Question

Quantum numbers arise naturally from the mathematics used to describe the possible states of an electron in an atom.

Answer 1

Answer: a) n= 3, l = 2, $m_l=-2$ , $m_s=-\frac{1}{2}$

d) n= 5, l= 4, $m_l=4$ , $m_s=+\frac{1}{2}$

e) n= 3, l= 3, $m_l=1$ , $m_s=+\frac{1}{2}$

Explanation:

Principle Quantum Numbers : It describes the size of the orbital and the energy level. It is represented by n. Where, n = 1,2,3,4....

Azimuthal Quantum Number : It describes the shape of the orbital. It is represented as 'l'. The value of l ranges from 0 to (n-1). For l = 0,1,2,3... the orbitals are s, p, d, f...

Magnetic Quantum Number : It describes the orientation of the orbitals. It is represented as . The value of this quantum number ranges from . When l = 2, the value of  will be -2, -1, 0, +1, +2.

Spin Quantum number : It describes the direction of electron spin. This is represented as  The value of this is $+\frac{1}{2}$ for upward spin and   $-\frac{1}{2}$ for downward spin.

a) n= 3, l = 2, $m_l=-2$ , $m_s=-\frac{1}{2}$

n=3  and l can have value 0 to (n-1) i.e. 0, 1 and 2.

$m_l$ can have values from -l to +l i.e for l= 2 , $m_l$ can have values -2, -1, 0 , +1 and +2. Thus is allowed.

b) n= 4, l= 0, $m_l=-1$ , $m_s=-\frac{1}{2}$

n= 4 and l can have value 0 to (n-1) i.e. 0, 1 , 2 and 3.

$m_l$ can have values from -l to +l. For l= 0 , $m_l$ can have value of 0 only. Thus is not allowed.

c)  n= 3 , l= -2, $m_l=-2$ , $m_s=+\frac{1}{2}$

n= 3 and l can have value 0 to (n-1) i.e. 0, 1 and 2.  Thus is not allowed.

d)  n= 5, l= 4, $m_l=4$ , $m_s=+\frac{1}{2}$

n= 5 and l can have value 0 to (n-1) i.e. 0, 1 , 2 ,3 and 4

$m_l$ can have values from -l to +l. For l= 4 , $m_l$ can have value of -4, -3, -2 , -1, 0 , +1, +2, +3 and +4. Thus is allowed.

e)  n= 3, l= 3, $m_l=1$ , $m_s=+\frac{1}{2}$

n= 3 and l can have value 0 to (n-1) i.e. 0, 1 and 2.

$m_l$ can have values from -l to +l. For l= 3 , $m_l$ can have value of -3, -2 , -1, 0 , +1, +2, and +3. Thus is allowed.