Problem 27 - University Physics with Modern Physics

Question

Make a list of the four quantum numbers \(n, l, m_{l},\) and \(m_{s}\) for each of the 10 electrons in the ground state of the neon atom Do not refer to Table 41.2 or \(41.3 .\)

Step-by-Step Solution

Answer

The quantum numbers for the 10 electrons are: \((1,0,0,+\frac{1}{2})\), \((1,0,0,-\frac{1}{2})\), \((2,0,0,+\frac{1}{2})\), \((2,0,0,-\frac{1}{2})\), \((2,1,-1,+\frac{1}{2})\), \((2,1,-1,-\frac{1}{2})\), \((2,1,0,+\frac{1}{2})\), \((2,1,0,-\frac{1}{2})\), \((2,1,1,+\frac{1}{2})\), \((2,1,1,-\frac{1}{2})\).

Step 1: Understanding Quantum Numbers

In quantum mechanics, each electron in an atom is described by a set of four quantum numbers: the principal quantum number \(n\), the azimuthal quantum number \(l\), the magnetic quantum number \(m_l\), and the spin magnetic quantum number \(m_s\). These numbers specify the electron's energy level, shape, orientation, and spin respectively.

Step 2: Identifying Quantum Numbers for Electrons

Neon has 10 electrons, which fill the \(1s\), \(2s\), and \(2p\) orbitals. For each orbital, we assign quantum numbers based on the electron configuration.

Step 3: Assigning Quantum Numbers for 1s Electrons

The first two electrons in neon are located in the \(1s\) orbital. For these electrons, \(n = 1\), \(l = 0\), \(m_l = 0\). The possible values for \(m_s\) are \(+\frac{1}{2}\) and \(-\frac{1}{2}\). Thus, the quantum numbers are: \((1, 0, 0, +\frac{1}{2})\) and \((1, 0, 0, -\frac{1}{2})\).

Step 4: Assigning Quantum Numbers for 2s Electrons

The next two electrons are in the \(2s\) orbital. For these electrons, \(n = 2\), \(l = 0\), \(m_l = 0\). The possible values for \(m_s\) are \(+\frac{1}{2}\) and \(-\frac{1}{2}\). Thus, the quantum numbers are: \((2, 0, 0, +\frac{1}{2})\) and \((2, 0, 0, -\frac{1}{2})\).

Step 5: Assigning Quantum Numbers for 2p Electrons

The remaining six electrons fill the \(2p\) orbital. For these electrons, \(n = 2\) and \(l = 1\). The possible values for \(m_l\) are \(-1\), \(0\), and \(1\). For each \(m_l\), \(m_s\) can be \(+\frac{1}{2}\) and \(-\frac{1}{2}\). The quantum numbers are: \((2, 1, -1, +\frac{1}{2})\), \((2, 1, -1, -\frac{1}{2})\), \((2, 1, 0, +\frac{1}{2})\), \((2, 1, 0, -\frac{1}{2})\), \((2, 1, 1, +\frac{1}{2})\), \((2, 1, 1, -\frac{1}{2})\).

Key Concepts

Quantum Mechanics

Quantum mechanics is the branch of physics that studies the behavior of very small particles, such as electrons, which do not follow the laws of classical physics. At this microscopic level, particles present unpredictable behaviors, and their positions and velocities cannot be determined simultaneously with absolute precision. Instead of having a specific place and momentum, particles are described by probabilities. This probabilistic nature is one of the core ideas of quantum mechanics.

In the context of chemistry, quantum mechanics provides us a way to understand how electrons are arranged in an atom, which is essential for comprehending chemical reactions and properties. Each electron is described using a set of quantum numbers, which serve as a unique address for the location and behavior of the electron within an atom.

• The principal quantum number \(n\) indicates the main energy level of the electron.
• The azimuthal quantum number \(l\) defines the shape of the electron's orbital within that energy level.
• The magnetic quantum number \(m_{l}\) describes the orientation of the orbital in space.
• The spin magnetic quantum number \(m_{s}\) represents the direction of the electron's spin.

These quantum numbers set the foundation for understanding electron arrangements and how atoms interact with one another.

Electron Configuration

Electron configuration is a way to describe the distribution of electrons in the orbitals of an atom. Knowing how these electrons are arranged allows us to predict the chemical properties of the element. It shows us which orbitals are occupied and uses a specific notation that encompasses the orbital name and the number of electrons in that orbital.

For example, the electron configuration of neon—an element with 10 electrons—is written as 1s\(^2\) 2s\(^2\) 2p\(^6\). This notation tells us that:

• Two electrons fill the 1s orbital,
• Two electrons fill the 2s orbital, and
• Six electrons fill the 2p orbital.

Following this configuration, electrons first fill lower-energy orbitals before higher-energy ones, in accordance with the Aufbau principle. The order of filling is 1s, 2s, 2p, 3s, 3p, and so on, progressing from lower to higher energy. Understanding electron configuration aids in grasping how elements form compounds and engage in various chemical reactions.

Atomic Orbitals

Atomic orbitals are regions around the nucleus of an atom where electrons are likely to be found. The shape and orientation of these orbitals are determined by the quantum numbers, specifically \(l\) and \(m_{l}\). Each type of orbital has a distinct shape:

• s-orbitals are spherical and non-directional.
• p-orbitals have a dumbbell shape with three possible orientations in space, corresponding to \(m_{l} = -1, 0,\) and \(+1\).
• d-orbitals and f-orbitals have more complex shapes and are not typically occupied in neon.

Orbital levels dictate how electrons are distributed. Lower energy orbitals like 1s and 2s are filled before the higher-energy 2p orbitals, as seen in neon's electron configuration. As these orbitals fill, they are populated by electrons according to Hund’s rule, which states that electrons will fill an empty orbital before they pair up. Understanding atomic orbitals is key in explaining the physical and chemical properties of various elements.