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.

Short answer

Neon has 10 electrons with quantum numbers: 1s: (1,0,0,+\(\frac{1}{2}\)), (1,0,0,-\(\frac{1}{2}\)); 2s: (2,0,0,+\(\frac{1}{2}\)), (2,0,0,-\(\frac{1}{2}\)); 2p: (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-by-step solution

Understand the Quantum Numbers

The four quantum numbers are: 1. Principal quantum number \( n \), which describes the energy level.2. Angular momentum quantum number \( l \), which describes the shape of the orbital.3. Magnetic quantum number \( m_l \), which describes the orientation of the orbital.4. Spin quantum number \( m_s \), which describes the spin of the electron and can be \( +\frac{1}{2} \) or \( -\frac{1}{2} \).

Determine Total Electrons

Neon has an atomic number of 10, meaning it has 10 electrons in its ground state.

Assign Quantum Numbers for Electrons in the 1s Orbital

The 1s orbital can hold 2 electrons. - Electron 1: \( n = 1, l = 0, m_l = 0, m_s = +\frac{1}{2} \)- Electron 2: \( n = 1, l = 0, m_l = 0, m_s = -\frac{1}{2} \)

Assign Quantum Numbers for Electrons in the 2s Orbital

The 2s orbital also holds 2 electrons. - Electron 3: \( n = 2, l = 0, m_l = 0, m_s = +\frac{1}{2} \)- Electron 4: \( n = 2, l = 0, m_l = 0, m_s = -\frac{1}{2} \)

Assign Quantum Numbers for Electrons in the 2p Orbital - First Pair

The 2p orbital can hold a total of 6 electrons, divided into 3 subshells. For the first subshell:- Electron 5: \( n = 2, l = 1, m_l = -1, m_s = +\frac{1}{2} \)- Electron 6: \( n = 2, l = 1, m_l = -1, m_s = -\frac{1}{2} \)

Assign Quantum Numbers for Electrons in the 2p Orbital - Second Pair

For the second subshell:- Electron 7: \( n = 2, l = 1, m_l = 0, m_s = +\frac{1}{2} \)- Electron 8: \( n = 2, l = 1, m_l = 0, m_s = -\frac{1}{2} \)

Assign Quantum Numbers for Electrons in the 2p Orbital - Final Pair

For the final subshell:- Electron 9: \( n = 2, l = 1, m_l = +1, m_s = +\frac{1}{2} \)- Electron 10: \( n = 2, l = 1, m_l = +1, m_s = -\frac{1}{2} \)

Key concepts

Neon Atom

Neon is a chemical element with an atomic number of 10. This means it has 10 protons and normally 10 electrons. Being a noble gas, neon is known for its lack of chemical reactivity, as it has a full outer electron shell. This full valence shell is a key reason why neon doesn't readily form compounds with other elements. In its atomic structure, neon has electrons arranged in such a way that the inner and outer shells are filled to capacity. Understanding neon's electron composition is essential for grasping its nearly inert nature.

Electron Configuration

Electron configuration refers to the arrangement of electrons in an atom's electron shells. For neon, with 10 electrons, the electrons fill the shells in a specified order: it has a complete 1s orbital with 2 electrons, and a complete 2s orbital with 2 electrons, plus the full 2p orbitals containing 6 electrons. In notation form, neon's electron configuration is written as:

• 1s²
• 2s²
• 2p⁶

This order of filling is due to the rules that govern how electrons occupy available atomic orbitals based on increasing energy, known as the Aufbau principle. The filled electron configuration of neon gives it a stable and low-energy state.

Atomic Orbitals

Atomic orbitals can be thought of as the regions around an atom's nucleus where the electrons are most likely to be found. They are defined by sets of quantum numbers. Each type of orbital has a specific shape and is named accordingly:

• 's' orbitals are spherical
• 'p' orbitals are dumbbell-shaped

The number of these orbitals increases with each energy level. In neon, the 1s orbital houses two electrons, and the 2s orbital also contains two electrons. The 2p orbital, made up of three subshells, contains six electrons. Each orbital's properties, such as its shape and number, influence how atoms interact to form molecules and how chemical reactions take place.

Ground State

An atom is in its ground state when its electrons exist in the lowest possible energy levels. For the neon atom, this is its most stable state since its 10 electrons fill the 1s, 2s, and 2p orbitals completely. This minimization of energy results in a lack of reactivity, characteristic of neon as a noble gas. Electrons in the ground state are not excited; thus, they are located in their respective atomic orbitals as per the specified quantum numbers:

• 1s:
  n = 1, l = 0, m_l = 0, m_s = +\( \frac{1}{2} \), -\( \frac{1}{2} \) for 2 electrons
• 2s:
  n = 2, l = 0, m_l = 0, m_s = +\( \frac{1}{2} \), -\( \frac{1}{2} \) for 2 electrons
• 2p:
  n = 2, l = 1, m_l = -1, 0, +1, m_s = +\( \frac{1}{2} \), -\( \frac{1}{2} \) for 6 electrons

Understanding an atom's ground state helps in visualizing its potential energy and chemical stability.