Problem 27 - Fundamentals of Physics

Question

Two of the three electrons in a lithium atom have quantum numbers \(\left(n, \ell, m_{\varepsilon}, m_{s}\right)\) of \(\left(1,0,0,+\frac{1}{2}\right)\) and \(\left(1,0,0,-\frac{1}{2}\right)\). What quantum numbers are possible for the third electron if the atom is (a) in the ground state and (b) in the first excited state?

Step-by-Step Solution

Answer

(a) Ground state: \((2, 0, 0, \pm \frac{1}{2})\). (b) First excited state: \((2, 1, -1, \pm \frac{1}{2})\), \((2, 1, 0, \pm \frac{1}{2})\), \((2, 1, +1, \pm \frac{1}{2})\).

Step 1: Understanding Quantum Numbers

Each electron in an atom is described by a set of four quantum numbers: the principal quantum number \(n\), the azimuthal or angular momentum quantum number \(\ell\), the magnetic quantum number \(m_{\ell}\), and the spin quantum number \(m_{s}\). These numbers describe the energy level, shape, orientation, and spin of the electron's orbital.

Step 2: Analyzing the Ground State of Lithium

In its ground state, a lithium atom has its electrons in the lowest possible energy levels. The first two electrons occupy the 1s orbital with quantum numbers \((n, \ell, m_{\ell}, m_{s}) = (1, 0, 0, +\frac{1}{2})\) and \((1, 0, 0, -\frac{1}{2})\). The third electron will occupy the next available orbital, which is 2s. Thus, it has the quantum numbers \((2, 0, 0, +\frac{1}{2})\) or \((2, 0, 0, -\frac{1}{2})\).

Step 3: Analyzing the First Excited State of Lithium

In the first excited state, one of the electrons moves to a higher energy orbital. Commonly, the third electron (which is in 2s in the ground state) will be excited to the 2p orbital. For a 2p orbital, \(\ell = 1\), so the possible quantum numbers \(m_{\ell}\) range from \(-1\) to \(+1\). The third electron can have quantum numbers like \((2, 1, -1, \pm \frac{1}{2})\), \((2, 1, 0, \pm \frac{1}{2})\), and \((2, 1, +1, \pm \frac{1}{2})\).

Step 4: Verifying Electron Distribution

For the possible quantum states of the third electron, check if their configurations respect the Pauli Exclusion Principle and Hund's Rule. The ground state configuration \((2, 0, 0, \, \pm \frac{1}{2})\) follows the exclusion principle since the 1s orbital is filled. In the excited state, the distribution where the electron moves to 2p is consistent with these principles, taking care to accommodate the available \(m_{\ell}\) values.

Key Concepts

Lithium Atom

A lithium atom is quite intriguing due to its simplicity and the complexity its electrons exhibit. Lithium, with an atomic number of 3, contains three electrons. These electrons are arranged in energy levels around the nucleus. Each electron can be characterized by a set of quantum numbers that defines its position and energy within the atom.
- The first two electrons in lithium occupy the 1s orbital, which is the lowest energy level and closest to the nucleus. - The third electron, in its ground state, takes a position in the 2s orbital, just a step above the first two. This configuration sets lithium apart as a typical atom in the alkali metals group, well-known for being highly reactive due to the presence of this one valence electron. Understanding how these electrons fill the 1s and 2s orbitals helps explain lithium's chemical properties.

Ground State

The ground state of an atom is its most stable, lowest-energy configuration. For lithium, this means the two electrons fill the 1s orbital entirely before the third electron starts to fill the next available orbital, which is the 2s.
- In this ground state, the third electron's quantum numbers become (2, 0, 0, +\(\frac{1}{2}\)) or (2, 0, 0, -\(\frac{1}{2}\)).- This configuration efficiently fills the 1s orbital first, minimizing the atom's energy level and maintaining stability.By aligning its electrons in this ordered manner, lithium exhibits predictable behavior that can be used to understand and predict its reactivity and interactions with other elements.

Excited State

When a lithium atom is in an excited state, one or more electrons have moved to a higher energy level than in the ground state. This transition occurs when the atom absorbs energy. In lithium's case, the third electron generally jumps from the 2s to the 2p orbital.
- For the 2p orbital, the azimuthal quantum number \(\ell\) is 1, and the magnetic quantum number \(m_\ell\) ranges from -1 to +1, providing a variety of new configurations.- The possible quantum numbers for this excited electron might be (2, 1, -1, \(\pm\frac{1}{2}\)), (2, 1, 0, \(\pm\frac{1}{2}\)), or (2, 1, +1, \(\pm\frac{1}{2}\)).In this higher energy state, the lithium atom becomes more reactive because its electrons are in a less stable configuration.

Pauli Exclusion Principle

The Pauli Exclusion Principle is a fundamental rule in quantum mechanics. It dictates that no two electrons in an atom can have the same set of all four quantum numbers. This principle explains the electron configurations in atoms, including lithium, and ensures stability.
- In the ground state of lithium, the first two electrons can share the same \(n\), \(\ell\), and \(m_\ell\) as they have opposite spins, denoted by \(m_s\) being +\(\frac{1}{2}\) and -\(\frac{1}{2}\).- As the electrons fill the lowest energy orbitals first, this principle helps prevent electrons from "crowding" into the same state.- For the excited state, even when electrons jump to higher orbitals, this rule ensures that they find distinct positions that do not replicate existing configurations.Understanding the Pauli Exclusion Principle aids in grasping how and why electrons configure themselves within an atom, guiding reactions and properties of elements.