Using Magnetic Moments to Assign Spin State

Introduction

Two iron complexes provide a clean magnetic comparison. [Fe(H₂O)₆]²⁺ and [Fe(CN)₆]⁴⁻ both contain formal Fe²⁺, hence d⁶, yet their ligands produce different octahedral splitting. Magnetic moments distinguish their high- and low-spin electron distributions more directly than a colour description alone.

Core explanation

Begin with charge bookkeeping. Water is neutral, so [Fe(H₂O)₆]²⁺ has Fe²⁺. Six CN⁻ ligands total −6; for [Fe(CN)₆]⁴⁻, x−6=−4, again giving Fe²⁺. The iron d count is six in both. Their coordination number is six and both are approximately octahedral in the standard description. The key variable is ligand field strength, not a change from Fe²⁺ to Fe³⁺.

Water is a comparatively weaker-field ligand. With a smaller Δₒ relative to pairing cost, the d⁶ high-spin filling is t₂g⁴e g². It has one electron pair in t₂g and four unpaired electrons overall. Its spin-only moment is μ so=√[4(4+2)]=√24≈4.90 BM. A measured effective moment around this range supports the high-spin assignment, subject to orbital contributions and measurement conditions.

Cyanide is a stronger-field ligand in this comparison. With a larger Δₒ, pairing in t₂g becomes worthwhile and low-spin d⁶ fills t₂g⁶e g⁰. All six d electrons are paired, so the spin-only moment is zero and the complex is diamagnetic. A nearly diamagnetic measurement supports this filling. The zero spin-only value does not imply no electron motion, no magnetic response at all or no chemical bonding.

Do not infer exact Δₒ or P from the two magnetic labels alone. Magnetism tells us which filling is populated under the measured conditions. The simplified model then says the relative energy balance lies on the corresponding side of the spin crossover, but precise gap values require spectroscopy or other methods. The measured moment may include a ligand/background diamagnetic correction; report data and conditions when making a real assignment.

Colour is complementary but not a direct spin counter. Different Δₒ values can shift absorption bands, and the observed colour is the light transmitted or reflected after absorption, not the absorbed wavelength itself. Charge-transfer bands and selection rules may contribute. Magnetic evidence is particularly strong here because high-spin d⁶ and low-spin d⁶ differ by four unpaired electrons, a large predicted contrast.

The example illustrates a general workflow. A ligand change can alter Δₒ while preserving metal oxidation state and d count. Other pairs may involve oxidation-state changes as well, so always calculate rather than assume. If geometry changes, its splitting pattern must also be redrawn before applying the moment table.

Step-by-step reasoning

Solve oxidation states for both complexes, obtaining Fe²⁺ d⁶. Confirm octahedral coordination. Draw t₂g⁴e g² and t₂g⁶ alternatives, count n=4 versus n=0, and calculate 4.90 versus 0 BM spin-only estimates. Compare with measured susceptibility or moment, then relate the assignment to weaker water versus stronger cyanide splitting.

Visual explanation

Draw two octahedral energy diagrams side by side. In the water diagram use a smaller Δₒ and put two electrons in e g; in the cyanide diagram use a larger Δₒ and fill all t₂g boxes as pairs. Add magnetic labels “four unpaired” and “zero unpaired” beneath them.

Real-world analogy

Six passengers can spread across a train with some paying for an upper deck, or share lower-deck seats if the upper-deck fare becomes very high. The passengers do not change identity; the price difference changes their arrangement. The iron remains Fe²⁺ while the ligand field changes filling.

Real-world example

Laboratory magnetic measurements on iron(II) salts and hexacyanoferrate(II) support the high-spin versus low-spin contrast. A high-spin aqua environment gives a sizable paramagnetic moment, while ferrocyanide is diamagnetic in the simple electronic picture.

Why?

Why does replacing water with cyanide not itself change the formal d count? Both complexes balance to Fe²⁺, despite different ligand charges and overall ion charges. Ligand identity changes the splitting energy; oxidation-state bookkeeping determines d⁶.

Common misconception

“The −4 charge of ferrocyanide means iron has four extra d electrons.” The −4 is the total complex charge after six −1 cyanide ligands and Fe²⁺ are combined. Iron remains d⁶, not d¹⁰.

Worked example

A six-coordinate Fe²⁺ complex has measured effective moment 4.8 BM. For high-spin d⁶, n=4 predicts 4.90 BM; low-spin d⁶, n=0, predicts zero. The value is close to high spin, so t₂g⁴e g² is supported. For a second Fe²⁺ complex with a diamagnetic response, t₂g⁶ is supported. The difference does not require a different metal oxidation state.

Quick check

1. What is the formal iron oxidation state in [Fe(CN)₆]⁴⁻? Answer: +2, because x+6(−1)=−4. 2. Which of the two named ions is low-spin d⁶? Answer: [Fe(CN)₆]⁴⁻ is low spin and t₂g⁶ in the standard description.

Exam focus

Show both oxidation-state equations and the two configurations. Quote moments as spin-only approximations, then use the large magnetic contrast as evidence for the spin assignment.

Advanced insight

Magnetic data establish populations of electronic states under particular conditions. In systems near a crossover, measured moments can vary with temperature as high- and low-spin populations change, so one room-temperature number need not describe every condition.

Summary

Both hexaaquairon(II) and hexacyanoferrate(II) are d⁶. Water supports a high-spin, four-unpaired state; cyanide supports a low-spin, paired state. Their contrasting moments test ligand-field predictions.

Practice questions

1. Calculate the spin-only moment of high-spin [Fe(H₂O)₆]²⁺. Answer: Fe²⁺ is d⁶, high-spin t₂g⁴e g² has n=4, so μ so=√24≈4.90 BM. 2. Why is [Fe(CN)₆]⁴⁻ diamagnetic in the elementary model? Answer: Fe²⁺ d⁶ is low spin with t₂g⁶e g⁰. Every d electron is paired, leaving n=0. 3. Can magnetic data alone give an exact numerical Δₒ for either complex? Answer: No. They support a spin-state filling; spectroscopy or more detailed analysis is needed to determine the splitting magnitude.