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Chapter 5: Problem 44

Assign a correct set of four quantum numbers for (a) Each electron in a nitrogen atom. (b) The valence electron in a sodium atom. (c) A \(3 d\) electron in a nickel atom.

Short Answer

Expert verified
For (a) nitrogen's 2p electron: (2, 1, +1, +1/2). For (b) sodium's valence electron: (3, 0, 0, +1/2). For (c) nickel's 3d electron: (3, 2, +2, +1/2).

Step by step solution

01

Understand Quantum Numbers

Quantum numbers describe the unique quantum state of an electron in an atom. The four quantum numbers are: 1. **Principal quantum number ( ) (n)**: Represents the energy level and size of the orbital. 2. **Azimuthal quantum number ( ) (l)**: Represents the shape of the orbital. 3. **Magnetic quantum number ( ) (m_l)**: Represents the orientation of the orbital. 4. **Spin quantum number ( ) (m_s)**: Represents the spin of the electron.
02

Solve for Nitrogen Atom

A nitrogen atom has 7 electrons. The electron configuration is 1s^2 2s^2 2p^3. - For the last electron (3rd in 2p): 1. : **2** (since 2p is the second shell) 2. : **1** (since p orbitals have =1) 3. : **-1, 0, or +1** (choosing **+1** for the last filled as it represents one of the three orbitals px, py, or pz) 4. : **+1/2** (assuming spin up) Thus, (2, 1, +1, +1/2) is one possible set of quantum numbers for one of the 2p electrons in nitrogen.
03

Solve for Sodium Valence Electron

A sodium atom has 11 electrons with electron configuration 1s^2 2s^2 2p^6 3s^1. The valence electron is the 3s^1 electron: - For this electron: 1. : **3** (since 3s is in the third shell) 2. : **0** (since s orbitals have =0) 3. : **0** (only one orientation for s, so =0) 4. : **+1/2** (assuming spin up) The quantum numbers for the valence electron in sodium are (3, 0, 0, +1/2).
04

Solve for Nickel's 3d Electron

A nickel atom has 28 electrons and configuration 1s^2 2s^2 2p^6 3s^2 3p^6 3d^8 4s^2. Focus on a 3d electron: - For a 3d electron: 1. : **3** (as it is in the third shell) 2. : **2** (since d orbitals have =2) 3. : **-2, -1, 0, +1, +2** (choosing one value, e.g., **+2**) 4. : **+1/2** The quantum numbers for one of nickel鈥檚 3d electrons might be (3, 2, +2, +1/2).

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Key Concepts

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

Principal Quantum Number
The principal quantum number, symbolized as \( n \), is fundamental in determining an electron's position within an atom. It highlights the energy level or shell where the electron resides. Think of \( n \) as the floor of a building where an electron 'lives.'
  • \( n \) ranges from 1 to infinity.
  • Higher values indicate electrons are further from the nucleus and possess more energy.
  • For example, if \( n = 2 \), the electron is in the second energy level.
This number significantly influences the size and energy of the orbital. At its core, \( n \) lays the groundwork for understanding how electrons are arranged within an atom, setting the stage for interpreting more complex quantum numbers.
Azimuthal Quantum Number
The azimuthal quantum number, also known as the angular momentum quantum number, is denoted by \( l \). It provides insights into the shape of an electron's orbital. This number helps us understand the variety of ways electrons can orbit the nucleus.
  • \( l \) can take on integer values from 0 to \( n-1 \).
  • Each \( l \) value corresponds to a specific type of orbital:
    • \( l = 0 \) represents an s orbital (spherical shape).
    • \( l = 1 \) indicates a p orbital (dumbbell shape).
    • \( l = 2 \) stands for a d orbital (clover shape).
    • \( l = 3 \) signifies an f orbital (complex shape).
For example, in a nitrogen atom with a 2p electron, \( l \) would be 1, indicating a p-shaped orbital within the second energy level.
Magnetic Quantum Number
The magnetic quantum number, represented as \( m_l \), describes the orientation of an electron's orbital in space. This helps pinpoint the exact 'seat' an electron occupies within its 'orbital classroom.'
  • \( m_l \) values can range from \(-l\) to \(+l\).
  • It determines the number of orbitals and their spatial orientation for a given \( l \).
    • For \( l = 1 \) (p orbitals), \( m_l \) can be \(-1, 0, +1\); these numbers refer to the px, py, and pz orbitals.
In the nitrogen example above, a 2p electron with \( m_l = +1 \) is located in one of those possible orientations. This specificity enables a more detailed mapping of where electrons might exist within an atom.
Spin Quantum Number
The spin quantum number, denoted as \( m_s \), is unique because it describes the intrinsic "spin" or rotation of an electron along its axis. Spin is a fundamental property that significantly impacts magnetic behavior.
  • This number has only two possible values: \(+\frac{1}{2}\) or \(-\frac{1}{2}\).
  • These values signify the electron's spin direction, either "spin up" (\(+\)1/2) or "spin down" (\(-\)1/2).
Imagine the spin quantum number like an electron's mood, only ever being 'positive' or 'negative,' never in-between. Spin forms the foundation of electron pairing, as electrons in the same orbital must have opposite spins to comply with the Pauli exclusion principle.

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Most popular questions from this chapter

Arrange these ions in order of decreasing size: \(\mathrm{Be}^{2+}\), \(\mathrm{Rb}^{+}, \mathrm{C}_{\mathrm{a}}^{2+}, \mathrm{Mg}^{2+}\)

Explain why nitrogen has a higher first ionization energy than does carbon, the element that precedes it in the periodic table.

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How many subshells are there in the electron shell with the principal quantum number \(n=4 ?\)

The ionization energy of \(\mathrm{H}(\mathrm{g})\) is \(1312 \mathrm{~kJ} / \mathrm{mol}\). The second ionization energy of \(\mathrm{He}(\mathrm{g})\) is almost exactly four times this value, and the third ionization energy of \(\mathrm{Li}(\mathrm{g})\) is almost exactly nine times that of hydrogen. What relationship, if any, do these data suggest between ionization energy and a fundamental characteristic of elements?
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