Question

The total spin resulting from a \(\mathrm{d}^{3}\) configuration is: (a) 3 (b) \(\frac{3}{2}\) (c) 1 (d) Zero

Short answer

The total spin for a \( \mathrm{d}^{3} \) configuration is \( \frac{3}{2} \).

Step-by-step solution

Understanding the context

When dealing with electronic configurations with multiple electrons, such as a \( \mathrm{d}^{3} \) configuration, the total spin refers to the total spin angular momentum of the electrons. Each electron can contribute with a spin of \( +\frac{1}{2} \) or \( -\frac{1}{2} \). The goal is to find the total spin quantum number \( S \) and thus \( 2S+1 \) for the configuration.

Identifying possible electron spins

In a \( \mathrm{d}^{3} \) electron configuration, there are three electrons to consider. Typically, electrons will fill orbitals with parallel spins (Hund's rule) to maximize total spin. Thus, each of these three electrons will be in a separate d orbital with a spin of \( +\frac{1}{2} \).

Calculating total spin angular momentum (S)

Sum up the spins from the configuration. With each electron contributing a spin of \(+\frac{1}{2}\), the total spin \( S \) for a \( \mathrm{d}^{3} \) configuration is: \[ S = \frac{1}{2} + \frac{1}{2} + \frac{1}{2} = \frac{3}{2} \]

Determining the Total Spin

The total spin \( S \) calculated in the previous step tells us the magnitude of the spin angular momentum. The multiplicity, which indicates the possible orientation of the spin, is \( 2S+1 \). For \( S = \frac{3}{2} \), this results in: \[ 2S + 1 = 2\left(\frac{3}{2}\right) + 1 = 4 \]This confirms that \( S = \frac{3}{2} \) is correct.

Key concepts

Total Spin

The total spin of an electron configuration is essentially the sum of the spin quantum numbers of all the electrons present. Each electron has a spin that can be either \(+\frac{1}{2}\) or \(-\frac{1}{2}\). For example, in a \(d^3\) configuration, there are three electrons. Aiming to maximize the total spin follows Hund's rule, meaning that electrons will initially fill available orbitals with parallel spins as much as possible, to avoid pairing the electrons prematurely. Therefore, each of the electrons contributes a spin of \(+\frac{1}{2}\). As such, the total spin \(S\) is calculated as the sum of these spins: \(S = \frac{1}{2} + \frac{1}{2} + \frac{1}{2} = \frac{3}{2}\). - In summary, the total spin represents the aggregate of all individual electron spins in a system. - It helps in understanding electron configuration in atoms, and predicting their behavior in magnetic fields.

Spin Angular Momentum

Spin angular momentum is a fundamental property of elementary particles, such as electrons. It relates to the intrinsic angular momentum carried by particles with spin. For electrons, this manifests as either a half-integer or integer value. In quantum mechanics, the spin angular momentum is quantized, meaning it takes specific discrete values.
Spin angular momentum for an atom can be derived from the total spin \(S\). For example, if you have determined the total spin \(S\) as \(\frac{3}{2}\), you can calculate the spin angular momentum using the formula:

• The magnitude of the spin angular momentum \( \left| \mathbf{S} \right| = \sqrt{S(S+1)} \hbar \)

Using this, for \(S = \frac{3}{2}\),

• \( \left| \mathbf{S} \right| = \sqrt{\frac{3}{2}\left(\frac{3}{2}+1\right)} \hbar = \sqrt{\frac{15}{4}} \hbar \)

This shows that spin angular momentum gives insight into how particles, like electrons, behave in the presence of external fields, such as magnetic fields, affecting their energy levels and placements in atoms.

Hund's Rule

Hund's Rule is crucial for understanding how electrons are distributed among orbitals in an atom. The rule states that electrons fill degenerate orbitals—those with the same energy level—in a way that maximizes the total spin. This means that electrons prefer to occupy separate orbitals with parallel spins before they pair up in any particular orbital.
Why is this rule important?

• It helps in predicting the electron configuration of atoms how energy will be minimized.
• It ensures the lowest energy state of an atom by reducing electron-electron repulsions, as electrons remain unpaired.

For a \(d^3\) configuration, Hund's Rule implies that the three electrons will each shake out into separate d orbitals with parallel spins (all \(+\frac{1}{2}\) in this case). This results in a higher total spin, which is more energetically favorable for the atom.

d orbital

The d orbitals are a set of five individually distinct orbitals found in the electron shell of an atom. They are crucial in the context of transition metals, which often fill these orbitals with their outer electrons. Each d orbital can hold a maximum of two electrons, according to the Pauli exclusion principle.

• The specific shapes of d orbitals contribute to their unique bonding and interaction characteristics.
• These orbitals have more complex shapes like the "cloverleaf," which influences the geometry of the molecules they are part of.

For a \(d^3\) configuration, the d orbitals will be half-filled, with each electron occupying its own orbital to maximize spin. Having multiple d orbitals available means that atoms can explore rich chemistry and participate in bonding and interactions that s and p orbitals alone cannot accommodate, ultimately affecting the electronic properties and color of compounds.