert 500 nm to cm⁻¹ 1/ ( 500 × 1 0 − 7 cm ) = 20000 cm − 1
t 2 g e g lower shelf of 3 orbitals vs upper shelf of 2 orbitals
Δ o energy gap between t 2 g e g
What a d–d transition is an electron absorbing energy Δ o t 2 g → e g
Why d 0 d 10 nothing to lift (d 0 d 10
Seen vs absorbed colour seen = complement of absorbed (opposite on the colour wheel)
ε molar absorptivity — how strongly (not which) light is absorbed
Why forbidden ≠ zero rules shrink ε
