Problem 73 - Chemistry An Atoms First Approach

Question

Which of the following sets of quantum numbers are not allowed in the hydrogen atom? For the sets of quantum numbers that are incorrect, state what is wrong in each set. a. \(n=3, \ell=2, m_{c}=2\) b. \(n=4, \ell=3, m_{\ell}=4\) c. \(n=0, \ell=0, m_{\ell}=0\) d. \(n=2, \ell=-1, m_{c}=1\)

Step-by-Step Solution

Answer

In conclusion: a. Allowed in the hydrogen atom. b. Not allowed, 𝑚_𝑙=4 is not within the range of -3 to 3. c. Not allowed, n=0 is not a positive integer (n must be greater than 0). d. Not allowed, 𝑙=-1 is not a non-negative integer.

Step 1: Check the principal quantum number (n).

Only positive integers are allowed for n. a. n=3, valid. b. n=4, valid. c. n=0, invalid. (n must be greater than 0) d. n=2, valid.

Step 2: Check the azimuthal quantum number (𝑙).

0 ≤ 𝑙 ≤ n-1 and it is an integer. a. 𝑙=2, valid. (0 ≤ 2 ≤ 3-1) b. 𝑙=3, valid. (0 ≤ 3 ≤ 4-1) c. 𝑙=0, invalid. (Since n is already invalid, there's no need to check 𝑙) d. 𝑙=-1, invalid. (𝑙 must be non-negative)

Step 3: Check the magnetic quantum number (𝑚_𝑙).

-𝑙 ≤ 𝑚_𝑙 ≤ 𝑙 and it is an integer. a. 𝑚_𝑐=2, valid. (-2 ≤ 2 ≤ 2) b. 𝑚_𝑙=4, invalid. (-3 ≤ 4 ≤ 3) c. 𝑚_𝑙=0, invalid. (Since n and 𝑙 are already invalid, there's no need to check 𝑚_𝑙) d. 𝑚_𝑐=1, invalid. (Since 𝑙 is invalid, there's no need to check 𝑚_𝑐) In conclusion: a. Allowed in the hydrogen atom. b. Not allowed, 𝑚_𝑙=4 is not within the range of -3 to 3. c. Not allowed, n=0 is not a positive integer (n must be greater than 0). d. Not allowed, 𝑙=-1 is not a non-negative integer.

Key Concepts

hydrogen atom

The hydrogen atom is the most basic atom and consists of only one proton and one electron. Understanding how electrons behave in the hydrogen atom is essential for quantum mechanics, which helps us predict atomic and molecular behaviors. In the context of quantum numbers, the electron's placement around the hydrogen nucleus is determined. These quantum numbers describe discrete energy levels and provide a framework for understanding the electron's properties. For the hydrogen atom, the Schrödinger equation can be used to explain the electron's motion. Since it is a system with rotational symmetry, the electron's state is best understood using quantum numbers, revealing various quantized states where the electron can exist. The study of the hydrogen atom is foundational in learning about quantum mechanics and atomic structures, offering insights into more complex systems.

principal quantum number

The principal quantum number, denoted as \( n \), indicates the main energy level or shell of an electron within an atom. This number must be a positive integer: \( n = 1, 2, 3, \ldots\). As \( n \) increases, the electron's energy level becomes higher, and it's located further from the nucleus.The value of \( n \) also determines the size of an electron's orbit:

• Lower \( n \) values indicate electrons closer to the nucleus with lower energy.
• Higher \( n \) values mean electrons are in outer shells with greater energy.

This quantum number plays a crucial role not only in defining the shell in which the electron resides but also influences other quantum numbers like the azimuthal quantum number.

azimuthal quantum number

The azimuthal quantum number, represented by \( \ell \), defines the shape of the electron's orbital. It is also known as the angular momentum quantum number. The value of \( \ell \) depends on the principal quantum number \( n \) and can take on integer values from 0 to \( n-1 \). For each value of \( n \), different values of \( \ell \) correspond to different subshells (or types of orbitals) like:

• \( \ell = 0 \): 's' orbital (spherical),
• \( \ell = 1 \): 'p' orbital (dumbbell-shaped),
• \( \ell = 2 \): 'd' orbital (cloverleaf-shaped),
• \( \ell = 3 \): 'f' orbital (complex shapes).

The azimuthal quantum number influences an electron's angular momentum and determines the number of angular nodes in the electron's wave function. The nodes are regions where the probability of finding an electron is zero.

magnetic quantum number

The magnetic quantum number, denoted by \( m_\ell \), specifies the orientation of an electron's orbital around the nucleus. It gives information about the directionality of the electron cloud and can be any integer value between and including \(-\ell \) and \( \ell \).For an azimuthal quantum number \( \ell \), the possible values of \( m_\ell \) are:

• -\( \ell \), -\( \ell + 1 \), ..., 0, ..., \( \ell - 1 \), \( \ell \)

These values determine the number of orbitals available within a subshell, influencing how electrons are distributed spatially. For instance, if \( \ell = 1 \) (a p orbital), \( m_\ell \) can be -1, 0, or 1, indicating the three orientations of the p orbitals in 3-dimensional space.Understanding \( m_\ell \) is crucial for visualizing the arrangement and behavior of electrons under magnetic fields, which is a core aspect of quantum mechanics.