Quantum numbers arise naturally from the mathematics used to describe the possible states of an electron in an atom. The four quantum numbers, the principal quantum number (n), the angular momentum quantum number (ℓ), the magnetic quantum number (mℓ), and the spin quantum number (ms) have strict rules which govern the possible values. Identify allowable combinations of quantum numbers for an electron. Select all that apply.
n = 3, ℓ= –1, mℓ= 0, ms= –1/2
n = 4, ℓ= 0, mℓ= 1, ms= 1/2
n = 5, ℓ= 5, mℓ= –1, ms= –1/2
n = 5, ℓ= 4, mℓ= 0, ms= 1/2
n = 3, ℓ= 0, mℓ= 0, ms= 1/2
n = 2, ℓ= 0, mℓ= 0, ms= 1
I really just don't understand this at all, anything helps! Thank you.
To identify the allowable combinations of quantum numbers for an electron, we need to follow the rules governing each quantum number:
1. Principal Quantum Number (n): It represents the energy level or shell of the electron and can have positive integer values (1, 2, 3, etc.).
2. Angular Momentum Quantum Number (ℓ): It specifies the shape of the orbital and can have values ranging from 0 to n-1. So, the valid range for ℓ is 0 to (n-1).
3. Magnetic Quantum Number (mℓ): It describes the orientation of the orbital and can have values ranging from -ℓ to ℓ. Therefore, the valid range for mℓ is from -ℓ to ℓ.
4. Spin Quantum Number (ms): It represents the spin of the electron and can have two values: +1/2 or -1/2.
Let's analyze each option:
1. n = 3, ℓ = -1, mℓ = 0, ms = -1/2: This combination is NOT valid because the angular momentum quantum number (ℓ) cannot be negative.
2. n = 4, ℓ = 0, mℓ = 1, ms = 1/2: This combination is VALID. It satisfies the rules for all four quantum numbers.
3. n = 5, ℓ = 5, mℓ = -1, ms = -1/2: This combination is NOT valid because the angular momentum quantum number (ℓ) cannot be greater than or equal to n.
4. n = 5, ℓ = 4, mℓ = 0, ms = 1/2: This combination is VALID. It satisfies the rules for all four quantum numbers.
5. n = 3, ℓ = 0, mℓ = 0, ms = 1/2: This combination is VALID. It satisfies the rules for all four quantum numbers.
6. n = 2, ℓ = 0, mℓ = 0, ms = 1: This combination is NOT valid because the spin quantum number (ms) can only be +1/2 or -1/2.
Therefore, the allowable combinations of quantum numbers for an electron are:
- n = 4, ℓ = 0, mℓ = 1, ms = 1/2
- n = 5, ℓ = 4, mℓ = 0, ms = 1/2
- n = 3, ℓ = 0, mℓ = 0, ms = 1/2
To determine the allowable combinations of quantum numbers for an electron, we need to follow the rules:
1. The principal quantum number (n) can have any positive integer value (1, 2, 3, ...).
2. The angular momentum quantum number (ℓ) must satisfy the condition 0 ≤ ℓ ≤ n - 1.
3. The magnetic quantum number (mℓ) must satisfy the condition -ℓ ≤ mℓ ≤ ℓ.
4. The spin quantum number (ms) can only have two values: 1/2 (representing spin-up) or -1/2 (representing spin-down).
Now, let's go through each combination provided:
n = 3, ℓ = -1, mℓ = 0, ms = -1/2
This combination is not valid because the angular momentum quantum number (ℓ) cannot be negative.
n = 4, ℓ = 0, mℓ = 1, ms = 1/2
This combination is valid. The principal quantum number (n) is 4, the angular momentum quantum number (ℓ) is 0, the magnetic quantum number (mℓ) is 1 (which satisfies -ℓ ≤ mℓ ≤ ℓ), and the spin quantum number (ms) is 1/2.
n = 5, ℓ = 5, mℓ = -1, ms = -1/2
This combination is not valid because the angular momentum quantum number (ℓ) cannot be greater than or equal to the principal quantum number (n).
n = 5, ℓ = 4, mℓ = 0, ms = 1/2
This combination is valid. The principal quantum number (n) is 5, the angular momentum quantum number (ℓ) is 4, the magnetic quantum number (mℓ) is 0, and the spin quantum number (ms) is 1/2.
n = 3, ℓ = 0, mℓ = 0, ms = 1/2
This combination is valid. The principal quantum number (n) is 3, the angular momentum quantum number (ℓ) is 0, the magnetic quantum number (mℓ) is 0, and the spin quantum number (ms) is 1/2.
n = 2, ℓ = 0, mℓ = 0, ms = 1
This combination is not valid because the spin quantum number (ms) can only have the values 1/2 or -1/2.
Therefore, the allowable combinations of quantum numbers for an electron are:
- n = 4, ℓ = 0, mℓ = 1, ms = 1/2
- n = 5, ℓ = 4, mℓ = 0, ms = 1/2
- n = 3, ℓ = 0, mℓ = 0, ms = 1/2
You need to know the rules. Here is how you do it. The rules are:
n = 1,2,3 etc in steps of whole numbers.
ell (I can't write the script l) is 0, 1, 2, etc in steps of whole numbers but never more than n-1.
or less than 0
mell = -ell to + ell in steps of whole numbers (including 0)
ms can be +1/2 or -1/2
So look at +1 in the question.
n = 3. that's permissible.
ell = -1---can be since ell can be 0, 1, 2 etc but no larger than n-1 and no less than 0.
Look at 2 in this problem.
n = 4. That's ok
ell = 0. That's ok
mell = 1--can't be. mell may be from -ell to + ell so it may be anything but not less than 0
You do each of these one by one. Post if you have questions. There is more than one answer that is correct.