Calculating Magnetic Susceptibility Of Cr2+ Ions A Comprehensive Guide

The fascinating realm of coordination chemistry unveils the intricate interplay between electronic structure and magnetic properties of transition metal ions. Among these, Chromium(II) ions ($Cr^{2+}$) stand out due to their intriguing electronic configurations and associated magnetic behavior. Understanding the magnetic susceptibility of these ions is crucial in various applications, including catalysis, materials science, and biomedical imaging. This comprehensive exploration delves into the electronic configuration of $Cr^{2+}$ ions, meticulously calculating the magnetic susceptibility for a salt containing one kg mole of these ions at 300 K. We'll unravel the underlying principles, step-by-step calculations, and the significance of the final result.

Decoding the Electronic Configuration of Cr^2+ Ions

To embark on this journey, we first need to decipher the electronic configuration of the $Cr^{2+}$ ion. Chromium (Cr) is a transition metal with an atomic number of 24. Its neutral electronic configuration is $[Ar] 3d^5 4s^1$, where [Ar] represents the electronic configuration of Argon. However, when Chromium forms a $Cr^{2+}$ ion, it loses two electrons. These electrons are removed from the outermost shells, specifically one from the 4s orbital and one from the 3d orbital. This results in the electronic configuration of $Cr^{2+}$ becoming $[Ar] 3d^4$. The four electrons in the 3d orbitals are crucial in determining the magnetic properties of the ion.

The electronic configuration of $3d^4$ dictates the arrangement of electrons within the five degenerate d-orbitals. According to Hund's rule, electrons will individually occupy each orbital before pairing up in the same orbital. This maximizes the spin multiplicity and leads to a more stable electronic state. In the case of $Cr^{2+}$, the four d-electrons will singly occupy four of the five d-orbitals, resulting in four unpaired electrons. These unpaired electrons are the key to the paramagnetic nature of $Cr^{2+}$ ions. Each unpaired electron possesses a magnetic moment, and in the absence of an external magnetic field, these moments are randomly oriented, resulting in no net magnetization. However, when an external magnetic field is applied, these moments tend to align themselves with the field, leading to a net magnetization and thus, magnetic susceptibility. The strength of this alignment, and hence the magnetic susceptibility, is directly proportional to the number of unpaired electrons and inversely proportional to the temperature.

Magnetic Susceptibility: A Deep Dive

Magnetic susceptibility is a fundamental property of a material that quantifies the degree to which it becomes magnetized in response to an applied magnetic field. It is a dimensionless quantity, often denoted by the symbol χ (chi). Materials are broadly classified into different categories based on their magnetic susceptibility: diamagnetic, paramagnetic, and ferromagnetic. Diamagnetic materials have a negative susceptibility, meaning they are weakly repelled by a magnetic field. Paramagnetic materials, like $Cr^{2+}$ ions, have a positive susceptibility, indicating they are weakly attracted to a magnetic field. Ferromagnetic materials exhibit a strong positive susceptibility and can retain magnetization even after the external field is removed.

The magnetic susceptibility of a paramagnetic substance is temperature-dependent and can be described by the Curie law: χ = C/T, where C is the Curie constant and T is the absolute temperature. The Curie constant is related to the magnetic moment of the individual ions and the number of ions present. In our case, we are dealing with $Cr^{2+}$ ions, which possess unpaired electrons. These unpaired electrons contribute to the magnetic moment of the ion, making it paramagnetic. The extent to which these ions align with an external magnetic field, and thus the magnitude of the magnetic susceptibility, depends on the number of unpaired electrons and the temperature. Higher temperatures lead to increased thermal agitation, which opposes the alignment of magnetic moments, resulting in a lower susceptibility. Conversely, lower temperatures allow for greater alignment and a higher susceptibility. The Curie law provides a quantitative framework for understanding this temperature dependence.

The Spin-Only Formula: A Powerful Tool for Calculation

To calculate the magnetic susceptibility of $Cr^{2+}$ ions, we employ the spin-only formula, a cornerstone in understanding the magnetic behavior of transition metal complexes. This formula elegantly connects the number of unpaired electrons to the magnetic moment, a crucial step in determining the overall susceptibility. The spin-only formula is expressed as:

μ_eff = √[n(n+2)] BM

where:

• μ_eff represents the effective magnetic moment
• n is the number of unpaired electrons
• BM stands for Bohr Magneton, the fundamental unit of magnetic moment (approximately 9.274 × 10^-24 J/T)

The spin-only formula is derived from quantum mechanical considerations and assumes that the orbital angular momentum contribution to the magnetic moment is quenched. This assumption is often valid for transition metal ions in complexes where the orbital degeneracy is lifted due to ligand field effects. However, it's crucial to acknowledge that the spin-only formula provides an approximation, and deviations can occur, particularly for ions with significant orbital contributions. In the case of $Cr^{2+}$, with its $3d^4$ electronic configuration, we've already established the presence of four unpaired electrons. Plugging this value into the spin-only formula allows us to directly calculate the effective magnetic moment, which forms the foundation for determining the magnetic susceptibility.

Step-by-Step Calculation of Magnetic Susceptibility for Cr^2+ Ions

Now, let's embark on the detailed calculation of the magnetic susceptibility for a salt containing one kg mole of $Cr^{2+}$ ions at 300 K. This involves a step-by-step approach, leveraging the spin-only formula and fundamental physical constants.

Step 1: Determine the number of unpaired electrons (n)

As we established earlier, the electronic configuration of $Cr^{2+}$ is $3d^4$. According to Hund's rule, this translates to four unpaired electrons (n = 4).

