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Qestion 1.) Which is not a permissible set of quantum numbers? Identify the subshell (if the quantum numbers identify a possible state).
I. n = 2, ℓ = 0, mℓ = 0
II. n = 3, ℓ = 2, mℓ = 2
III. n = 2, ℓ = 1, mℓ = –1
IV. n = 3, ℓ = 3, mℓ = 0
V. n = 4, ℓ = 3, mℓ = –3
Answer:
- The combination that is not permissible is IV. n = 3, ℓ = 3, mℓ = 0.
- Se below to identify the subshells.
Explanation:
The electrons are identified by a set of four quantum numbers.
The first quantum number, n, is the principal quantum number and it tells the shell. The second quantum number,ℓ , is the azymuthal quantum number and it tells the subshell.
The letters used to indicate the subshells are:
s: ℓ = 0
p: ℓ = 1
d: ℓ = 2
f: ℓ = 3
The third and fourth quantum numbers are mℓ (magnetic quantum number) and s (spin).
The rules that apply to predict which quantum numbers are possible are:
- n: 1, 2, 3, 4, 5, 6, 7 (an integer greater than 0)
- ℓ: 0, 1, 2, 3, ..., n-1 (an integer less than n)
- mℓ: an integer from - ℓ to + ℓ
- s: - 1/2 or +1/2
Two electrons in an atom cannot have the same set of 4 quantum numbers.
With that:
I. n = 2, ℓ = 0, mℓ = 0
- This combination of three quantum numbers is permissible, since n is a positive integer, ℓ is less than n, and mℓ is in the interval +ℓ to - ℓ.
- The combination n = 2 and ℓ = 0 means the subshell is 2p.
II. n = 3, ℓ = 2, mℓ = 2
- This combination of three quantum numbers is permissible, since n is a positive integer, ℓ is less than n, and mℓ is in the interval +ℓ to - ℓ.
- The combination n = 3 and ℓ = 2 means the subshell is 3d.
III. n = 2, ℓ = 1, mℓ = –1
- This combination of three quantum numbers is permissible, since n is a positive integer, ℓ is less than n, and mℓ is in the interval +ℓ to - ℓ.
- The combination n = 2 and ℓ = 1 means the subshell is 2p.
IV. n = 3, ℓ = 3, mℓ = 0
- This set of three quantum numbers is not permissible, since ℓ = 3 is not less than n = 3.
V. n = 4, ℓ = 3, mℓ = –3
- This set of three quantum numbers is permissible, since n is a positive integer, ℓ is less than n, and mℓ is in the interval +ℓ to - ℓ.
- The combination n = 4 and ℓ = 3 means the subshell is 4f.
Question 2.)What is the difference between the 2pz and the 3pz orbitals? Which quantum numbers in the orbital designation are different? Which will be the same? What does this indicate about the orbitals?
Answer:
The difference between and
(note that the letter z is a subscript) is in the first quantum number.
The first quantum number indicates the main energy level and so it is related with the size of the orbital.
So, the 3pz orbital is bigger than the 2pz orbital.
The second quantum number is related to the letter p, so the same letter indicates the same shape of the orbital. Remember the table for the letters used to indicate the subshells are:
s: ℓ = 0
p: ℓ = 1
d: ℓ = 2
f: ℓ = 3
So, the scond quantum number for the two orbitals is ℓ = 1.
The subscript indicates the space orientation. So, since both orbitals have the same subscript, z, they have the same orientation.
In conclusion, the only difference between those orbitals is the size of the orbitals, but they have the same shape and orientation.
3.)What is the maximum number of electrons that can have n = 3 and ms = + ½ ?
Answer:
- 9 electrons
Explanation:
Using the rules, for n = 3
- ℓ can be 0, 1, or 2;
- mℓ can be 0 for ℓ = 0,
- mℓ can be -1, 0, or -1 for for ℓ = 1, and
- mℓ can be -2, -1, 0, +1, or +2 for ℓ = 2,
You can get the possible sets of quantum numbers (with n = 3):
- (3, 0, 0, +1/2)
- (3, 0, 0, -1/2)
- (3, 1, 0, +1/2)
- (3, 1, 0, -1/2)
- (3, 1, 1, +1/2)
- (3, 1, 1, -1/2)
- (3, 1, -1, +1/2)
- (3, 1, -1, -1/2)
- (3, 2, 0, +1/2)
- (3, 2, 0, -1/2)
- (3, 2, -2, +1/2)
- (3, 2, -2, -1/2)
- (3, 2, -1, +1/2)
- (3, 2, -1, -1/2)
- (3, 2, 1, +1/2)
- (3, 2, 1, -1/2)
- (3, 2, 2, +1/2)
- (3, 2, 2, -1/2)
So, those are a total of 18 electrons from which half have n = 3 and ms = +1/2.
Hence, 9 electrons can have n = 3 and ms = +1/2.
