Write the condensed electron configurations for the following atoms and indicate how many unpaired electrons each has: $(\mathbf{a}) \mathrm{Mg},(\mathbf{b}) \mathrm{Ge},(\mathbf{c}) \mathrm{Br},(\mathbf{d}) \mathrm{V},(\mathbf{e}) \mathrm{Y},(\mathbf{f}) \mathrm{Lu} .$

Using the periodic table, find the atomic numbers of the given elements: a) Mg: 12 b) Ge: 32 c) Br: 35 d) V: 23 e) Y: 39 f) Lu: 71

For each element, build the electron configuration following the Aufbau principle, Hund's rule, and Pauli Exclusion Principle. a) Mg: \(1s^2 \, 2s^2 \, 2p^6 \, 3s^2\) b) Ge: \(1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 4s^2 \, 3d^{10} \, 4p^2\) c) Br: \(1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 4s^2 \, 3d^{10} \, 4p^5\) d) V: \(1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 4s^2 \, 3d^3\) e) Y: \(1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 4s^2 \, 3d^{10} \, 4p^6 \, 5s^2 \, 4d^1\) f) Lu: \(1s^2 \, 2s^2 \, 2p^6 \, 3s^2 \, 3p^6 \, 4s^2 \, 3d^{10} \, 4p^6 \, 5s^2 \, 4d^{10} \, 5p^6 \, 6s^2 \, 4f^{14} \, 5d^1\)

Using the electron configurations from step 2, write the condensed electron configurations by indicating the nearest noble gas in brackets and the remaining configuration: a) Mg: \([\text{Ne}] \, 3s^2\) b) Ge: \([\text{Ar}] \, 4s^2 \, 3d^{10} \, 4p^2\) c) Br: \([\text{Ar}] \, 4s^2 \, 3d^{10} \, 4p^5\) d) V: \([\text{Ar}] \, 4s^2 \, 3d^3\) e) Y: \([\text{Kr}] \, 5s^2 \, 4d^1\) f) Lu: \([\text{Xe}] \, 6s^2 \, 4f^{14} \, 5d^1\)

Now, examine the electron configurations and count the unpaired electrons: a) Mg: 0 unpaired electrons (fully filled 3s orbital) b) Ge: 0 unpaired electrons (fully filled 4s and 3d orbitals, two paired electrons in 4p orbital) c) Br: 1 unpaired electron (5 electrons in 4p orbital) d) V: 3 unpaired electrons (3 electrons in 3d orbital) e) Y: 1 unpaired electron (1 electron in 4d orbital) f) Lu: 1 unpaired electron (1 electron in 5d orbital) So, the condensed electron configurations and the number of unpaired electrons for each element are: a) Mg: \([\text{Ne}] \, 3s^2\) (0 unpaired electrons) b) Ge: \([\text{Ar}] \, 4s^2 \, 3d^{10} \, 4p^2\) (0 unpaired electrons) c) Br: \([\text{Ar}] \, 4s^2 \, 3d^{10} \, 4p^5\) (1 unpaired electron) d) V: \([\text{Ar}] \, 4s^2 \, 3d^3\) (3 unpaired electrons) e) Y: \([\text{Kr}] \, 5s^2 \, 4d^1\) (1 unpaired electron) f) Lu: \([\text{Xe}] \, 6s^2 \, 4f^{14} \, 5d^1\) (1 unpaired electron)