Question

How many electrons in an atom can have each of the following quantum number designations? (a) \(\mathbf{n}=2, \ell=1, \mathbf{m}_{\ell}=0\) (b) \(\mathbf{n}=2, \ell=1, \mathbf{m}_{\ell}=-1\) (c) \(\mathrm{n}=3, \ell=1, \mathrm{~m}_{\ell}=0, \mathrm{~m}_{\mathrm{s}}=+\frac{1}{2}\)

Short answer

a) n=2, ℓ=1, mℓ=0 b) n=2, ℓ=1, mℓ=-1 c) n=3, ℓ=1, mℓ=0, ms=+1/2 Answer: a) 2 electrons b) 2 electrons c) 1 electron

Step-by-step solution

Part (a) - n = 2, ℓ = 1, mℓ = 0

For this part, we are given the principle quantum number (n), the angular momentum quantum number (ℓ) and the magnetic quantum number (mℓ) as 2, 1, and 0 respectively. The spin quantum number, ms, is not given. We know that ms can have two possible values: +1/2 and -1/2. So, for the given set of quantum numbers, there can be two electrons with two possible ms values. Number of electrons for part (a) = 2

Part (b) - n = 2, ℓ = 1, mℓ = -1

In this case, we are given n, ℓ, and mℓ as 2, 1, and -1, respectively. Again, the ms value is not given. So we have two possible ms values: +1/2 and -1/2. Therefore, there can be two electrons with the given set of quantum numbers. Number of electrons for part (b) = 2

Part (c) - n = 3, ℓ = 1, mℓ = 0, ms = +1/2

For part (c), we are given the quantum numbers n, ℓ, mℓ, and ms as 3, 1, 0, and +1/2, respectively. Since all the quantum numbers are specified, this set of quantum numbers corresponds to a single unique electron state. Number of electrons for part (c) = 1 In conclusion: Number of electrons for part (a) = 2 Number of electrons for part (b) = 2 Number of electrons for part (c) = 1