The magnitudes of the resultant spins of the electrons of the elements B through Ne when in the ground state are: \(\sqrt{3} \hbar / 2, \quad \sqrt{2} \hbar, \quad \sqrt{15} \hbar / 2, \quad \sqrt{2} \hbar, \quad \sqrt{3} \hbar / 2, \quad\) and \(\quad 0\) respectively. Argue that these spins are consistent with Hund's rule.

First, let's identify the electron configurations of elements B through Ne: 1. Boron (B) - 1s²2s²2p¹ 2. Carbon (C) - 1s²2s²2p² 3. Nitrogen (N) - 1s²2s²2p³ 4. Oxygen (O) - 1s²2s²2p⁴ 5. Fluorine (F) - 1s²2s²2p⁵ 6. Neon (Ne) - 1s²2s²2p⁶

Now, let's use the electron configurations to determine the total spin angular momentum for each element and compare it to the spins given in the exercise. 1. Boron (B) - 1s²2s²2p¹ For Boron, the single electron in the 2p subshell will have a spin angular momentum of either +1/2 or -1/2. The total spin angular momentum will thus be \(\sqrt{3}\hbar/2\), which matches the value given in the exercise. 2. Carbon (C) - 1s²2s²2p² There are two unpaired electrons in the 2p subshell, each with a spin angular momentum of either +1/2 or -1/2. Following Hund's rule, the electrons will have parallel spins, so the total spin angular momentum is \(\sqrt{2}\hbar\), matching the value given in the exercise. 3. Nitrogen (N) - 1s²2s²2p³ There are three unpaired electrons in the 2p subshell, each with a spin angular momentum of either +1/2 or -1/2. Following Hund's rule, each electron will have parallel spins, so the total spin angular momentum is \(\sqrt{15}\hbar/2\), matching the value given in the exercise. 4. Oxygen (O) - 1s²2s²2p⁴ There are four electrons in the 2p subshell, two of them will be paired with opposite spins (+1/2 and -1/2). The other two electrons will be unpaired and have parallel spins. So the total spin angular momentum is \(\sqrt{2}\hbar\), matching the value given in the exercise. 5. Fluorine (F) - 1s²2s²2p⁵ There are five electrons in the 2p subshell, two of them will be paired with the opposite spins (+1/2 and -1/2), and the other three electrons will be unpaired with parallel spins. Thus, the total spin angular momentum is \(\sqrt{3}\hbar/2\), which matches the value given in the exercise. 6. Neon (Ne) - 1s²2s²2p⁶ All electrons in the 2p subshell are paired, so the total spin angular momentum is zero, matching the value given in the exercise.

The provided magnitudes of the resultant spins for the electrons of the elements B through Ne are consistent with Hund's rule. In each case, the total spin angular momentum calculated using Hund's rule matches the values given in the exercise.