Question

For which of the following ions would you expect the spin-only formula to give reasonable cstimates of the magnetic moment: (a) \(\left[\mathrm{Cr}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+}\) (b) \(\left[\mathrm{V}\left(\mathrm{OH}_{2}\right)_{6}\right]^{3+}\) (c) \(\left[\mathrm{CoF}_{6}\right]^{3-}\) ? Rationalize your answer.

Short answer

The spin-only formula gives reasonable estimates for \([\mathrm{Cr(NH}_3)_6]^{3+}\) and \([\mathrm{V(OH}_2)_6]^{3+}\), but not for \([\mathrm{CoF}_6]^{3-}\).

Step-by-step solution

Understanding spin-only formula

The spin-only formula for calculating the magnetic moment, \( \mu \), in units of Bohr magnetons (B.M.) is given by \( \mu = \sqrt{n(n+2)} \), where \( n \) is the number of unpaired electrons in the d-orbitals.

Calculating unpaired electrons for Cr

The ion \( [\mathrm{Cr(NH}_3)_6]^{3+} \) has a chromium ion in the +3 oxidation state, meaning it has \( [\mathrm{Ar}]~3d^3 \) configuration. There are 3 unpaired electrons, so using the formula: \( \mu = \sqrt{3(3+2)} = \sqrt{15} \).

Calculating unpaired electrons for V

The ion \( [\mathrm{V(OH}_2)_6]^{3+} \) has a vanadium ion in the +3 oxidation state, meaning it has \( [\mathrm{Ar}]~3d^2 \) configuration. There are 2 unpaired electrons, so using the formula: \( \mu = \sqrt{2(2+2)} = \sqrt{8} \).

Calculating unpaired electrons for Co

The ion \( [\mathrm{CoF}_6]^{3-} \) has a cobalt ion in the +3 oxidation state, meaning it has \( [\mathrm{Ar}]~3d^6 \) configuration. The strong field ligand F usually leads to pairing up of the electrons, leaving 1 unpaired electron. However, with a low-spin complex, we use 1 unpaired electron for estimation: \( \mu = \sqrt{1(1+2)} = \sqrt{3} \).

Assessing applicability of the spin-only formula

The spin-only formula gives good estimates when there is negligible contribution from orbital behavior. For high spin complexes without pairing influence like \( [\mathrm{Cr(NH}_3)_6]^{3+} \) and \( [\mathrm{V(OH}_2)_6]^{3+} \), it gives reasonable estimates. However, \( [\mathrm{CoF}_6]^{3-} \) can form a low-spin complex in the presence of strong field ligands, making the formula less applicable due to significant orbital contributions.

Key concepts

Spin-Only Formula

The spin-only formula is a handy tool for estimating magnetic moments of transition metal ions. It's especially useful in predicting the magnetic properties of complexes. To arrive at the magnetic moment using this formula, you need to know the number of unpaired electrons in the d-orbitals of the metal ion. The formula is given by: \[ \mu = \sqrt{n(n+2)} \]where \( n \) is the number of unpaired electrons. The result is expressed in Bohr magnetons (B.M.). The spin-only formula tends to give more accurate estimates in cases where orbital contributions to the magnetic moment are minimal. This usually occurs in high-spin complexes where the energies of the d-orbitals closely align, preventing significant pairing. In these scenarios, the unpaired electrons dominate the magnetic behavior, making the spin-only calculation reliable. However, in low-spin complexes or cases where ligands induce significant d-orbital splitting and pairing, the orbital contribution can become significant, reducing the formula's reliability.

Unpaired Electrons

Unpaired electrons play a crucial role in determining the magnetic properties of transition metals and their complexes. These electrons, which do not have a partner to pair with in their atomic or molecular orbitals, are responsible for the magnetism observed in many compounds. In a transition metal complex, whether unpaired electrons exist depends on the electron configuration of the metal ion as well as the influence of surrounding ligands. For example:

• In a high-spin complex, the energy difference between the d-orbitals is small, allowing electrons to remain unpaired to minimize repulsion.
• In a low-spin complex, the significant energy gap leads to electron pairing within the lower-energy d-orbitals.

The count of unpaired electrons is extracted by examining the d-electron configuration of the metal ion and evaluating how they are influenced by the crystal field created by ligands. Calculating this helps us apply the spin-only formula, as it forms the very basis of the formula \( \mu = \sqrt{n(n+2)} \), where \( n \) represents the number of unpaired electrons.

Transition Metal Complexes

Transition metal complexes are fascinating chemical species composed of a central metal ion bound to surrounding molecules or ions called ligands. These ligands can significantly affect the properties of the metal ion, altering features such as color, reactivity, and magnetism. The d-orbitals of transition metals are particularly suited for forming bonds with ligands. These bonding interactions and the resultant splitting of the d-orbitals determine whether the complex is high-spin or low-spin, thus affecting the magnetic properties.

• High-spin complexes occur when the crystal field splitting is small, leading to a greater number of unpaired electrons.
• Low-spin complexes occur when the field splitting is large, resulting in more paired electrons due to occupation of lower energy levels first.

The nature of the ligands plays a critical role in this behavior. Strong field ligands, like cyanide or carbonyl, tend to cause greater splitting, often resulting in low-spin complexes. Conversely, weak field ligands, such as halides, allow for smaller splitting, typically resulting in high-spin complexes.

Oxidation State and Electron Configuration

The oxidation state and electron configuration are foundational concepts in understanding transition metal complexes. The oxidation state of a metal in a complex informs us of its electron count and ultimately its electronic behavior.Let's explore this further:

• Oxidation state is the charge assigned to an atom in a molecule/ion, assuming that electrons shared in bonds are owned by the more electronegative atom. It helps us deduce the number of electrons lost to form the complex.
• Electron configuration describes the distribution of electrons among the orbitals of an atom or ion. It provides insight into how many unpaired electrons are present, which influences magnetic properties and reactivity.

For instance, take the chromium ion \([\mathrm{Cr(NH}_3)_6]^{3+}\). Here, the +3 oxidation state indicates that chromium has lost three electrons, providing it with a \([\mathrm{Ar}]~3d^3\) configuration. From this, we derive that it has 3 unpaired electrons, crucial for calculating the magnetic moment using the spin-only formula.