Which elements have the quantum numbers (4,3,-2,+1/2) and (5,2,-1,-1/2)

The elements are samarium (Sm) and silver (Ag).

Quantum numbers (4,3,-2,+½)

n = 4: Principal quantum number = 4.

l = 3: Element has 4f electrons

It is conventional to list quantum numbers in decreasing order.

Hund's rule states that all the orbitals must be half-filled with electrons having the same spin before any can receive a second electron.

We get the following table:

Element n l mₗ mₛ

La 4 3 3 + ½

Ce 4 3 2 + ½

Pr 4 3 1 + ½

Nd 4 3 0 + ½

Pm 4 3 - 1 + ½

Sm 4 3 - 2 + ½

The element is samarium, Sm.

Quantum numbers (5,2,-1,-½)

n = 5: Principal quantum number = 5.

l = 2: Element has 5d electrons

We get the following table:

Element n l mₗ mₛ

Y 5 2 2 + ½

Zr 5 2 1 + ½

Nb 5 2 0 + ½

Mo 5 2 - 1 + ½

Tc 5 2 - 2 + ½

Ru 5 2 2 - ½

Rh 5 2 1 - ½

Pd 5 2 0 - ½

Ag 5 2 - 1 - ½

The element is silver, Ag.

Note: These assignments assume that there are no exceptions in the Periodic Table.