Magnetic Susceptibility: Gouy Method and Effective Moment

By Aniket Bhardwaj · 10 November 2026 · Inorganic

Magnetic susceptibility measurements provide direct experimental evidence for the electronic structure of coordination compounds. In competitive examinations like CSIR NET, GATE, and IIT-JAM, numerical problems regularly test how to find gram susceptibility from Gouy balance data, apply Pascal diamagnetic corrections, and determine the effective magnetic moment (μ_eff). These calculations connect laboratory balance readings directly to oxidation states, electron configurations, and ligand field splitting.

Magnetic Susceptibility: Definitions and Unit Systems

When placed in an external magnetic field H, a material acquires an induced magnetization I. The volume magnetic susceptibility is κ = I / H. In CGS-emu, κ has units of emu·cm⁻³; in SI, κ is dimensionless.

Chemists express susceptibility per gram or per mole:

Gram susceptibility (χ_g): χ_g = κ / ρ, where ρ is density. Units are cm³·g⁻¹ (CGS) or m³·kg⁻¹ (SI).
Molar susceptibility (χ_M): χ_M = χ_g × M, where M is molar mass. Units are cm³·mol⁻¹ (CGS) or m³·mol⁻¹ (SI).

Textbooks differ between CGS and SI conventions: χ_M (SI) = 4π × 10⁻⁶ × χ_M (CGS). In coordination chemistry, CGS units (cm³·mol⁻¹) are standard.

Key Susceptibility and Moment Formulas:
χ_M = χ_g × M (units: cm³·mol⁻¹)
χ_M^corr = χ_M^obs − χ_dia = χ_M^obs + |χ_dia|
μ_eff = 2.828 √(χ_M^corr × T) B.M. (CGS, with T in kelvin)
μ_eff = √[3kTχ_M / (μ₀N_Aβ²)] ≈ 797.7 √(χ_M^corr × T) B.M. (SI)

Principle of the Gouy Balance and Calibration

In the Gouy method, a uniform cylinder of sample of length l and cross-sectional area A hangs vertically between electromagnet poles. The bottom experiences a strong magnetic field H, while the top remains in zero field. Paramagnetic samples are drawn into the field, showing an apparent mass increase (Δm > 0). Diamagnetic samples are repelled, showing an apparent mass loss (Δm < 0).

The magnetic force is F = Δm × g = (1/2) × κ × A × H². Tubes are calibrated with a known standard, typically mercury tetrathiocyanatocobaltate(II), Hg[Co(NCS)₄], with χ_g,std = 16.44 × 10⁻⁶ cm³·g⁻¹ at 293 K.

For an unknown sample packed to the same height as the calibrant:

χ_g,unk = χ_g,std × (Δm_unk / Δm_std) × (m_std / m_unk)

Pascal's Constants and Diamagnetic Corrections

Every atom has closed electron shells producing an underlying diamagnetic susceptibility (χ_dia), which is negative and temperature-independent. The observed molar susceptibility is χ_M^obs = χ_M^para + χ_dia.

To obtain the true paramagnetic susceptibility χ_M^corr of the metal centre, we add the diamagnetic contribution calculated from empirical Pascal's constants: χ_M^corr = χ_M^obs − χ_dia = χ_M^obs + |χ_dia|.

Curie-Weiss Law and Effective Magnetic Moment

Isolated paramagnetic ions obey Curie's law, χ_M^corr = C / T. With magnetic exchange coupling between centres, compounds obey the Curie-Weiss law: χ_M^corr = C / (T − θ), where C is the Curie constant and θ is the Weiss constant. Positive θ denotes ferromagnetic coupling, while negative θ denotes antiferromagnetic coupling. When written as (T + θ), positive θ denotes antiferromagnetism.

The effective magnetic moment μ_eff is compared with the spin-only value μ_s = √[n(n + 2)] B.M. For A and E ground terms (such as octahedral d³), orbital angular momentum is quenched, so μ_eff ≈ μ_s. For T ground terms (such as octahedral d⁶), orbital contribution is unquenched, making μ_eff exceed μ_s.

Exam details

Source: practice problem written for this article (not a past-paper question)
Relevant exams: CSIR NET, GATE Chemistry, IIT-JAM, CUET-PG
Topic: Magnetic Susceptibility, Gouy Method and Effective Moment
Difficulty: Advanced

Question: A Gouy tube is calibrated at 293 K with solid Hg[Co(NCS)₄] (χ_g,std = 16.44 × 10⁻⁶ cm³·g⁻¹). A mass of 1.500 g of standard gives a mass gain of Δm_std = +24.66 mg. The same tube is packed to identical height with 1.250 g of a nickel(II) complex (molar mass = 288.0 g·mol⁻¹), registering a mass gain of Δm_unk = +20.00 mg. Calculate: (a) χ_g,unk and (b) χ_M^obs.

