Question

What is the correct set of quantum numbers $ ( $ n, l, $ {m_l} $ , $ {m_s}) $ for the highest energy electron in the ground state of tin, Sn $ ? $

Answer 1

Hint : Tin $ ( $ Sn $ ) $ belongs to the carbon family with atomic number $ 50 $ . It belongs to the group $ 14 $ and period $ 5 $ . The quantum numbers n, l, $ {m_l} $ and $ {m_s} $ gives us the information about the spin and location of the electrons in Sn. Tin is a p block element and therefore the outermost electrons will be present in the p orbital and they possess the highest energy.

Complete Step By Step Answer:
To find out the set of quantum numbers, we must know the electronic configuration of Sn.
Sn $ = [Kr]4{d^{10}}5{s^2}5{p^2} $
The two electrons in p orbital have the highest energy and we must find its quantum numbers.
The principal quantum number ,n, describes the energy level or the electron shell of the atom. Here, n $ = 5 $ (for the highest energy electrons).
The azimuthal quantum number, l, also known as the orbital or angular quantum number describes the subshell of the electron. It has values from $ 0 \to (n - 1) $ . Here, n $ = 5 $ and therefore l can take values $ 0,1,2,3,4 $ .
l $ = 0 $ denotes s subshell
l $ = 1 $ denotes p subshell
l $ = 2 $ denotes d subshell
l $ = 3 $ denotes f subshell
l $ = 4 $ denotes g subshell
As the electrons lie in the p subshell, l $ = 1 $ .
The magnetic quantum number $ {m_l} $ describes the projection of orbital angular momentum along an axis. It gives an idea about where exactly (in which orbital) you can find an electron. Its values range from $ - l \to + l $ . Here, l $ = 1 $ . Therefore, $ {m_l} = - 1,0, + 1 $ . These are the three p orbitals.
 $ {m_l} = - 1 = 5{p_x} $
 $ {m_l} = 0 = 5{p_y} $
 $ {m_l} = + 1 = 5{p_z} $
According to Hund's rule, pairing of electrons occurs only after all the orbitals are singly occupied. Also, the two electrons must occupy two distinct orbitals. Therefore filling occurs initially at $ {m_l} = - 1 $ and $ {m_l} = 0 $ at $ 5{p_x} $ and $ 5{p_y} $ orbitals respectively.
The spin quantum number $ {m_s} $ describes the projection of spin angular momentum along an axis. We get to know the spin of the electron within the orbital. It can take only two values, $ {m_s} = \pm \dfrac{1}{2} $ .
( $ + $ value $ = $ spin up and $ - $ value $ = $ spin down) . Here, both electrons have the same spin up values. (Hund's rule).
The two possible sets of quantum numbers for the two p electrons are :
 $ n = 5,l = 1,{m_l} = - 1,{m_s} = + \dfrac{1}{2} $ : $ \,5{p_x} $
 $ n = 5,l = 1,{m_l} = 0,{m_s} = + \dfrac{1}{2} $ $ :\,\,5{p_y} $

Note :
Hund’s rule says that orbitals must be filled by electrons in a manner such that they must be singly occupied before getting paired. Also, the unpaired electrons must have the same spin (parallel).
Pauli’s exclusion principle suggests that no two electrons can have the same set of quantum numbers. This is clearly evident in the above question.