Problem 45 - Chemistry The Molecular Science

Question

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) \(3,2,0,-\frac{1}{2}\) (c) 1,0,0,1 (d) \(3,3,2,-\frac{1}{2}\) $$ \text { (e) } 2,0,0,+\frac{1}{2} $$

Step-by-Step Solution

Answer

Sets (a), (c), and (d) are invalid. Sets (b) and (e) are valid.

Step 1: Quantum Numbers Review

Understand each quantum number: - **Principal quantum number** \( n \) describes the energy level and must be a positive integer (\( n \geq 1 \)).- **Azimuthal quantum number** \( \ell \) describes the shape of the orbital and can have integer values from 0 to \( n-1 \).- **Magnetic quantum number** \( m_{\ell} \) describes the orientation of the orbital and ranges from \( -\ell \) to \( \ell \).- **Spin quantum number** \( m_{s} \) has only two possible values: \( +\frac{1}{2} \) or \( -\frac{1}{2} \).

Step 2: Analyze Set (a)

The set is \((2, 1, 2, +\frac{1}{2})\). Check the values: - \( n = 2 \), \( \ell = 1 \) is valid since \( \ell < n \). - \( m_{\ell} = 2 \) is invalid because \( m_{\ell} \) must be \(-1, 0, 1\) for \( \ell = 1 \). Set (a) is invalid due to \( m_{\ell} \).

Step 3: Analyze Set (b)

The set is \((3, 2, 0, -\frac{1}{2})\). Check the values:- \( n = 3 \), \( \ell = 2 \) is valid since \( \ell < n \).- \( m_{\ell} = 0 \) and is within the valid range \(-2, -1, 0, 1, 2\) for \( \ell = 2 \).- \( m_{s} = -\frac{1}{2} \) is valid. Set (b) is valid.

Step 4: Analyze Set (c)

The set is \((1, 0, 0, 1)\). Check the values:- \( n = 1 \), \( \ell = 0 \) is valid because \( \ell < n \).- \( m_{\ell} = 0 \) is valid for \( \ell = 0 \).- \( m_{s} = 1 \) is invalid as \( m_{s} \) can only be \( +\frac{1}{2} \) or \( -\frac{1}{2} \). Set (c) is invalid due to \( m_{s} \).

Step 5: Analyze Set (d)

The set is \((3, 3, 2, -\frac{1}{2})\). Check the values:- \( n = 3 \), \( \ell = 3 \) is invalid because \( \ell < n \) must hold (\( \ell \leq n-1 \)). Set (d) is invalid due to \( \ell \).

Step 6: Analyze Set (e)

The set is \((2, 0, 0, +\frac{1}{2})\). Check the values:- \( n = 2 \), \( \ell = 0 \) is valid as \( \ell < n \).- \( m_{\ell} = 0 \) is valid for \( \ell = 0 \).- \( m_{s} = +\frac{1}{2} \) is valid. Set (e) is valid.

Key Concepts

Principal Quantum Number

The principal quantum number, denoted as \( n \), is crucial in understanding the energy levels of an electron in an atom. It essentially tells us the

• size
• energy level

of an atomic orbital. This quantum number must be a positive integer, starting from 1 and going upwards: \( n = 1, 2, 3, \) and so forth.

An electron with a higher principal quantum number \( n \) resides farther from the nucleus and consequently has higher energy. Consider it as a floor in a building: the higher you go, the further you are from the ground level. All other quantum numbers depend on the value of \( n \) to define the precise configuration of an electron.

Azimuthal Quantum Number

Often called the angular or orbital quantum number, the azimuthal quantum number \( \ell \) defines the shape of the electron's orbital. It varies between 0 and \( n-1 \), meaning for any given principal quantum number \( n \), \( \ell \) can assume integer values such as 0, 1, 2, ..., until \( n-1 \).

Each value of \( \ell \) symbolizes a different orbital type:

• \( \ell = 0 \) is an s orbital (spherical shape).
• \( \ell = 1 \) signifies a p orbital (dumbbell shape).
• \( \ell = 2 \) corresponds to a d orbital (cloverleaf shape).
• \( \ell = 3 \) represents an f orbital (complex shape).

The azimuthal quantum number is fundamental in determining the electron's orbital shape and thus influences the atom's chemical bonding and physical properties.

Magnetic Quantum Number

The magnetic quantum number, represented as \( m_{\ell} \), reflects the orientation of the atomic orbital around the nucleus. Based on the azimuthal quantum number \( \ell \), it ranges from \( -\ell \) to \( \ell \).

For example, if \( \ell = 2 \) (a d orbital), then the possible values for \( m_{\ell} \) are -2, -1, 0, 1, and 2. These values showcase the different orientations an orbital can have within a given subshell.

This quantum number plays a key role in explaining how magnetic fields affect electrons and their trajectories within an atom, hence its name 'magnetic' quantum number.

Spin Quantum Number

The spin quantum number \( m_{s} \) adds another layer of complexity. It's crucial because it represents the intrinsic angular momentum or 'spin' of the electron in an orbital. Unlike the other quantum numbers, \( m_{s} \) has only two allowed values: \( +\frac{1}{2} \) and \( -\frac{1}{2} \).

These values are often thought of as the two opposites of spinning: up and down. Electron spin is a fundamental property, not caused by actual spinning, but more a type of angular momentum.

The concept of electron spin is vital in the Pauli exclusion principle, which stipulates that no two electrons in the same atom can have the identical set of all four quantum numbers. Hence, each electron in the same orbital must have opposite spins, providing crucial insights into the chemical behavior of elements.