Question

Give the maximum number of electrons in an atom that can have these quantum numbers: a. \(n=0, \ell=0, m_{\ell}=0\) b. \(n=2, \ell=1, m_{\ell}=-1, m_{s}=-\frac{1}{2}\) c. \(n=3, m_{s}=+\frac{1}{2}\) d. \(n=2, \ell=2\) e. \(n=1, \ell=0, m_{\ell}=0\)

Short answer

a. 0 electrons (invalid quantum numbers) b. 1 electron c. 5 electrons d. 0 electrons (invalid quantum numbers) e. 2 electrons

Step-by-step solution

a. n=0, ℓ=0, mℓ=0

For this case, we see that the principal quantum number, n, is given as 0. However, n must be a positive integer (n=1,2,3,...). Therefore, this set of quantum numbers is invalid, and no electrons can have these values.

b. n=2, ℓ=1, mℓ=-1, ms=-1/2

This set of quantum numbers seems valid; let's verify them. The principal quantum number n=2 implies that (n-1) shells are present. The azimuthal quantum number ℓ is given as 1, which is valid because 0 ≤ ℓ ≤ (n-1). The magnetic quantum number mℓ is -1, which is valid because -ℓ ≤ mℓ ≤ ℓ. The spin quantum number ms has two values amongst electrons: +1/2 or -1/2, which is consistent with the given value -1/2. Therefore, these quantum numbers are valid, and there is one electron with these values.

c. n=3, ms=+1/2

In this case, we are only given n and ms. Since ms can have two values (+1/2 and -1/2) and ms is given as +1/2, we count the maximum number of electrons in the n=3 shell having ms=+1/2. The total number of orbitals in the n=3 shell is \(3^2 = 9\). Since each orbital can hold two electrons (one with ms=+1/2 and one with ms=-1/2), there can be 9/2=4.5 orbitals holding electrons with ms=+1/2, which we can round up to 5. Therefore, the maximum number of electrons with these quantum numbers is 5.

d. n=2, ℓ=2

In this case, n is given as 2, and ℓ is given as 2. However, since 0 ≤ ℓ ≤ (n-1), the maximum possible value for ℓ in this case is 1. Therefore, this set of quantum numbers is invalid, and no electrons can have these values.

e. n=1, ℓ=0, mℓ=0

For this case, all the given quantum numbers are valid: n=1 is the first shell, ℓ=0 is valid within the range of 0 ≤ ℓ ≤ (n-1), and mℓ=0 is also valid within the range of -ℓ ≤ mℓ ≤ ℓ. There is only one orbital (s-orbital) in the first shell. Two electrons can occupy the s-orbital, one with ms=+1/2 and one with ms=-1/2. Therefore, the maximum number of electrons with these quantum numbers is 2.