Problem 93 - Chemistry: The Central Science in SI Units, Global Edition

Question

Determine whether each of the following sets of quantum numbers for the hydrogen atom are valid. If a set is not valid, indicate which of the quantum numbers has a value that is not valid: (a) \(n=3, l=3, m_{l}=2, m_{5}=+\frac{1}{2}\) (b) \(n=4, l=3, m_{l}=-3, m_{\mathrm{s}}=+\frac{1}{2}\) (c) \(n=3, l=1, m_{l}=2, m_{s}=+\frac{1}{2}\) (d) \(n=5, l=0, m_{l}=0, m_{\mathrm{s}}=0\) (e) \(n=2, l=1, m_{l}=1, m_{s}=-\frac{1}{2}\)

Step-by-Step Solution

Answer

(a) Not valid, l is invalid. (b) Valid. (c) Not valid, \(m_l\) is invalid. (d) Not valid, \(m_s\) is invalid. (e) Valid.

Step 1: (a) Check the set for n = 3, l = 3, m_l = 2, m_s = +1/2

For n = 3, l must range from 0 to 2 (0 ≤ l < n). However, l = 3 is given, which is outside this range. Therefore, this set of quantum numbers is not valid due to an incorrect value for `l`.

Step 2: (b) Check the set for n = 4, l = 3, m_l = -3, m_s = +1/2

For n = 4, l must range from 0 to 3 (0 ≤ l < n). Since l = 3, it is valid. Now, for l = 3, m_l must range from -3 to +3 (-3 ≤ m_l ≤ 3). Since m_l = -3, it is also valid. Finally, the spin quantum number m_s can take values of either +1/2 or -1/2, so m_s = +1/2 is valid. Therefore, this set of quantum numbers is valid.

Step 3: (c) Check the set for n = 3, l = 1, m_l = 2, m_s = +1/2

For n = 3, l must range from 0 to 2 (0 ≤ l < n). Since l = 1, it is valid. However, for l = 1, m_l must range from -1 to +1 (-1 ≤ m_l ≤ 1). Since m_l = 2 is given, which is outside this range, this set of quantum numbers is not valid due to an incorrect value for `m_l`.

Step 4: (d) Check the set for n = 5, l = 0, m_l = 0, m_s = 0

For n = 5, l must range from 0 to 4 (0 ≤ l < n). Since l = 0, it is valid. For l = 0, m_l must be equal to 0, so m_l = 0 is also valid. However, the spin quantum number m_s can take values of either +1/2 or -1/2; m_s = 0 is not a valid value. Therefore, this set of quantum numbers is not valid due to an incorrect value for `m_s`.

Step 5: (e) Check the set for n = 2, l = 1, m_l = 1, m_s = -1/2

For n = 2, l must range from 0 to 1 (0 ≤ l < n). Since l = 1, it is valid. For l = 1, m_l must range from -1 to +1 (-1 ≤ m_l ≤ 1). Since m_l = 1, it is also valid. Finally, since m_s = -1/2, it is also valid (as m_s can take values of either +1/2 or -1/2). Therefore, this set of quantum numbers is valid.

Key Concepts

Hydrogen Atom

The hydrogen atom is the simplest atom found in nature. It consists of only one proton and one electron. This basic configuration makes it an ideal model for studying atomic theory and quantum mechanics. In quantum mechanics, electrons are described as existing in orbitals, which are defined by quantum numbers.
These quantum numbers offer a way to determine the size, shape, and orientation of the electron's orbital. For the hydrogen atom, quantum numbers are crucial as they help describe and predict how the electron behaves around the nucleus. The Bohr model initially explained hydrogen's spectrum, but quantum mechanics provided a more accurate picture using quantum numbers.

Validity of Quantum Numbers

Quantum numbers describe the unique quantum state of an electron in an atom and have specific rules they must follow. There are four types of quantum numbers:

• Principal quantum number ( ): determines the energy level
• Angular momentum quantum number ( ): defines the shape of the orbital
• Magnetic quantum number ( ): gives the orientation of the orbital
• Spin quantum number ( ): specifies the electron's spin

Each quantum number must fall within valid ranges to correctly describe an electron's state. For instance, the angular momentum quantum number ( ) must be less than the principal quantum number ( ), and the magnetic quantum number ( ) must fall between and . These rules ensure that quantum numbers accurately represent the possible states of an electron in an atom.

Spin Quantum Number

The spin quantum number ( ) describes the intrinsic angular momentum, or "spin," of an electron. Unlike orbital quantum numbers, which are spatial, the spin quantum number represents a fundamental property of electrons not associated with any movement in space.
Electrons can have one of two possible spin states, denoted as or . This property is critical in the context of quantum mechanics and atomic theory because it influences electronic configurations and the chemical behavior of atoms. Despite being a small value, the spin of an electron plays a significant role in quantum mechanics rules, impacting the overall magnetic properties of atoms and molecules.

Principal Quantum Number

The principal quantum number ( ) plays a vital role in determining the overall energy and size of an electron's orbital within an atom. Represented by , it takes positive integer values such as 1, 2, 3, and so on. With each increase in , electrons are located in higher energy levels further away from the nucleus.
This quantum number is linked closely to the energy of an electron's orbital, meaning higher values correspond to higher energy levels. For instance, for a hydrogen atom, the energy required to move an electron from to would be less than from to . Thus, the principal quantum number is essential for understanding the energy levels in atoms and how electrons transition between these levels.