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Lesson 4 · 30 min

Magnetism: counting unpaired electrons with a balance

Iron(II) sulfate is pulled into a magnetic field; potassium ferrocyanide is not. Test Curie’s law on Curie’s own 1895 measurements of oxygen and ferrous sulfate, compute effective moments, and weigh the difference between high and low spin on a Gouy balance.

In this lesson

  • Explain where paramagnetism and diamagnetism come from, and describe the temperature dependence of paramagnetic susceptibility with Curie’s law χ = C/T
  • Compute the effective moment μ_eff = 2.828√(χ_M T) from a susceptibility, compare it with spin-only values and deduce the number of unpaired electrons
  • Understand how a Gouy balance measures susceptibility, and use magnetism to tell high-spin from low-spin complexes

Unpaired electrons are invisible, but you can weigh them. Hang a tube of iron(II) sulfate below a balance with its lower end between the poles of an electromagnet; switch on the current and the reading rises by tens of milligrams — the sample is pulled into the field. With potassium ferrocyanide the reading hardly moves. Both salts contain Fe²⁺; the difference is how the electrons are arranged: the first has 4 unpaired electrons, the second none.

Being drawn into a field is paramagnetism, and it comes from the magnetic moments of unpaired electrons. Every substance also has a weak diamagnetism that pushes it out of a field, from the orbital motion of paired electrons in the field. In his 1895 doctoral thesis Pierre Curie measured how the magnetism of many substances changes with temperature and found that paramagnetic susceptibility is inversely proportional to absolute temperature — Curie’s law.

Predict first

Oxygen is paramagnetic. Curie heated a bulb of oxygen from 20 °C to about 450 °C (the absolute temperature rises about 2.5-fold). What happens to its specific susceptibility?

Pick an answer and lock it in before you look. A wrong guess teaches more than no guess.

From the Curie constant to unpaired electrons

A field gives the moments μ a slight preference in orientation, of order μB/kT — weaker the hotter it is. Averaging over a mole of free moments gives the molar susceptibility χ_M = N_Aμ²/(3kT). So the Curie constant tells us each particle’s moment: μ_eff = √(3kC_M/N_A), which in Bohr magnetons μ_B is 2.828√(χ_M T).

If the moment comes from electron spin alone, n unpaired electrons give √(n(n+2)) μ_B (the spin-only formula). Compare a measured μ_eff with 1.73, 2.83, 3.87, 4.90 and 5.92 and you can count unpaired electrons. Oxygen gives about 2.9 μ_B: two unpaired electrons — exactly what molecular-orbital theory predicts and no Lewis structure can show.

χM=CMT,μeff=3k CMNAμB2=2.828χMT,μso=n(n+2)\chi_\mathrm{M} = \frac{C_\mathrm{M}}{T},\qquad \mu_\mathrm{eff} = \sqrt{\frac{3k\,C_\mathrm{M}}{N_\mathrm{A}\mu_\mathrm{B}^2}} = 2.828\sqrt{\chi_\mathrm{M}T},\qquad \mu_\mathrm{so} = \sqrt{n(n+2)}
χ_M in the cgs unit cm³ mol⁻¹ (the convention in the chemical literature), μ in Bohr magnetons μ_B.
χM\chi_\mathrm{M}
molar susceptibility
CMC_\mathrm{M}
molar Curie constant
nn
number of unpaired electrons
Original data from Curie’s 1895 thesis · cgs units

Curie’s law: susceptibility inversely proportional to temperature

250300350400T / K051015202530χ_g T / (10⁻³ cm³ K g⁻¹)

C = 24.16 × 10⁻³ cm³ K g⁻¹ × M = 151.91 g/mol → C_M = 3.669 cm³ K mol⁻¹; μ_eff = 2.828√C_M = 5.42 μ_B, which the spin-only formula would match with n = 4.5.

Fe²⁺ is d⁶, with 4 unpaired electrons when high spin (spin-only 4.90 μ_B). The measured value is larger because electrons in the t₂g orbitals keep part of their orbital moment.

