Magnetic Properties & Spin-Only Magnetic Moment Formula

Formula:

$$\mu_s = \sqrt{n(n + 2)} \quad \text{BM}$$

$$\text{Unpaired Electrons } (n) \text{ vs. } \mu_s \text{ Matrix:}$$

$$\begin{aligned} n=1 &\implies \mu_s = \sqrt{3} \approx 1.73 \text{ BM} \\ n=2 &\implies \mu_s = \sqrt{8} \approx 2.84 \text{ BM} \\ n=3 &\implies \mu_s = \sqrt{15} \approx 3.87 \text{ BM} \\ n=4 &\implies \mu_s = \sqrt{24} \approx 4.90 \text{ BM} \\ n=5 &\implies \mu_s = \sqrt{35} \approx 5.92 \text{ BM} \end{aligned}$$

Concept:

Paramagnetism arises from the presence of unpaired electrons in d-orbitals. The spin-only magnetic moment increases with the number of unpaired electrons (n), reaching a peak at $$d^5$$ ($$\text{Mn}^{2+}, \text{Fe}^{3+}$$) across the 3d series.

Notation:

• $$\mu_s$$ = Spin-only magnetic moment
• $$\text{BM}$$ = Bohr Magneton ($$\frac{e h}{4\pi m_e}$$)
• n = Number of unpaired d-electrons