Which of the following set of quantum numbers (ordered n, ℓ, mℓ, ms) are possible for an electron in an atom?
Check all that apply.
2, -3, 1/2
-3, 2, 2, -1/2
5, 2, 1, -1/2
3, 2, 2, -1/2
4, 3, 4, -1/2
3, 2, 1, -1
4, 3, -2, 1/2
3, 3, 1, -1/2
As a humorous bot, let me make this quantum leap for you:
Out of the given options, the possible set of quantum numbers for an electron in an atom are:
3, 2, 2, -1/2: This one is feeling positive in its moods!
3, 2, 1, -1/2: It's getting closer to the nucleus, so it's a bit negative.
3, 3, 1, -1/2: Just like a third wheel, this electron has a third value!
So, the options 3, 2, 2, -1/2; 3, 2, 1, -1/2; and 3, 3, 1, -1/2 are possible for an electron in an atom. Keep those quantum ducks in a row!
To determine which set of quantum numbers are possible for an electron in an atom, we need to check for the following conditions:
1. The principal quantum number (n) must be a positive integer.
2. The azimuthal quantum number (ℓ) must be an integer between 0 and (n-1).
3. The magnetic quantum number (mℓ) must be an integer between -ℓ and ℓ.
4. The spin quantum number (ms) must be either +1/2 or -1/2.
Let's check each set of quantum numbers one by one:
1. Set: 2, -3, 1/2
- n = 2 (valid)
- ℓ = -3 (invalid, should be an integer between 0 and (n-1))
- mℓ = 1/2 (invalid, should be an integer)
- ms = Not given
2. Set: -3, 2, 2, -1/2
- n = -3 (invalid, should be a positive integer)
- ℓ = 2 (valid)
- mℓ = 2 (valid, should be between -ℓ and ℓ)
- ms = -1/2 (valid)
3. Set: 5, 2, 1, -1/2
- n = 5 (valid)
- ℓ = 2 (valid)
- mℓ = 1 (valid, should be between -ℓ and ℓ)
- ms = -1/2 (valid)
4. Set: 3, 2, 2, -1/2
- n = 3 (valid)
- ℓ = 2 (valid)
- mℓ = 2 (valid, should be between -ℓ and ℓ)
- ms = -1/2 (valid)
5. Set: 4, 3, 4, -1/2
- n = 4 (valid)
- ℓ = 3 (valid)
- mℓ = 4 (invalid, should be between -ℓ and ℓ)
- ms = -1/2 (valid)
6. Set: 3, 2, 1, -1
- n = 3 (valid)
- ℓ = 2 (valid)
- mℓ = 1 (valid, should be between -ℓ and ℓ)
- ms = -1 (invalid, should be either +1/2 or -1/2)
7. Set: 4, 3, -2, 1/2
- n = 4 (valid)
- ℓ = 3 (valid)
- mℓ = -2 (valid, should be between -ℓ and ℓ)
- ms = 1/2 (valid)
8. Set: 3, 3, 1, -1/2
- n = 3 (valid)
- ℓ = 3 (invalid, should be between 0 and (n-1))
- mℓ = 1 (valid, should be between -ℓ and ℓ)
- ms = -1/2 (valid)
Based on the analysis above, the sets of quantum numbers that satisfy all the conditions are:
- Set: 5, 2, 1, -1/2
- Set: 3, 2, 2, -1/2
- Set: 4, 3, -2, 1/2
To determine which set of quantum numbers are possible for an electron in an atom, we need to use the following rules:
1. The principal quantum number (n) represents the energy level of the electron and must be a positive integer (1, 2, 3, ...).
2. The azimuthal quantum number (ℓ) represents the shape of the electron's orbital and can range from 0 to n - 1. It is often represented using letters: s (0), p (1), d (2), f (3), and so on.
3. The magnetic quantum number (mℓ) represents the orientation of the orbital and can range from -ℓ to ℓ.
4. The spin quantum number (ms) represents the spin orientation of the electron and has two possible values: +1/2 for spin up and -1/2 for spin down.
Let's evaluate each set of quantum numbers to see if they follow these rules:
1. 2, -3, 1/2 -> The principal quantum number (n) is valid because 2 is a positive integer. The azimuthal quantum number (ℓ) is invalid because it is negative. Therefore, this set is not possible.
2. -3, 2, 2, -1/2 -> Both the principal quantum number (n) and the azimuthal quantum number (ℓ) are invalid because they are negative. Therefore, this set is not possible.
3. 5, 2, 1, -1/2 -> The principal quantum number (n) is valid because 5 is a positive integer. The azimuthal quantum number (ℓ) is valid because it is less than n. The magnetic quantum number (mℓ) is valid because it falls within the range of -ℓ to ℓ (-1 to 1 in this case). The spin quantum number (ms) is valid because it is either +1/2 or -1/2. Therefore, this set is possible.
4. 3, 2, 2, -1/2 -> The principal quantum number (n) is valid because 3 is a positive integer. The azimuthal quantum number (ℓ) is valid because it is less than n. The magnetic quantum number (mℓ) is valid because it falls within the range of -ℓ to ℓ (-2 to 2 in this case). The spin quantum number (ms) is valid. Therefore, this set is possible.
5. 4, 3, 4, -1/2 -> The principal quantum number (n) is valid because 4 is a positive integer. The azimuthal quantum number (ℓ) is invalid because it is greater than or equal to n. Therefore, this set is not possible.
6. 3, 2, 1, -1 -> The principal quantum number (n) is valid because 3 is a positive integer. The azimuthal quantum number (ℓ) is valid because it is less than n. The magnetic quantum number (mℓ) is valid because it falls within the range of -ℓ to ℓ (-1 to 1 in this case). The spin quantum number (ms) is invalid because it is not +1/2 or -1/2. Therefore, this set is not possible.
7. 4, 3, -2, 1/2 -> The principal quantum number (n) is valid because 4 is a positive integer. The azimuthal quantum number (ℓ) is valid because it is less than n. The magnetic quantum number (mℓ) is valid because it falls within the range of -ℓ to ℓ (-3 to 3 in this case). The spin quantum number (ms) is valid. Therefore, this set is possible.
8. 3, 3, 1, -1/2 -> The principal quantum number (n) is valid because 3 is a positive integer. The azimuthal quantum number (ℓ) is invalid because it is greater than or equal to n. Therefore, this set is not possible.
Based on the evaluation, the sets of quantum numbers that are possible for an electron in an atom are:
- 5, 2, 1, -1/2
- 3, 2, 2, -1/2
- 4, 3, -2, 1/2
Here are the rules. If you don't know them you should.
n = 1,2,3,4, etc..........
l = 0,1,2, etc up to n-1
ml = -l in increments of 1 to 0 and up to +l; i.e., for l = 2 then ml can be -2,-1,0 ,+1, +2
ms = only +1/2 or -1/2
Here are your first 3 answers which I will call A,B,C.
A. 2, -3, 1/2
B -3, 2, 2, -1/2
C 5, 2, 1, -1/2
A can't be correct because l can't be negative.
B can't be right because n can't be negative.
C is possible. The rules (read them) allow these numbers.
You do the others.