Problem 43

Question

Assign a correct set of four quantum numbers for (a) Each electron in a boron atom. (b) The $3 s$ electrons in a magnesium atom. (c) A $3 d$ electron in an iron atom.

Step-by-Step Solution

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Answer

Quantum numbers are specific to orbital configurations: Boron (5 electrons), Magnesium 3s, Iron 3d.

1Step 1: Understand Atomic Structure and Electron Configuration

Boron has an atomic number of 5, meaning it has 5 electrons. Magnesium has an atomic number of 12, thus it has 12 electrons, and iron has an atomic number of 26, with 26 electrons.

2Step 2: Write Electron Configurations

The electron configuration for boron is \[1s^2 2s^2 2p^1\]. For magnesium, it's \[1s^2 2s^2 2p^6 3s^2\]. For iron, it's \[1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^6\].

3Step 3: Assign Quantum Numbers for Boron Electrons

Quantum numbers ( , l, m_l, m_s) for boron: - 1s electrons: 1: (1, 0, 0, +1/2) and 2: (1, 0, 0, -1/2) - 2s electrons: 3: (2, 0, 0, +1/2) and 4: (2, 0, 0, -1/2) - 2p electron: 5: (2, 1, -1, +1/2)

4Step 4: Assign Quantum Numbers for Magnesium 3s Electrons

The magnesium 3s orbital can hold two electrons. Their quantum numbers are: - First 3s electron: (3, 0, 0, +1/2) - Second 3s electron: (3, 0, 0, -1/2)

5Step 5: Assign Quantum Numbers for an Iron 3d Electron

For the 3d orbital of iron: - Consider one 3d electron: Using example of (3, 2, +2, +1/2) Note: There are multiple configurations possible for 3d orbitals.

Key Concepts

Boron atom electron configurationMagnesium atom electron configurationIron atom electron configuration

Boron atom electron configuration

To understand the electron configuration of a boron atom, we start by noting that boron has an atomic number of 5. This means it has 5 electrons. The distribution of these electrons follows the Aufbau principle, which dictates that electrons fill orbitals starting from the lowest energy levels.

The electron configuration for boron is given as:

$1s^2$, which means the first energy level ($n=1$) has a filled $s$ orbital with 2 electrons.
$2s^2$, indicating that the second energy level ($n=2$) also has a filled $s$ orbital.
$2p^1$, showing that there is one electron in the $p$ orbital of the second energy level.

For each electron in a boron atom, we assign quantum numbers:

1s electrons: (1, 0, 0, +1/2) and (1, 0, 0, -1/2)
2s electrons: (2, 0, 0, +1/2) and (2, 0, 0, -1/2)
2p electron: (2, 1, -1, +1/2)

This setup helps understand the specific energy state of each electron. Each set of quantum numbers describes an electron's energy (), angular momentum ($l$), magnetic orientation ($m_l$), and spin ($m_s$).

Magnesium atom electron configuration

Magnesium has an atomic number of 12, meaning it possesses 12 electrons. The electron configuration follows the same Aufbau principle method used for boron.

The electron configuration for magnesium is:

$1s^2$ - 2 electrons in the first energy level.
$2s^2$ and $2p^6$ - a total of 8 electrons in the second energy level.
$3s^2$ - 2 electrons in the third energy level.

Considering the 3s electrons, we assign the quantum numbers:

For the first 3s electron: (3, 0, 0, +1/2)
For the second 3s electron: (3, 0, 0, -1/2)

The $n=3$ makes these electrons part of the third energy level. The $l=0$ indicates they are in an $s$ orbital. The $m_s$ terms, +1/2 and -1/2, represent the spin orientations of these two electrons.

Iron atom electron configuration

With an atomic number of 26, an iron atom includes 26 electrons. Following the Aufbau principle, these electrons fill in various energy levels and orbitals, represented as:\[1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^6\]

Focus on the 3d electrons, which are among the last filled.

$3d^6$ indicates the 3d subshell has 6 electrons.

Assigning quantum numbers for one of the 3d electrons in iron, you might see:

(3, 2, +2, +1/2) - This set tells us:
$n=3$, denoting the third energy level.
$l=2$, showing it is a $d$ type orbital.
$m_l=+2$, indicating its magnetic orientation.
$m_s=+1/2$, represents its spin direction.

Several combinations of $m_l$ and $m_s$ values are possible due to the multiple available spots within the $3d$ orbital. All these variations still beautifully align to define the electrons' precise characteristics and positions in the iron atom.

Problem 40

Problem 44

Other exercises in this chapter

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Arrange these items from largest to smallest de Broglie wavelength: basketball; proton; potassium atom; the planet Venus; soccer ball.

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Problem 40

Compare and contrast the representations of electron density in an atom provided by dot-density diagrams and boundary-surface diagrams.

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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 ni

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Problem 45

Some of these sets of quantum numbers $\left(n, \ell, m_{\ell}, m_{s}\right)$ could not occur. Explain why. $$ \text { (a) } 2,1,2,+\frac{1}{2} $$ (b) \(3,2,0

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