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Chapter 41: Problem 27

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

Expert verified
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

01

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 \( -\frac{1}{2} \).
02

Determine Total Electrons

Neon has an atomic number of 10, meaning it has 10 electrons in its ground state.
03

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: \( n = 1, l = 0, m_l = 0, m_s = -\frac{1}{2} \)
04

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: \( n = 2, l = 0, m_l = 0, m_s = -\frac{1}{2} \)
05

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: \( n = 2, l = 1, m_l = -1, m_s = -\frac{1}{2} \)
06

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: \( n = 2, l = 1, m_l = 0, m_s = -\frac{1}{2} \)
07

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: \( n = 2, l = 1, m_l = +1, m_s = -\frac{1}{2} \)

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Key Concepts

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

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:
  • 1蝉虏
  • 2蝉虏
  • 2辫鈦�
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.

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Most popular questions from this chapter

Assume that the researchers place an atom in a state with \(n\) = 100, \(l\) = 2. What is the magnitude of the orbital angular momentum \(L\) associated with this state? (a) \(\sqrt{2} \space\hslash \); (b) \(\sqrt{6} \space\hslash\); (c) \(\sqrt{200}\space \hslash\); (d) \(\sqrt{10,100}\space \hslash \).

(a) Write out the ground-state electron configuration (\(1s^2, 2s^2,\dots\)) for the carbon atom. (b) What element of nextlarger \(Z\) has chemical properties similar to those of carbon? Give the ground-state electron configuration for this element.

A hydrogen atom in a particular orbital angular momentum state is found to have \(j\) quantum numbers \({7\over2}\) and \({9\over2}\) . (a) What is the letter that labels the value of \(l\) for the state? (b) If \(n = 5\), what is the energy difference between the \(j = {7\over2}\) and \(j = {9\over2}\) levels?

A particle is described by the normalized wave function \(\psi$$(x, y, z)\) = \(Axe$${^-}{^a}{^x}^2$$e$${^-}{^\beta}$${^y}^2$$e$${^-}{^y}^z$$^2\), where \(A\), \(\alpha\),\(\beta\), and \(\gamma\) are all real, positive constants. The probability that the particle will be found in the infinitesimal volume \(dx\) \(dy\) \(dz\) centered at the point \((x_0\), \(y_0\), \(z_0\)) is \(\mid$$\psi$$(x_0\), \(y_0\), \(z_0\))\(\mid$$^2\) \(dx\) \(dy\) \(dz\). (a) At what value of \(x_0\) is the particle most likely to be found? (b) Are there values of \(x_0\) for which the probability of the particle being found is zero? If so,at what \(x_0$$?\)

You are studying the absorption of electromagnetic radiation by electrons in a crystal structure. The situation is well described by an electron in a cubical box of side length \(L\). The electron is initially in the ground state. (a) You observe that the longest-wavelength photon that is absorbed has a wavelength in air of \(\lambda\) = 624 nm. What is \(L\)? (b) You find that \(\lambda\) = 234 nm is also absorbed when the initial state is still the ground state. What is the value of \(n$$^2\) for the final state in the transition for which this wavelength is absorbed, where \(n$$^2\) = \(n$$_X^2\) + \(n$$_y^2\) + \(n$$_z^2\) ? What is the degeneracy of this energy level (including the degeneracy due to electron spin)?
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