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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.

The Curie constant measures the squared magnetic moment averaged over all orientations of the $J$ multiplet, givingFor the usual positive Landé g-factor of the ion here, the absolute-value sign can be omitted. This effective paramagnetic moment differs from the maximum projected or saturation magnetic moment, ${\mu }_{\mathrm{sat}}=|\tilde{g}|J{\mu }_B$.

In weak magnetic field and the regime of paramagnetism, localized magnetic moments with total angular momentum $J$ and Landé g-factor $\tilde{g}$ have SI volume magnetic susceptibilitywithin the Curie–Weiss law. Here $n_m$ counts magnetic ions per volume, ${\mu }_0$ is the vacuum permeability, ${\mu }_B$ is the Bohr magneton, and $\Theta$ is the Weiss temperature. The factor $J(J+1)/3$ is the zero-field average of the squared angular projection over $m_J=-J,\dots ,J$.

For one magnetic ion per formula unit, a molar volume $V_m$ contains $N_A$ magnetic ions, where $N_A$ is the Avogadro constant. Consequently $n_m{V}_m=N_A$, and the molar magnetic susceptibility isDefining the dimensionless effective paramagnetic moment $m_{\mathrm{eff}}$ by ${\mu }_{\mathrm{eff}}=m_{\mathrm{eff}}{\mu }_B$, numerical substitution givesThe normalization is per mole of formula units, which here also means per mole of magnetic ions.

Write the fitted inverse molar magnetic susceptibility asThe vertical scale is in $10^6\mathrm{mol}{\mathrm{m}}^{-3}$, not unscaled inverse SI magnetic susceptibility. The source line is approximately $8.8\times 10^6\mathrm{mol}{\mathrm{m}}^{-3}$ at $100\mathrm{K}$ and $19\times 10^6\mathrm{mol}{\mathrm{m}}^{-3}$ at $300\mathrm{K}$. These readings giveTherefore the effective paramagnetic moment and Weiss temperature areEquivalently, ${\mu }_{\mathrm{eff}}\simeq (3.5\text{–}3.6){\mu }_B$. An uncertainty of order $10\mathrm{K}$ in the graphical Weiss temperature is reasonable; the plotted fit does not justify exact integer parameters. Its positive intercept at $T=0$ requires a negative extrapolated temperature-axis intercept. Thus the dominant interactions have the antiferromagnetic sign in the interpretation of part (b).

With $J=4$, the measured effective paramagnetic moment gives the Landé g-factorFor the free-ion Hund's rules values $S=1$, $L=5$, $J=4$, the stated formula yieldsThusThe source fit is consistent with the unquenched $4f$ orbital magnetic moment of the free praseodymium ion, within graphical accuracy. Small deviations need not indicate a change of electron configuration: crystal field splitting, interactions and the fit range can affect a measured magnetic susceptibility.

A chemical element of atomic number 59. Its trivalent ion has a $4f^2$ open-shell electron configuration. In LS coupling, Hund's rules select $L=5$, $S=1$, $J=4$, the ${}^3{H}_4$ term symbol. The corresponding free-ion Landé g-factor is $4/5$ and its effective paramagnetic moment is $(4/5)\sqrt{20}{\mu }_B\simeq 3.58{\mu }_B$.