SCH4U - Chemistry 12 (2024-25) - A

A. Introduction:

In the last activity, we learned about the first 2 quantum numbers n and ℓ. In this lesson we will learn about the last 2 quantum numbers which tell us how many of each type of orbital exists within a shell (m_l) as well as the spin of a given electron within a shell (m_s).

B. The Magnetic Quantum Number (m_l)

Magnetic Quantum Number (m_l) describes the orientation of an orbital in space (ie what plane does it lie in on a 3-D axis).  This number tells us how many of a particular type of orbital are in each shell (for example there are 3 p-orbitals in every shell from n = 2 up).  Values for m_l range from  –ℓ to +ℓ , including 0.  For example if the secondary quantum number (ℓ) = 2, the (m_l) has values that range from -2 to +2, therefore there are 5 different d-orbitals in a given shell ( -2, -1, 0, 1, 2).

As summarized in the chart below, when ℓ = 2, this corresponds to the d-orbital, which has 5 different possible orientations (look in the last column, 3^rd row), which are described by (m_l)

When ℓ = 0: (m_l) values range from 0 to 0; therefore there is 1 s-orbital in every shell

When ℓ = 1:(m_l) values range: -1, 0, 1; therefore there are 3 p-orbitals in every shell from n = 2 and above

When ℓ = 2:(m_l) values range: -2, -1, 0, 1, 2: therefore there are 5 d-orbitals in every shell from n = 3 and above

When ℓ = 3:(m_l) values range from -3, -2, -1, 0, 1, 2, 3: therefore there are 7 f-orbitals in every shell from n = 4 and above

Check your understanding:

Question 1-2C-1: When ℓ = 3, how many m_l values exist for these orbitals?

a) 3

b) 6

c) 7

d) 4

Answer 1-2C:1

Question 1-2C-2: How many p-orbitals exist in any shell in any shell above the n = 1 energy level?

a) 1

b) 2

c) 3

d) 4

Answer 1-2C:2

To summarize everything we have learned so far in one tidy table:

Example

Looking at the table above, we can see that for example, when n = 3:

• • ℓ-values range from 0 to 2,
  • this shell would have 1 s-orbital, 3 p-orbitals and 5 d-orbitals
  • the shell would have 9 orbitals total and could contain up to 18 total electrons (2 per orbital)

Remember we said earlier, every shell (n) can have n² orbitals and every orbital can hold 2 electrons, therefore every shell (n) can hold 2n² electrons

C. The Spin Quantum Number (m_s)

4. The Spin Quantum Number (m_s) describes the direction a given electron is spinning. Electrons have “spin” values associated with them that represent the direction that a given electron spins. For the m_s number there are only 2 possible values +1/2 and -1/2. An arrow pointing up or down can also represent the spin of an electron. 

As we learned earlier each orbital can only contain 2 electrons, and it turns out that if an orbital has 2 electrons they must spin in opposite directions, therefore, the Pauli Exclusion Principle states: no 2 electrons in the same atom can have the same 4 quantum numbers. 

To recap, we can describe an electron in an orbital using 4 quantum numbers:

1. Principal Quantum Number (n): Describes the distance of a shell from the nucleus
2. Secondary Quantum Number (ℓ): Describes the shape of an orbital (subshell)
3. Magnetic Quantum Number (m_l): describes the orientation of the orbital in space (ie. what axis does it lie on)
4. Spin Quantum Number (m_s): describes the direction an electron is spinning

Check your understanding:

Question 1-2C-3 How many possible orientations of the f-orbital exist?

a) 3

b) 7

c) 4

d) 1

Check Answer 1-2C:3

Question 1-2C-4 How many electrons can reside in the n = 2 shell?

a) 8

b) 4

c) 16

d) 2

Check Answer 1-2C:4

D. Summary

From these 2 sections on quantum numbers the most important information to remember is the following:

• n = 1 shell has 1 s-orbital
• n = 2 shell has 1 s-orbital and 3 p-orbitals
• n = 3 shell has 1 s-orbital, 3 p-orbitals and 5 d-orbitals
• n = 4 shell has 1 s-orbital, 3 p-orbitals, 5 d-orbitals and 7 f-orbitals
• Each shell can have n² orbitals and hold 2n² electrons
• Each orbital can hold 2 electrons, with opposite spin values
• The Pauli Exclusion Principle tells us that 2 electrons in the same orbital cannot have the same spin value.