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A $3d^9$ shell has one hole in the otherwise filled $d$ shell. Hund's rules give $S=1/2$, $L=2$, and, for a more-than-half-filled shell, $J=L+S=5/2$: the free-ion term is ${}^2{D}_{5/2}$. Its Landé factor isDistinguish the Curie effective moment from a saturated projection:The question then explicitly asks us to use orbital quenching in the crystal. Ligand fields split and mix orbital states, suppressing their freely orientable angular momentum. The spin-only approximation has $g\simeq 2$, $S=1/2$, so ${\mu }_{\mathrm{eff}}=\sqrt{3}{\mu }_B$ but the two field-axis projections are $\pm {\mu }_B$. The latter, not $\sqrt{3}{\mu }_B$, belong in the tanh argument and saturation magnetization.

The sulfate and water environment supplies mainly closed-shell, weakly magnetic surroundings, stabilizes localized Cu2+ sites and separates their moments so that exchange can be weak on the temperature scale considered. Paramagnetism here is an approximation of weakly interacting local spins, not a claim that dilution guarantees no magnetic ordering at any temperature. Elemental copper has a filled $3d$ band and itinerant conduction electrons rather than one localized Cu2+ spin per atom. Pauli spin response and orbital/core contributions then replace a local-spin Curie law; ordinary pure copper has a weak net diamagnetic response.^[ref]

There is one Cu2+ per formula unit, hence $n_s=N_A/V_m=5.5148\times 10^{27}$ m${}^{-3}$. The spin-only weak-field SI volume susceptibility isThese neglect exchange, anisotropic $g$ factors and background diamagnetism. At high field and low temperature, saturation requires the nonlinear response in a; a field-independent Curie value then describes only the zero-field slope.

For a free transition-metal ion, degeneracy among different orbital states can support an orbital contribution to its magnetic moment. In a crystal, the electric fields and bonding of the surrounding ligands split those states. If the orbital ground state becomes nondegenerate and is separated from excited orbital states, its wavefunction can usually be chosen real in zero field. Since the orbital angular momentum operator is imaginary in a real-orbital basis, its expectation value then vanishes.

This suppression is orbital quenching. The remaining leading magnetic moment is the spin contribution, approximately $g{\mu }_B\mathbf{S}$ with $g\simeq 2$. Spin-orbit coupling admixes excited orbital states and produces residual anisotropy and shifts in the g-factor; quenching does not mean that every orbital matrix element or every higher-order orbital response vanishes. A degenerate ground orbital multiplet need not be quenched until a further distortion removes its degeneracy.