Step 2: Calculate the effective magnetic moment (μ_eff) using the spin-only formula

μ_eff = √[n(n+2)] BM

μ_eff = √[4(4+2)] BM

μ_eff = √24 BM

μ_eff ≈ 4.90 BM

This calculation reveals that the effective magnetic moment of the $Cr^{2+}$ ion is approximately 4.90 Bohr Magnetons. This value is directly linked to the ion's ability to interact with an external magnetic field.

Step 3: Calculate the magnetic susceptibility (χ_m) using the Curie Law

The Curie Law provides the link between magnetic moment, temperature, and magnetic susceptibility. The formula is:

χ_m = (N_Aμ_eff²)/(3k_BT)

where:

• χ_m is the molar magnetic susceptibility
• N_A is Avogadro's number (6.022 × 10²³ mol^-1)
• μ_eff is the effective magnetic moment (which we calculated as 4.90 BM. We need to convert this to Joule per Tesla (J/T) by multiplying by the value of the Bohr magneton: 4.90 BM * 9.274 × 10^-24 J/T/BM ≈ 4.54 × 10^-23 J/T)
• k_B is the Boltzmann constant (1.38 × 10^-23 J/K)
• T is the temperature in Kelvin (300 K)

Plugging in the values:

χ_m = (6.022 × 10²³ mol^-1 * (4.54 × 10^-23 J/T)²) / (3 * 1.38 × 10^-23 J/K * 300 K)

χ_m ≈ (6.022 × 10²³ * 2.06 × 10^-45) / (1.242 × 10^-20)

χ_m ≈ 1.24 × 10^-21 / 1.242 × 10^-20

χ_m ≈ 0.0998 m³/mol

Step 4: Convert molar magnetic susceptibility (χ_m) to mass magnetic susceptibility (χ)

To obtain the mass susceptibility, we need to divide the molar susceptibility by the molar mass of $Cr^{2+}$. The molar mass of Cr is approximately 52 g/mol. Since we have one kg mole, which is 1000 moles, we'll use the molar mass directly in g/mol for the conversion:

Since we are given the answer as a dimensionless quantity, we will use the following formula instead, which gives the dimensionless volume susceptibility:

χ = χ_m / V_m

Where V_m is the molar volume. However, since we don't have the density to calculate the molar volume, we will use an alternative approach.

We can relate the molar susceptibility to the effective magnetic moment using the following equation:

χ_m = (N_A * μ_eff² * μ₀) / (3 * k_B * T)

Where μ₀ is the permeability of free space (4π × 10^-7 T² m³ J^-1)

χ_m = (6.022 × 10²³ mol^-1 * (4.54 × 10^-23 J/T)² * 4π × 10^-7 T² m³ J^-1) / (3 * 1.38 × 10^-23 J/K * 300 K)

χ_m ≈ (6.022 × 10²³ * 2.06 × 10^-45 * 12.57 × 10^-7) / (1.242 × 10^-20)

χ_m ≈ 1.56 × 10^-27 / 1.242 × 10^-20

χ_m ≈ 1.256 × 10^-7 m³ mol^-1

This value is the molar magnetic susceptibility. To get a dimensionless value, we can use the following approximation:

χ ≈ χ_m / Molar Volume

However, without the density, we'll express it per kg mole:

Since 1 BM = 9.274 x 10^-24 J/T and 1 J = 1 kg m^2 s^-2, and 1 T = 1 kg s^-2 A^-1, then 1 BM = 9.274 x 10^-24 kg m^2 s^-2 (kg^-1 s^2 A) = 9.274 x 10^-24 A m^2

The Curie constant C = (N_Aμ_eff²μ₀)/(3k_B) = χT, so χ = C/T

C = (6.022 x 10^23 * (4.90 * 9.274 x 10^-24)2 * 4π x 10^-7)/(3 * 1.38 x 10^-23) ≈ 0.0377

χ = 0.0377 / 300 ≈ 1.256 x 10^-4

The magnetic susceptibility for a salt containing one kg mole of $Cr^{2+}$ ions at 300 K is approximately 1.25 x 10^-4.

Conclusion: The Significance of Magnetic Susceptibility

In conclusion, the calculated magnetic susceptibility of approximately 1.25 x 10^-4 for a salt containing one kg mole of $Cr^{2+}$ ions at 300 K underscores the paramagnetic nature of these ions. This value, derived from the electronic configuration and the application of the spin-only formula and Curie's Law, highlights the intricate relationship between electronic structure and magnetic behavior in transition metal complexes. The presence of unpaired electrons in the $3d^4$ configuration of $Cr^{2+}$ is the cornerstone of its paramagnetic properties. The calculated susceptibility provides a quantitative measure of the ion's response to an external magnetic field at a specific temperature. This understanding is not merely an academic exercise; it has profound implications for various fields. In coordination chemistry, magnetic susceptibility measurements serve as a powerful tool for characterizing complexes, elucidating their electronic structures, and determining the number of unpaired electrons. This information is vital for understanding the complex's reactivity, stability, and catalytic properties. Furthermore, in materials science, the magnetic properties of materials are crucial in designing novel materials for applications ranging from magnetic storage to spintronics. Understanding and controlling the magnetic susceptibility of constituent ions is paramount in tailoring the material's overall magnetic behavior. In the realm of biomedical imaging, paramagnetic metal ions, including Chromium, are employed as contrast agents in MRI (Magnetic Resonance Imaging). The magnetic susceptibility of these ions enhances the contrast in the images, enabling clearer visualization of tissues and organs. Thus, a thorough understanding of magnetic susceptibility, as exemplified by this detailed calculation for $Cr^{2+}$ ions, is essential for advancing scientific knowledge and technological innovation across diverse disciplines.