Problem 64 - Chemistry: The Central Science

Question

What is the maximum number of electrons in an atom that can have the following quantum numbers: (a) \(n=2\), \(m_{s}=-\frac{1}{2}\), (b) \(n=5, l=3 ;\) (c) \(n=4, l=3, m_{l}=-3\) (d) \(n=4, l=1, m_{l}=1\) ?

Step-by-Step Solution

Answer

The maximum number of electrons for each case are: (a) 4, (b) 14, (c) 2, and (d) 2.

Step 1: Case (a): \(n=2\), \(m_{s}=-\frac{1}{2}$$tag_content#In this case, we only have the principal quantum number and the spin quantum number. Since \)n=2\(, the possible values for \)l\( are 0 and 1. For each \)l\( value, there is a corresponding range for the magnetic quantum number \)m_l$: - If \(l=0\), then \(m_l=0\) (1 orbital). - If \(l=1\), then \(m_l\) takes 3 possible values: \(-1, 0, 1\) (3 orbitals). In total, we have 4 orbitals for every different \(n, l, m_l\) combination. Since \(m_s\) is fixed at \(-\frac{1}{2}\), there can only be one electron with that quantum number for each orbital. So, the maximum number of electrons for case (a) is 4.

Case (b): \(n=5, l=3$$tag_content#In this case, we have the principal and angular momentum quantum numbers. For this specific combination, there are 7 possible values of the magnetic quantum number: \)m_l = -3, -2, -1, 0, 1, 2, 3$. In other words, we have 7 orbitals. Since for each orbital, there can be one electron with an \(m_s\) value of \(\frac{1}{2}\) and one with \(-\frac{1}{2}\), there can be a maximum of 2 electrons per orbital. So, the maximum number of electrons for case (b) is 14.

Step 2: Case (c): $n=4, l=3, m_{l}=-3$$tag_content#In this case, we have specific values for the principal, angular momentum, and magnetic quantum numbers. With these specified, we can say that there is only 1 orbital that meets these conditions. Since in this orbital, there can be one electron with an \(m_s\) value of \(\frac{1}{2}\) and one with \(-\frac{1}{2}\), there can be a maximum of 2 electrons. So, the maximum number of electrons for case (c) is 2.

Case (d): $n=4, l=1, m_{l}=1$$tag_content#In this case, we also have specified values for the principal, angular momentum, and magnetic quantum numbers. This means we have only one orbital for that particular combination. There can be one electron with an \(m_s\) value of \(\frac{1}{2}\) and one with \(-\frac{1}{2}\) in that single orbital. So, the maximum number of electrons for case (d) is 2.

Key Concepts

Principal Quantum Number

The principal quantum number, denoted by the symbol \( n \), is a key identifier in describing the electron's energy level in an atom. It primarily indicates the relative size and energy level of the atomic orbitals. Integer values starting from 1 (\( n = 1, 2, 3, \ldots \)) are used. As \( n \) increases, the electron's distance from the nucleus usually increases as well, meaning higher energy levels and larger orbitals.

• \( n = 1 \) represents the first energy level, the closest to the nucleus.
• \( n = 2 \) represents the second energy level, farther from the nucleus.

In practical application, as can be seen in our original exercise, \( n \) values are vital for understanding how many orbitals are available and how these impact the maximum number of electrons. For example, with \( n = 2 \), we have different possible angular momentum quantum numbers (\( l \)) indicating various orbitals.

Angular Momentum Quantum Number

The angular momentum quantum number is labeled as \( l \) and it defines the shape of the electron's orbital. For any given principal quantum number \( n \), \( l \) can take integer values ranging from 0 up to \( n-1 \). The value of \( l \) also corresponds to different subshells, such as s, p, d, and f.

• \( l = 0 \) is called an s orbital.
• \( l = 1 \) is a p orbital.
• \( l = 2 \) is a d orbital.
• \( l = 3 \) is an f orbital.

The angular momentum quantum number importantly determines the number of subdivisions an energy level has, which influence electron configurations. For instance, in case (b) of our exercise, \( n = 5, l = 3 \), the orbitals shaped as f can hold multiple electrons since \( l = 3 \) has 7 possible magnetic quantum number values.

Magnetic Quantum Number

A magnetic quantum number, referenced as \( m_l \), specifies the orientation of an orbital in space relative to the other orbitals, and it defines the number of orbitals and their orientation within a subshell. The values that \( m_l \) can take range from \(-l\) to \(+l\), including zero.This means:

• For \( l = 0 \), \( m_l = 0 \).
• For \( l = 1 \), \( m_l = -1, 0, 1 \).
• For \( l = 2 \), \( m_l = -2, -1, 0, 1, 2 \).
• For \( l = 3 \), \( m_l = -3, -2, -1, 0, 1, 2, 3 \).

In our exercise, this concept helps determine how many different spatial configurations of the orbitals there are. In cases (c) and (d), specific \( m_l \) values narrow down the electron configurations to very particular orientations.

Spin Quantum Number

The spin quantum number, symbolized by \( m_s \), describes the intrinsic spin property of an electron within an orbital, which can be understood as a form of angular momentum. Electrons can have a spin of either \(+\frac{1}{2}\) or \(-\frac{1}{2}\). This property is crucial as it differentiates the electrons occupying the same orbital.

• The \(+\frac{1}{2}\) represents an "up" spin.
• The \(-\frac{1}{2}\) represents a "down" spin.

This is especially significant in quantum mechanics because only two electrons can exist in one orbital, each with opposing spins as demonstrated in our exercise. In case (a), we observed how the spin quantum number distinctly identifies the electron configuration in the available orbitals for \( m_s = -\frac{1}{2} \). Hence, the spin helps determine maximum electron allocations in different configurations.