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Chemistry Question on The Lanthanoids

The magnetic moment of lanthanide ions is determined from which one of the following relation?

A

μ  = n(n+2)\mu \, \, = \, \sqrt{n(n+2)}

B

μ =gJ(J+1)\mu \, = g \sqrt{J(J+1)}

C

μ=gn(n+1)\mu=g\sqrt{n(n+1)}

D

μ=2n(n+1)\mu = 2 \sqrt{n(n+1)}

Answer

μ =gJ(J+1)\mu \, = g \sqrt{J(J+1)}

Explanation

Solution

In case of lanthanoids, 4f orbitals lie too deep and hence the magnetic effect of the motion of the electron in its orbital is not quenched out. Here spin contribution S and orbital contribution L couple together to give a new quantum number J. Thus magnetic moment of lanthanoids is given by, μ =gJ(J+1)\mu \, = g \sqrt{J(J+1)} where J = L - S when the shell is less than half fill J = L + S when the shell is more than half fill and g=112+S(S+1)−L(L+1)2J(J+1)g=1\frac{1}{2}+ \frac{S(S+1)-L(L+1)}{2J(J+1)}