Problem 30 - Physics for Scientists and Engineers with Modern Physics

Question

(I) How many electrons can be in the \(n=6, \ell=4\) subshell?

Step-by-Step Solution

Answer

The \( n=6, \ell=4 \) subshell can hold 18 electrons.

Step 1: Determine the subshell

The subshell is determined by the values of the principal quantum number \( n \) and the azimuthal quantum number \( \ell \). In this exercise, \( n = 6 \) indicates the 6th shell, and \( \ell = 4 \) designates the subshell as a \( g \)-subshell, because \( \ell = 4 \) corresponds to the \( g \) orbital type.

Step 2: Calculate the number of orbitals in the subshell

The number of orbitals within a subshell is given by the formula \( (2\ell + 1) \). For \( \ell = 4 \), the calculation is \( 2\times4 + 1 = 9 \) orbitals.

Step 3: Determine the maximum number of electrons per orbital

Each orbital can hold a maximum of 2 electrons. This is because each orbital holds one electron for each of the two possible spin states.

Step 4: Calculate the total number of electrons in the subshell

With 9 orbitals and each orbital holding 2 electrons, the total number of electrons in the \(n=6, \ell=4\) subshell is calculated as \( 9 \times 2 = 18 \) electrons.

Key Concepts

Understanding Quantum Numbers

Quantum numbers are like an address system for electrons in an atom. They help us pinpoint the position, shape, orientation, and spin of an electron's orbit in an atom. These numbers are crucial for understanding electron configuration.

There are four main quantum numbers:

• Principal quantum number \( n \): Indicates the energy level or shell of an electron. For example, \( n = 1, 2, 3, \) etc. Higher values of \( n \) mean electrons are further from the nucleus and possess more energy.
• Azimuthal quantum number \( \ell \): Defines the shape of the electron's subshell. It can take integer values from \( 0 \) to \( n-1 \). Each value of \( \ell \) corresponds to different subshells: \( s, p, d, f, g, \) and so on.
• Magnetic quantum number \( m_\ell \): Specifies the orientation of an orbital within a subshell. It has a range from \( -\ell \) to \( +\ell \).
• Spin quantum number \( m_s \): Represents the two possible spins of an electron, either \( +\frac{1}{2} \) or \( -\frac{1}{2} \).

These quantum numbers collectively define the "address" of an electron, helping us understand where an electron "lives" in an atom.

Identifying Subshells

Subshells are specific regions within energy levels where electrons are likely to be found. The azimuthal quantum number \( \ell \) is key in identifying these subshells. For each principal quantum number \( n \), there are \( n \) possible subshells.

Each value of \( \ell \) corresponds to a subshell and determines its shape:

• When \( \ell = 0, \) it forms an \( s \)-subshell (spherical shape).
• When \( \ell = 1, \) it forms a \( p \)-subshell (dumbbell shape).
• When \( \ell = 2, \) it forms a \( d \)-subshell (cloverleaf shape).
• When \( \ell = 3, \) it forms an \( f \)-subshell (complex shape).
• When \( \ell = 4, \) it forms a \( g \)-subshell, which follows even more complex shapes.

In the problem you worked on, \( \ell = 4 \) was given, indicating a \( g \)-subshell. Knowing the subshell helps calculate how many orbitals exist within it and thereby, determine the electron capacity of that subshell.

Understanding Orbitals

Orbitals are regions within subshells where electrons are most likely to be found. Each orbital can hold a maximum of two electrons.

The number of orbitals in a subshell is governed by the azimuthal quantum number \( \ell \), specifically through the formula \( 2\ell + 1 \). This equation tells us how many unique orientations (or orbitals) exist within each subshell type.

For example:

• For \( \ell = 0 \) (s-subshell), there is \( 1 \) orbital.
• For \( \ell = 1 \) (p-subshell), there are \( 3 \) orbitals.
• For \( \ell = 2 \) (d-subshell), there are \( 5 \) orbitals.
• For \( \ell = 3 \) (f-subshell), there are \( 7 \) orbitals.
• For \( \ell = 4 \) (g-subshell), as in the original question, there are \( 9 \) orbitals.

Given that each orbital can contain 2 electrons, a g-subshell with 9 orbitals can accommodate a total of \( 18 \) electrons. This understanding aids in predicting how elements will interact based on their electron arrangements.