Calculating Spin-Only Moments

Introduction

The spin-only formula becomes most useful when its common values are familiar. For one through five unpaired electrons, the predicted moments are about 1.73, 2.83, 3.87, 4.90 and 5.92 BM. A measurement near one of these values can support a proposed d-electron filling, but the table is an aid to reasoning rather than a substitute for oxidation-state and geometry analysis.

Core explanation

Calculate each value from μ so=√[n(n+2)] BM. At n=1, the product is 1×3=3, so μ≈1.73 BM. At n=2, 2×4=8, so μ≈2.83 BM. At n=3, 3×5=15, giving 3.87 BM. At n=4, 4×6=24, giving 4.90 BM. At n=5, 5×7=35, giving 5.92 BM. For n=0 the spin-only result is zero. The sequence rises with n, but not by exactly one BM at each step.

The table does not by itself identify a metal. Many distinct formal d counts can have the same number of unpaired electrons. High-spin octahedral d⁴ and high-spin d⁶ each have n=4, so both have the same spin-only estimate. Octahedral d³ and high-spin d⁷ each have n=3. A moment near 3.87 BM therefore narrows a set of electronic configurations but cannot uniquely give oxidation state or chemical formula.

Geometry can also change n for a fixed metal ion. Four-coordinate Ni²⁺ is d⁸; tetrahedral e⁴t₂⁴ has two unpaired electrons and a 2.83 BM spin-only estimate. Square-planar d⁸ often has all eight electrons paired and an estimate near zero. Before interpreting the magnetic result, identify likely geometry from coordination number, ligand type and structure where possible.

If an exercise gives a measured moment, choose the nearest plausible integer n only after considering experimental and model uncertainty. A value of 4.7 BM might support a four-unpaired arrangement, but one should not solve n(n+2)=4.7² and announce n=3.82 electrons as a physical occupancy. Deviations arise from orbital contributions, spin–orbit coupling, magnetic exchange, temperature or imperfect sample correction. The inversion formula can be used as a mathematical guide: n≈−1+√(1+μ²) for μ in BM, but the final electronic count must be an integer supported by chemistry.

Measured moments may exceed spin-only predictions, especially for some Co²⁺ complexes where orbital angular momentum contributes. A small observed moment can also result from antiferromagnetic coupling between centres rather than each centre having zero unpaired electrons. A table comparison is strongest for isolated mononuclear complexes at a stated temperature and with known oxidation state.

The numerical values are independent of Δₒ once n is fixed. Changing a ligand may change Δₒ enough to switch spin state, causing a jump between table entries. If it does not change n, the spin-only formula gives the same value, even though colour or bond lengths may change. This distinction keeps an energy gap from being confused with a magnetic unit.

Step-by-step reasoning

Compute oxidation state and d count, draw a geometry-specific diagram and count n. Look up or calculate √[n(n+2)] BM. If a measured value is supplied, compare it with plausible table entries, then test candidates against d count, charge and geometry. Report any material deviation from spin-only behaviour as a limit of the model.

Visual explanation

Draw a staircase with steps labelled n=0 through n=5. Write moments 0, 1.73, 2.83, 3.87, 4.90 and 5.92 BM beside the steps. On a separate row show that d⁴ and d⁶ can both point to the n=4 step, illustrating why the table is not a unique d-count decoder.

Real-world analogy

A shoe size can narrow which person a footprint belongs to, but many people share that size. Magnetic moment similarly narrows the unpaired count, while oxidation state, ligands and geometry provide the remaining identification clues.

Real-world example

The approximately five-unpaired spin pattern of high-spin Mn²⁺ often gives a moment near the 5.92 BM spin-only value in suitable mononuclear complexes. The actual measured value and temperature should still be reported, rather than rounding every Mn(II) sample to the table entry.

Why?

Why can one moment correspond to more than one d count? The formula uses only n, the number of unpaired electrons, not the total number of d electrons. Different orbital fillings can have the same number of unpaired arrows.

Common misconception

“A measured 3.9 BM proves d³.” It supports approximately three unpaired electrons in a spin-only interpretation, but high-spin d⁷ and other configurations can also have three unpaired electrons.

Worked example

A known octahedral Fe²⁺ complex has a reported moment of 5.0 BM. Fe²⁺ is d⁶. High-spin d⁶ has n=4 and predicts √24=4.90 BM; low-spin d⁶ has n=0 and predicts zero. The measured value is close to the high-spin estimate, so high spin is supported. The 0.1 BM difference is not grounds for changing the integer unpaired count.

Quick check

1. What moment corresponds to five unpaired electrons in the spin-only model? Answer: √35≈5.92 BM. 2. Which two high-spin octahedral d counts share the n=4 table entry? Answer: d⁴ and d⁶.

Exam focus

Show the calculation rather than quoting a memorised number alone. Use an integer electron count justified by a configuration, and mention that measured effective moments can differ from spin-only values.

Advanced insight

The inverse relation n≈−1+√(1+μ²) is algebraically convenient but chemically underdetermined. It does not identify oxidation state, geometry or magnetic coupling; those require independent experimental evidence and a plausible electronic structure.

Summary

Spin-only moments for n=1–5 are approximately 1.73, 2.83, 3.87, 4.90 and 5.92 BM. They help test unpaired counts, while multiple d configurations and non-spin contributions limit unique identification.

Practice questions

1. Calculate the spin-only moment for n=3. Answer: μ=√[3(3+2)]=√15≈3.87 BM. 2. A measured moment near 2.8 BM is consistent with how many unpaired electrons? Answer: About two in a spin-only mononuclear model, because n=2 predicts √8≈2.83 BM. Oxidation state and geometry remain to be checked. 3. Can the spin-only table distinguish high-spin d⁴ from high-spin d⁶ by itself? Answer: No. Both have four unpaired electrons and predict about 4.90 BM; another chemical or spectroscopic observation is needed.