To determine the number of unpaired electrons in the $Mn^{2+}$ ion, let's first consider the electron configuration of a neutral manganese (Mn) atom.
Manganese (Mn) has an atomic number of 25, so the electron configuration of neutral Mn is:
$[Ar] 4s^2 3d^5$
When manganese forms a $Mn^{2+}$ ion, it loses two electrons. These electrons are removed first from the 4s orbital because it is higher in energy than the 3d orbital.
So, the electron configuration of $Mn^{2+}$ is:
$[Ar] 3d^5$
The 3d subshell has 5 electrons in the case of $Mn^{2+}$, and since the 3d orbitals can hold a maximum of 10 electrons, the 5 electrons will occupy the 5 available d orbitals with each orbital having 1 electron (this is the maximum number of unpaired electrons).
Therefore, the number of unpaired electrons in $Mn^{2+}$ is:
Answer: 5
A beam of light of wavelength \(\lambda\) falls on a metal having work function \(\phi\) placed in a magnetic field \(B\). The most energetic electrons, perpendicular to the field, are bent in circular arcs of radius \(R\). If the experiment is performed for different values of \(\lambda\), then the \(B^2 \, \text{vs} \, \frac{1}{\lambda}\) graph will look like (keeping all other quantities constant).