Problem 76 - Chemistry: Principles and Reactions

Question

The energy of any one-electron species in its nth state \((\mathbf{n}=\) principal quantum number) is given by \(E=-B Z^{2} / \mathbf{n}^{2}\), where \(Z\) is the charge on the nucleus and \(B\) is \(2.180 \times 10^{-18} \mathrm{~J}\). Find the ionization energy of the \(\mathrm{Li}^{2+}\) ion in its first excited state in kilojoules per mole.

Step-by-Step Solution

Answer

Answer: The ionization energy of the Li²⁺ ion in its first excited state is approximately -267.0 kJ/mol.

Step 1: Identify the values for the Li²⁺ ion

The Li²⁺ ion has a charge on the nucleus (Z) of 3, as it is lithium with two electrons removed. The first excited state means that the principal quantum number (n) is 2. The constant B is given as 2.180 × 10⁻¹⁸ J.

Step 2: Calculate the energy in the n=1 state

Using the given formula, we plug in the values for Z, n, and B: E₁ = -B * Z² / n² E₁ = -2.180 × 10⁻¹⁸ J * (3²) / (1²) E₁ = -5.914 × 10⁻¹⁸ J

Step 3: Calculate the energy in the n=2 state

Now we find the energy in the n=2 state: E₂ = -B * Z² / n² E₂ = -2.180 × 10⁻¹⁸ J * (3²) / (2²) E₂ = -1.479 × 10⁻¹⁸ J

Step 4: Calculate the ionization energy

The ionization energy is the energy required to remove the electron, which is the difference between the energy in the n=2 state and n=1 state: Ionization energy = E₁ - E₂ Ionization energy = (-5.914 × 10⁻¹⁸ J) - (-1.479 × 10⁻¹⁸ J) Ionization energy = -4.435 × 10⁻¹⁸ J

Step 5: Convert joules to kilojoules per mole

To convert the ionization energy in joules to kilojoules per mole, we first divide by 1000 to get the energy in kilojoules: Ionization energy = -4.435 × 10⁻²¹ kJ Next, we multiply by Avogadro's number (6.022 × 10²³) to convert to kilojoules per mole: Ionization energy = -4.435 × 10⁻²¹ kJ * (6.022 × 10²³) Ionization energy = -267.0 kJ/mol Thus, the ionization energy of the Li²⁺ ion in its first excited state is approximately -267.0 kJ/mol.

Key Concepts

Principal Quantum Number

In quantum mechanics, the concept of the principal quantum number, often denoted as \( n \), helps us understand the energy levels of electrons in an atom. Imagine it as a ladder, where each rung represents a specific energy level or shell. The principal quantum number is a positive integer, starting from 1 and going upwards. Each number indicates an energy level that's farther from the nucleus.
The principal quantum number determines:

• The size of the electron's orbit, with larger values of \( n \) representing larger orbits.
• The energy state of the electron, where higher \( n \) values correspond to higher energy levels.

This concept is essential when calculating the energy states of electrons, as seen in the calculations for one-electron species like the \( ext{Li}^{2+} \) ion, where \( n \) identifies the electron's position in an energy level like the first excited state (\( n=2 \)). The level of energy increases as \( n \) increases. This stepping up in energy is critical for computing ionization energies, which require knowledge of the specific \( n \) level an electron occupies.

One-Electron Species

One-electron species refers to atoms or ions that have a single electron around the nucleus. Such species are simplified systems that are easier to study and calculate, making them ideal for understanding fundamental quantum mechanics principles.
Examples of one-electron species include:\( ext{H}^0\) (a hydrogen atom), \( ext{He}^+\) (helium with one electron stripped away), and \( ext{Li}^{2+}\) (lithium stripped of two electrons).

• In one-electron species, calculations for energy states rely heavily on quantum mechanical models, because there's no electron-electron interaction to consider.
• The focus is on the interaction between the single electron and the nucleus, making it an excellent model for learning about atomic structures and behaviors.

The calculation of the energy for a one-electron species in different states helps illustrate the broader concept of energy transitions and ionization energies, using clear and relevant mathematical representations.

Nucleus Charge

The nucleus charge \( Z \) is a fundamental aspect in calculating the energy levels and ionization energies of atoms and ions. It refers to the effective nuclear charge that the electron feels due to the protons in the nucleus. For a given element, \( Z \) corresponds to the number of protons in the nucleus.
In the context of one-electron ions like \( ext{Li}^{2+} \), the nucleus charge is still the same as that of a neutral Li atom, which is 3 protons. This charge affects the electron's binding energy level:

• The formula \( E = -B Z^2 / n^2 \) uses \( Z \) to determine how tightly the electron is bound to the nucleus.
• Higher nucleus charges result in a stronger attraction, increasing the energy required to remove an electron.

Thus, understanding nucleus charge is key when using the formula to find the energy states of electrons, especially for calculating the ionization energy of ions in different states or configurations.

Energy States

Energy states are the various fixed energy levels that an electron in an atom or ion can occupy. Each energy state corresponds to a principal quantum number, \( n \), and represents a distinct level of energy the electron can have. These states play a crucial role in the quantum description of atoms.
The energy of an electron in a given state is influenced by:

• The principal quantum number, \( n \), as it's related to the shell or level the electron occupies.
• The nucleus charge, which affects how much energy is needed to maintain the electron in that state.

The significance of energy states is evident in processes such as ionization, where electrons are moved from one state to another, such as from the "first excited state" \( n=2 \) to the ionization limit where the electron is free from the atom. Calculating changes between these states helps us understand the amount of energy needed for ionization and other electronic transitions.