Step-by-step solution:

Step 1: Write the Gouy calibration equation for equal tube height:

χ_g,unk = χ_g,std × (Δm_unk / Δm_std) × (m_std / m_unk)

Step 2: Evaluate the mass change ratio and sample mass ratio:

Δm_unk / Δm_std = 20.00 mg / 24.66 mg = 0.81103

m_std / m_unk = 1.500 g / 1.250 g = 1.200

Step 3: Multiply the standard susceptibility by the ratios:

χ_g,unk = 16.44 × 10⁻⁶ × 0.81103 × 1.200 = 1.600 × 10⁻⁵ cm³·g⁻¹

Step 4: Verify with the tube constant K (Route 2):

K = (χ_g,std × m_std) / Δm_std = (16.44 × 10⁻⁶ × 1.500) / 0.02466 = 1.000 × 10⁻³ cm³

χ_g,unk = (K × Δm_unk) / m_unk = (1.000 × 10⁻³ × 0.02000) / 1.250 = 1.600 × 10⁻⁵ cm³·g⁻¹

Step 5: Calculate observed molar magnetic susceptibility:

χ_M^obs = χ_g,unk × M = 1.600 × 10⁻⁵ × 288.0 = 4.608 × 10⁻³ cm³·mol⁻¹

Answer: χ_g,unk = 1.600 × 10⁻⁵ cm³·g⁻¹; χ_M^obs = 4.608 × 10⁻³ cm³·mol⁻¹

Question: An octahedral chromium(III) complex gives an observed molar susceptibility χ_M^obs = 6.040 × 10⁻³ cm³·mol⁻¹ at 300.0 K. The Pascal diamagnetic correction is χ_dia = −2.10 × 10⁻⁴ cm³·mol⁻¹. Determine: (a) χ_M^corr, (b) μ_eff in Bohr Magnetons, and (c) the number of unpaired electrons by comparing with the spin-only formula.

Step-by-step solution:

Step 1: Write χ_dia in common scientific notation:

χ_dia = −0.210 × 10⁻³ cm³·mol⁻¹

Step 2: Calculate corrected molar susceptibility:

χ_M^corr = χ_M^obs − χ_dia = 6.040 × 10⁻³ − (−0.210 × 10⁻³) = 6.250 × 10⁻³ cm³·mol⁻¹

Step 3: Calculate product χ_M^corr × T:

χ_M^corr × T = 6.250 × 10⁻³ × 300.0 = 1.875 cm³·K·mol⁻¹

Step 4: Compute μ_eff using √(8 × χ_M^corr × T):

μ_eff = √(8 × 1.875) = √(15.00) = 3.873 B.M.

Step 5: Verify using the 2.828 factor (Route 2):

μ_eff = 2.8284 × √(1.875) = 2.8284 × 1.3693 = 3.873 B.M.

Step 6: Compare with spin-only formula μ_s = √[n(n + 2)]:

For Cr(III), a d³ ion, n = 3 unpaired electrons. μ_s = √[3(5)] = 3.873 B.M. The experimental μ_eff equals μ_s, confirming complete quenching of orbital moment in the ⁴A_2g ground state.

Answer: χ_M^corr = 6.250 × 10⁻³ cm³·mol⁻¹; μ_eff = 3.87 B.M. (matches n = 3 unpaired electrons)

Common Student Mistakes in Magnetic Susceptibility Calculations

Subtracting Instead of Adding Diamagnetic Corrections: Because χ_dia is negative, χ_M^corr is larger than χ_M^obs (χ_M^corr = χ_M^obs − χ_dia = χ_M^obs + |χ_dia|). Subtracting the magnitude lowers paramagnetic susceptibility.
Confusing CGS and SI Units: The relation μ_eff = 2.828 √(χ_M × T) requires CGS units (cm³·mol⁻¹). Using SI values directly produces errors by several orders of magnitude.
Neglecting Uneven Sample Packing: The Gouy method requires uniform tube packing. Uneven packing produces erratic balance readings and incorrect mass changes.
Assuming μ_eff Always Equals μ_s: In complexes with T ground terms, unquenched orbital angular momentum increases μ_eff substantially above √[n(n + 2)].

Competitive Examination	Typical Question Format	Key Concepts Tested
CSIR NET (Chemical Sciences)	Multi-step numerical and conceptual problems	Gouy calibration, Pascal corrections, Curie-Weiss plots, orbital quenching
GATE (Chemistry)	Numerical Answer Type (NAT) and multiple-choice questions	Calculating μ_eff from χ_M, finding unpaired electrons, temperature dependence
IIT-JAM (Chemistry)	Objective questions on transition metal complexes	Spin-only versus effective moment, converting χ_g to χ_M, electronic ground terms
CUET-PG	Formula-based and conceptual multiple-choice questions	Curie law trends, Bohr Magneton units, principles of Gouy balance

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