GATE 2021 PH – Question 62
An atomic gas at 300 K has number density $n=10^{20}$ m⁻³ and ground-state valence configuration $f^7$. Its paramagnetic susceptibility is $\chi=m\times10^{-11}$. Find $m$, rounded to two decimal places. Given $\mu_0=4\pi\times10^{-7}$ H m⁻¹, $\mu_B=9.274\times10^{-24}$ A m² and $k_B=1.3807\times10^{-23}$ J K⁻¹.
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Correct answer: 5.40 to 5.50
Explanation
Hund’s rules for the half-filled f shell give $S=7/2,L=0,J=7/2$, and Landé factor $g_J=2$. Hence $\mu_{\rm eff}^2=g_J^2J(J+1)\mu_B^2=63\mu_B^2$. Curie’s law gives $\chi=\mu_0n\mu_{\rm eff}^2/(3k_BT)$. Inserting the stated constants yields $\chi=5.4796\times10^{-11}$, so $m=5.48$.
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