Oxygen: Table I of Curie’s thesis (p. 52), hard-glass bulb at about 18 atm; the 249 °C row is printed “61,2”, but the table’s difference and t′ columns both correspond to 66.2, used here. Ferrous sulfate: p. 60, a 5.35 % solution, corrected for the diamagnetism of water (−0.79 × 10⁻⁶) and referred to the anhydrous salt. Curie used T = 273 + t. The fit is least squares through the origin, χ = C/T; the molecules’ own weak diamagnetism is not subtracted, which lowers C slightly. The earlier measurements on oxygen that Curie lists (10⁶χ from 87 to 125 at 20 °C) disagree widely: the uncertainty of the absolute value is much larger than the statistical scatter here.
Switch to “χT against T”: both data sets lie on horizontal lines. Ferrous sulfate gives about 5.4 μ_B — more than the 4.90 of four unpaired electrons, less than the 5.92 of five. Octahedral Fe²⁺ (d⁶) can have only 4 or 0 unpaired electrons; the excess comes from an orbital moment that is not completely quenched.

The Gouy balance

The most direct laboratory method is the Gouy balance: a long sample tube hangs with its lower end where the field is strongest and its upper end outside the field, so the force F = χ_g(m/L)B²/2 shows up as a change in the balance reading. A paramagnet is pulled down and “gains weight”; a diamagnet is pushed out and “loses weight”. Weigh Δm, compute χ_g, and multiply by the molar mass for χ_M. In practice the balance is calibrated with a standard of known susceptibility, such as Hg[Co(NCS)₄], and the diamagnetism of the tube and the ligands is subtracted.

0.500 g of sample filling a 5.00 cm tube · 20 °C · paramagnetism only

The Gouy balance: weighing unpaired electrons

The apparatus (sketch)
12.8406 gNS

Magnet off: the balance reads 12.8406 g. Switch the magnet on and watch the reading.

The force is F = χ_g (m/L) B²/2 (cgs), with the lower end of the sample in the field and the upper end outside; a paramagnet is drawn into the field and the balance reading rises. Iron(II) sulfate uses the Curie constant Curie measured for FeSO₄ (10⁶χT = 24 200, per anhydrous salt); the copper and manganese salts are assumed to have spin-only moments of 1 and 5 unpaired electrons — a model estimate, not a measurement; potassium ferrocyanide is set to zero. The diamagnetism of ligands and lattice water is ignored for all samples, as is the susceptibility of the air in the tube. In the sketch the tube’s drop is an exaggerated cue: a real balance barely moves and measures a force.
Switch on the magnet, read the gain, then change the sample. Double the field and the gain quadruples (F ∝ B²). In potassium ferrocyanide the strong-field ligand CN⁻ pushes Fe²⁺ to low-spin t₂g⁶, and the reading hardly moves.

Worked example

Counting electrons from Curie’s data

Curie measured the specific susceptibility of oxygen at 20.5 °C (T = 293.5 K) as χ_g = 114.8 × 10⁻⁶ cm³ g⁻¹. Find the effective moment of O₂ and the number of unpaired electrons.

Concept check

[Fe(H₂O)₆]²⁺ and [Fe(CN)₆]⁴⁻ both contain Fe²⁺ (d⁶), yet their magnetism is completely different. Why?

Try the calculation

A Mn²⁺ salt has a molar susceptibility of 1.47 × 10⁻² cm³ mol⁻¹ at 293 K. What is its effective moment (μ_B)?

μ_B

Key takeaways

  • Unpaired electrons give paramagnetism (attraction into a field); paired electrons give only weak diamagnetism. Curie’s law χ = C/T describes the paramagnetism of independent moments.
  • μ_eff = 2.828√(χ_M T); comparing with the spin-only √(n(n+2)) counts unpaired electrons: Curie’s data give about 2.9 μ_B for O₂ (two) and about 5.4 μ_B for FeSO₄ (four, plus an orbital contribution).
  • The Gouy balance turns susceptibility into weighing, F = χ_g(m/L)B²/2, and tells high-spin from low-spin complexes directly.
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Data and references

  1. P. Curie, Propriétés magnétiques des corps à diverses températures, thesis, Paris (1895), Tableau I (p. 52) and pp. 49–50, 60–61.
  2. L. Pauling, C. D. Coryell, “The magnetic properties and structure of hemoglobin, oxyhemoglobin and carbonmonoxyhemoglobin”, Proc. Natl. Acad. Sci. USA 22, 210–216 (1936).
  3. N. N. Greenwood, A. Earnshaw, Chemistry of the Elements, 2nd ed., Butterworth-Heinemann (1997), ch. 19 (magnetic properties of complexes).