Question

The magnetic moment of a transition metal ion is \[\sqrt {15} \] BM. Therefore, the number of unpaired electrons present in it is:
(A) 4
(B) 1
(C) 2
(D) 3

Answer 1

Hint: Magnetic moment is defined as the magnetic strength and orientation of a magnet or any other substance that produces a magnetic field. The magnetic dipole moment is defined in terms of the torque that the object experiences in a magnetic field.

Formula used: Magnetic moment of the transition metals is calculated using the formula –
\[\mu {\text{ }} = {\text{ }}\sqrt {n(n + 2)} \]
where,
\[\mu {\text{ }} = {\text{ }}\] magnetic moment
\[n{\text{ }} = \] number of unpaired electrons

Complete step by step answer:
Transition metals have valence electrons that are present in the \[(n{\text{ }} - {\text{ }}1)\]and \[{\text{ns}}\] orbitals. These orbitals have very little energy gap between them.
Given:
Magnetic moment, \[{\text{\mu = }}\]\[\sqrt {15} \] BM
To find: The number of unpaired electrons, \[{\text{n}}\]
\[\mu {\text{ }} = {\text{ }}\sqrt {n(n + 2)} \]
Substituting the value of \[{\text{\mu }}\]in the above equation,
\[\sqrt {15} {\text{ }} = {\text{ }}\sqrt {n{\text{ }}(n{\text{ }} + {\text{ }}2)} \]
Squaring both sides,
\[15{\text{ }} = {\text{ }}n(n{\text{ }} + {\text{ }}2)\]
\[ \Rightarrow {\text{ }}15{\text{ }} = {\text{ }}{n^2}{\text{ }} + {\text{ }}2n\]
\[ \Rightarrow {\text{ }}{n^2}{\text{ }} + 2n{\text{ }} - {\text{ }}15{\text{ }} = {\text{ }}0\]
\[ \Rightarrow n(n{\text{ }} - {\text{ }}3){\text{ }} + {\text{ }}5(n{\text{ }} - {\text{ }}3){\text{ }} = {\text{ }}0\]
\[ \Rightarrow (n{\text{ }} - {\text{ }}3){\text{ }}(n{\text{ }} + {\text{ }}5){\text{ }} = {\text{ }}0\]
\[ \Rightarrow n{\text{ }} - {\text{ }}3{\text{ }} = {\text{ }}0{\text{ }}or{\text{ }}n{\text{ }} + {\text{ }}5{\text{ }} = {\text{ }}0\]
\[ \Rightarrow n{\text{ }} = {\text{ }}3{\text{ }}or{\text{ }}n{\text{ }} = {\text{ }} - {\text{ }}5\]
Since, the number of unpaired electrons cannot be negative, \[ - {\text{ }}5\] is discarded.
Therefore, the number of unpaired electrons is 3.

Hence, the correct answer is (D) i.e., 3

Note: The direction of the moment is from the south pole to the north pole. In nonmagnetic materials the electron moments cancel since there is random ordering to the direction of the electron spins.