Intuition The one core ideaA transition-metal complex is coloured because six (or four) ligands squeeze the metal's five d-orbitals into two energy shelves , and a d-electron hops between them by swallowing exactly one colour of light. Everything on the parent page is just the vocabulary needed to say how big the shelf-gap is and which colour gets swallowed — this page builds that vocabulary from nothing.
Before you can read the parent note, you must own every symbol it throws at you . Below, each one is built in order: plain words → a picture → why the topic needs it. Nothing is used before it is defined.
Definition Metal ion, ligand, complex
A metal ion is a transition-metal atom that has lost some electrons, so it carries positive charge — e.g. T i 3 +
A ligand is a small molecule or ion (like H 2 O N H 3 C N −
A complex is the metal ion surrounded by its ligands, written in square brackets, e.g. [ T i ( H 2 O ) 6 ] 3 + T i 3 +
Intuition Read the formula like a sentence[ metal T i ( ligand H 2 O ) how many 6 ] 3 + total charge 6 counts the ligands; the superscript 3+ is the leftover charge on the whole package. The picture is a metal ball at the centre with six little water arms pointing at it.
We need this because which ligands and how many is exactly what decides the colour later.
An orbital is a region of space where an electron is likely to be found — think of it as a shelf an electron can sit on. Electrons fill shelves labelled s p d f d five of them.
Definition The five d-orbitals and their shapesEach of the five d-orbitals has a distinct shape and points in a distinct direction :
d x y , d y z , d z x between the coordinate axes.d x 2 − y 2 along the x y d z 2 z
The subscript is just a name tag telling you which way the orbital points (x y x 2 − y 2
Why does the topic need the direction each orbital points? Because the ligands sit on the axes . An orbital pointing straight at a ligand feels more repulsion than one pointing into the gaps — and that difference is the whole source of colour.
Common mistake The superscript is not a powerd x 2 − y 2 x 2 − y 2 just a shape label , not "x squared minus y squared" being calculated. It reminds you the lobes lie along the x and y axes.
d n d n n n 0 10
T i 3 + d 1 C u 2 + d 9 Z n 2 + d 10 S c 3 + d 0
d n jump from a low shelf to a high shelf. That requires two things at once : at least one electron to do the jumping (n ≥ 1 n ≤ 9 d 1 d 9 d 0 d 10
The parent's Worked Examples 1–3 are literally just this rule: d 1 d 10 d 0
Two or more orbitals are degenerate when they have exactly the same energy . In a lonely gaseous metal ion (no ligands), all five d-orbitals are degenerate — five shelves at the same height , so an electron doesn't care which it sits on.
Picture five books resting on one flat table: same height everywhere. The topic needs this as the "before" picture — colour appears only once this flatness is broken .
Now the ligands approach. Their electron pairs repel the metal's d-electrons (like charges push apart). Orbitals pointing straight at the ligands get shoved up in energy; orbitals pointing into the gaps get pushed down. The flat table breaks into a high group and a low group .
Δ e g t 2 g
Δ energy gap between the high group and the low group. It is measured in energy units (joules, or kJ mol⁻¹).Octahedral (six ligands on the axes): high group is called e g { d x 2 − y 2 , d z 2 } t 2 g { d x y , d y z , d z x } Δ o Tetrahedral (four ligands): the gap is Δ t e g t 2 g group name-tags from symmetry — read them as "the high two" and "the low three".
Intuition Why the split, not just "all pushed up"?Every d-orbital is repelled, so all rise a bit — but the ones aimed at the ligands rise more . It's the difference in height , Δ Δ
The subscript on Δ Δ o Δ t geometry , because geometry changes the gap size — the parent's rule Δ t = 9 4 Δ o
To connect a gap to a colour , we need the language of light.
Definition The light symbols
λ wavelength , the length of one wave. Short λ λ 1 nm = 1 0 − 9 m ν frequency , how many waves pass per second. Measured in hertz (per second).c speed of light , ≈ 3 × 1 0 8 m s − 1 h Planck's constant , ≈ 6.626 × 1 0 − 34 J s E energy of one photon (one packet of light), in joules.
Intuition Why two formulas, and why combine them?E = h ν frequency , but chemists label colours by wavelength λ ν c / λ E = λ h c Δ o Δ o = λ h c "which colour fits the gap."
Intuition The inverse relationship (avoid the classic trap)In λ = Δ o h c Δ o denominator . So bigger gap ⇒ smaller λ (more energetic, bluer light). "Strong field, short wave." Don't let "bigger" trick you into "longer wavelength".
Definition Complementary colourWhite light is all colours mixed. When the complex swallows one colour, your eye receives "white minus that colour." The leftover mix is called the complementary colour of what was absorbed. Green-yellow absorbed → you see purple; orange absorbed → you see blue.
The topic needs this because we can only measure the absorbed wavelength, but we see its complement. See Complementary Colours & the Colour Wheel .
Metal ion + ligands = complex
Five d-orbitals point in set directions
Degenerate before ligands arrive
Ligands repel -- shelves split by gap Delta
Count electrons with d-to-the-n
Electron jumps low group to high group
Light symbols lambda nu c h
E equals h nu and c equals lambda nu
Combine to E equals hc over lambda
Photon energy equals Delta
Absorbed colour fixed by Delta
Seen colour is the complement
Cover the right side and test yourself — you are ready for the parent note when every one is instant.
What does the subscript in [ T i ( H 2 O ) 6 ] 3 + The number of ligands (here, 6 water molecules) attached to the metal.
What does the superscript 3 + The total charge on the whole complex ion.
How many d-orbitals are there, and what does d n Five d-orbitals; d n n n = 0 10
What does "degenerate" mean? Two or more orbitals having exactly the same energy.
Why do the d-orbitals split when ligands approach? Ligand electron pairs repel the metal d-electrons; orbitals pointing at ligands rise more than those pointing between them.
What is Δ Δ o The energy gap between the upper (e g t 2 g
Which orbitals are e g t 2 g e g = d x 2 − y 2 , d z 2 t 2 g = d x y , d y z , d z x
What are λ ν c h Wavelength, frequency, speed of light, Planck's constant.
State the two light relations. c = λ ν E = h ν E = h c / λ
Does a bigger Δ Shorter (more energetic), since λ = h c /Δ
Which two d n d 0 d 10
What is a complementary colour? The colour you see = white light minus the absorbed colour.
Parent: Colour of complexes — d-d transitions
Crystal Field Theory Octahedral vs Tetrahedral Splitting Spectrochemical Series Complementary Colours & the Colour Wheel Planck's Equation E=hν Charge-Transfer Spectra (why KMnO4 is intensely coloured) Magnetic Properties of Complexes