Curie Constant Calculator (C = χT, effective magnetic moment and unpaired electrons)

Understanding the Curie constant calculator — concept and principle

In a paramagnetic material, magnetic susceptibility falls as temperature rises. The Curie constant captures that behaviour and links it to the effective magnetic moment of the ions. This calculator finds the Curie constant from a measured susceptibility and estimates the number of unpaired electrons.

C = 0.0146 × 300 = 4.38; μeff = 2.828√C = 5.92 μB, the spin-only value for five unpaired electrons.

How it is calculated

Curie law: χ = C ÷ T, so C = χT.
Curie–Weiss law: χ = C ÷ (T − θ), so C = χ(T − θ).
Effective moment: μeff ≈ 2.828 √(χm T) Bohr magnetons, with χm in CGS units (cm³/mol).
Spin-only formula: μ = √(n(n + 2)) μB, used to estimate the number of unpaired electrons n.
SI molar susceptibility = 4π × 10⁻⁶ × CGS value.

The principle

Unpaired electrons behave like tiny magnets. A magnetic field tries to align them, while thermal motion scrambles them. The higher the temperature, the weaker the net alignment, so susceptibility falls as 1 ÷ T. Langevin's theory relates the Curie constant to the square of the magnetic moment, so measuring χ at a known temperature reveals the moment, and the spin-only formula translates it into a number of unpaired electrons.

Worked example

Default inputs: χm = 0.0146 cm³/mol, T = 300 K, θ = 0.

C = 0.0146 × 300 = 4.38 cm³·K/mol.
μeff = 2.828 × √4.38 ≈ 5.918 μB.
The spin-only value closest to this is n = 5: √(5 × 7) ≈ 5.916 μB, so 5 unpaired electrons.
SI molar susceptibility = 4π × 10⁻⁶ × 0.0146 ≈ 1.835 × 10⁻⁷ m³/mol.

Each point is the calculator re-run with only “Molar magnetic susceptibility χₘ (cm³/mol, CGS)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Molar magnetic susceptibility χₘ (cm³/mol, CGS)	Curie constant C (cm³·K/mol)
0.00365	1.095
0.0073	2.19
0.0109	3.285
0.0146	4.38
0.0182	5.475
0.0219	6.57
0.0292	8.76

How to read the result

Five unpaired electrons is typical of high-spin d⁵ ions such as Mn²⁺ and Fe³⁺. When measured moments exceed the spin-only value, orbital contributions are present (common for Co²⁺ and many lanthanides). A non-zero θ indicates interactions between neighbouring moments: negative for antiferromagnetic tendencies, positive for ferromagnetic.

Common mistakes

Mixing CGS and SI susceptibility units.
Using mass or volume susceptibility instead of molar susceptibility.
Forgetting diamagnetic corrections from the ligands and core electrons.
Applying the Curie law near a magnetic ordering temperature.

Keep in mind: Assumes isolated paramagnetic ions. Precise magnetochemistry requires diamagnetic corrections and temperature-dependent measurements.

Frequently asked questions

What is the Curie constant?

The constant C in χ = C ÷ T, proportional to the square of the magnetic moment.

How many unpaired electrons give 5.92 μB?

Five, by the spin-only formula √(n(n + 2)).

What is the difference between the Curie and Curie–Weiss laws?

Curie–Weiss adds θ to account for interactions between neighbouring magnetic moments.

Sources: Kittel, C., Introduction to Solid State Physics, 8th ed. Wiley (paramagnetism) · Housecroft, C. E. & Sharpe, A. G., Inorganic Chemistry, 5th ed. Pearson (magnetic moments, spin-only formula)

Checked 9 Nov 2026. LibreTexts, “Map: Inorganic Chemistry (Housecroft)”, §20.10C “The Effects of Temperature on Magnetic Moment” (Curie law χ = C/T, Curie–Weiss law, μ = 2.828√(χ_mT) B.M. from χ_m = Nμ²/3kT); spin-only formula μ = √(n(n+2)) μ_B and its values: Wikipedia, “Magnetochemistry” (secondary). k_B, N_A exact (SI 2019); μ_B: standard value.

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