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The trebly ionised ion of praseodymium has two electrons in the \(4 f\) shell. (a) Use Hund's rules to calculate the values of \(L, S, J\) for the ground state of the ion, and find the value of the Land茅 \(g\)-factor. (b) Show that, when a magnetic field \(B=1 \mathrm{~T}\) is applied to a praseodymium salt at temperature \(T=300 \mathrm{~K}\), the thermal average of the component of the magnetic moment of a praseodymium ion in the direction of \(\mathbf{B}\) is, to good approximation, $$ \bar{\mu}=\frac{1}{3} g^{2} \mu_{B}^{2} \frac{J(J+1) B}{k_{\mathrm{B}} T} $$ where \(\mu_{\mathrm{B}}\) is the Bohr magneton. (c) Estimate the magnetic susceptibility of the salt per mole of praseodymium for these values of \(B\) and \(T\).

Step by step solution

01

Identify Electron Configuration

The trebly ionized praseodymium ion (Pr鲁鈦�) has its outer electrons removed, leaving two electrons in the 4f shell, which contributes to the ion's magnetic properties.

02

Apply Hund's Rules 鈥� Determine L, S, J

1. **Maximize Spin (S):** Each f-electron has parallel spins, so for two unpaired electrons, S = 1. 2. **Maximize Orbital (L):** Electrons in the f orbital have values l = 3, and with two unpaired electrons, they maximize L = 5. 3. **Calculate J:** For less than half-filled shell, J = |L - S| = 4.

03

Calculate the Land茅 g-factor

The Land茅 g-factor formula is: \[ g = 1 + \frac{J(J + 1) + S(S + 1) - L(L + 1)}{2J(J + 1)} \]Substitute L = 5, S = 1, J = 4 to find \[ g = 1 + \frac{4(5) + 1(2) - 5(6)}{2(4)(5)} = 0.8 \]

04

Verify Expression for Thermal Average of Magnetic Moment

The thermal average of the magnetic moment component can be obtained using the formula \[ \bar{\mu} = \frac{1}{3} g^{2} \mu_{B}^{2} \frac{J(J+1) B}{k_{\mathrm{B}} T} \]Plugging known values of g, J, B, and T into this equation matches the given expression.

05

Estimate Magnetic Susceptibility

Magnetic susceptibility 蠂 is related by\[ \chi = \frac{N \mu_0 \mu_B^2 g^2 J(J+1)}{3k_B T} \]where N is Avogadro's number. Calculate 蠂 using given values of T = 300 K, and established parameters for B and other constants.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hund's Rules

Hund鈥檚 rules are a set of guidelines that help us determine the electronic configuration of atoms and ions, specifically in terms of their energy levels, based on spin multiplicity and orbital filling. Here are some key points to understand Hund's rules:

• **Maximize Spin (S):** The ground state of an atom or ion is such that the spin multiplicity is maximized. In simple terms, this means that the number of unpaired electrons (having parallel spins) should be maximized. For example, in the praseodymium ion case, two electrons in the 4f orbital both have parallel spins, leading to a total spin angular momentum (S) of 1.
• **Maximize Orbital (L):** The electrons should fill the orbitals in a way that the total orbital angular momentum (L) is maximized. For f-electrons, the orbital angular momentum quantum number (l) is 3. For the praseodymium ion with two electrons, L is calculated to be 5.
• **Determine Total Angular Momentum (J):** Finally, for less than half-filled shells (as with praseodymium), we calculate J as the absolute difference between L and S: \( J = |L - S| \). Thus, J is 4 in this scenario.

These rules help in predicting the energy states of electrons and are important for understanding atomic spectra and magnetic properties.

Land茅 g-factor

The Land茅 g-factor is a dimensionless quantity that describes the ratio between the magnetic moment and the angular momentum of an atom or ion. It essentially defines how a magnetic moment interacts with an external magnetic field. Here's how the Land茅 g-factor is calculated and why it matters:

• The general formula to calculate the Land茅 g-factor is: \[ g = 1 + \frac{J(J + 1) + S(S + 1) - L(L + 1)}{2J(J + 1)} \]
• This formula takes into account the contributions from both the orbital angular momentum (L) and the spin angular momentum (S), along with the total angular momentum (J).
• In the case of the praseodymium ion, substituting the known values (L = 5, S = 1, and J = 4), the Land茅 g-factor is found to be 0.8.

The Land茅 g-factor is crucial in calculating the effects of magnetic fields on atomic ions, such as determining the energy level splitting (Zeeman effect) and predicting the magnetic behavior of the ions.

Magnetic Susceptibility

Magnetic susceptibility is a measure of how much a material will become magnetized in an applied magnetic field. It's an important concept in understanding the magnetic behavior of materials. Here's what you need to know:

• Magnetic susceptibility \( \chi \) is essentially the ratio of the material's magnetization to the applied magnetic field strength.
• For a paramagnetic material like the praseodymium salt in question, an equation used to relate susceptibility is: \[ \chi = \frac{N \mu_0 \mu_B^2 g^2 J(J+1)}{3k_B T} \]
• Where \( N \) is Avogadro's number, \( \mu_0 \) is the permeability of free space, \( \mu_B \) is the Bohr magneton, \( g \) is the Land茅 g-factor, \( J \) is the total angular momentum quantum number, and \( T \) is the temperature in Kelvins.

Using known values of temperature (300 K), the Land茅 g-factor calculated earlier, and constants, you can estimate the magnetic susceptibility of praseodymium ions at given conditions. This helps in understanding how strongly the material will respond to magnetic fields.

Electron Configuration

Electron configuration refers to the arrangement of electrons in an atom or ion, detailing how electrons occupy the various orbitals. Understanding this concept is key to predicting chemical properties, bonding, and magnetism.

• The electron configuration is determined by following the **Aufbau Principle**, Hund's Rules, and the Pauli Exclusion Principle.
• For trebly ionized praseodymium (Pr鲁鈦�), electrons have been removed such that there are only two electrons left in the 4f shell.
• The **Aufbau Principle** suggests filling orbitals starting from the lowest energy level up. However, many transition metals and lanthanides can have electrons in higher subshells, reflecting their complex electron-electron interactions.
• Understanding electron configurations also provides insights into magnetic and spectra properties, allowing for determination of properties like oxidation states and reactivity.

By understanding electron configurations, we can predict and explain the unique magnetic properties seen in elements like praseodymium, especially in its various ionized forms.