Problem 10 - Fundamentals of Physics

Question

An electron is in a state with \(n=3\). What are (a) the number of possible values of \(\ell,(\mathrm{b})\) the number of possible values of \(m_{e},(\mathrm{c})\) the number of possible values of \(m_{s},(\mathrm{~d})\) the number of states in the \(n=3\) shell, and (e) the number of subshells in the \(n=3\) shell?

Step-by-Step Solution

Answer

(a) 3, (b) 9, (c) 2, (d) 18, and (e) 3 subshells.

Step 1: Understanding the Quantum Numbers

Quantum numbers are used to describe the properties of an electron in an atom. They include the principal quantum number ( ), the azimuthal quantum number ( ℓ), the magnetic quantum number ( m_e), and the electron spin quantum number ( m_s). Here, n=3 is given, so we will determine the possible values for each of the other quantum numbers based on n=3.

Step 2: Determine Possible Values for ℓ

The azimuthal quantum number ( ℓ) can take integer values from 0 to n-1. For n=3, ℓ=0, 1, and 2 are the possible values. Therefore, there are 3 possible values for ℓ.

Step 3: Calculate Possible Values for m_e

The magnetic quantum number ( m_e) can have integer values ranging from - ℓ to + ℓ for each ℓ value. Thus, for ℓ=0, m_e=0; for ℓ=1, m_e=-1, 0, +1; and for ℓ=2, m_e=-2, -1, 0, +1, +2. Consequently, the possible values for m_e are obtained by adding the possibilities for each ℓ: 1+3+5=9 possible values.

Step 4: Consider the Possible Values for m_s

The electron spin quantum number ( m_s) can take two possible values: +1/2 and −1/2. Hence, there are always 2 possible values for m_s.

Step 5: Count the Number of States in the n=3 Shell

For each electron in the n=3 shell, you can have different combinations of ℓ, m_e, and m_s. The total number of states is the product of the possible values for ℓ, m_e, and m_s. Therefore, there are 3 ℓ values, 9 m_e values, and 2 m_s values, which gives 3x9x2=18 states in total.

Step 6: Identify the Number of Subshells in the n=3 Shell

Each different ℓ value represents a subshell. Since ℓ can be 0, 1, or 2, there are 3 subshells in the n=3 shell.

Key Concepts

Principal Quantum Number

The Principal Quantum Number, denoted by the symbol \( n \), is a key concept in understanding atomic structure. It indicates the relative size and energy level of an electron's orbital. In layman's terms, think of the principal quantum number as a layer around the nucleus, with higher numbers indicating layers that are further out.

• \( n \) is always a positive integer, meaning it can be 1, 2, 3, etc.
• The larger the value of \( n \), the higher the energy level and the larger the orbit.
• In the context of our given problem \( n=3 \), this indicates that the electron is in the third energy level or shell.

Understanding the principal quantum number is foundational because it sets the stage for determining other quantum numbers, which specify the shape, orientation, and spin characteristics of the electron orbitals.

Azimuthal Quantum Number

Often represented by \( \ell \), the azimuthal quantum number describes the shape of the electron's orbital. It is sometimes called the angular momentum quantum number. This number is crucial because it defines subshells within a given principal shell \( n \).

• \( \ell \) can take on any integer value from 0 to \( n-1 \).
• For example, when \( n=3 \), possible \( \ell \) values are 0, 1, and 2, representing the subshells: s, p, and d.
• Each choice of \( \ell \) corresponds to a different type of orbital shape, such as spherical or dumbbell-shaped.

The azimuthal quantum number is vital for distinguishing the electron clouds within the same energy level, allowing us to visualize where electrons are likely to be found.

Magnetic Quantum Number

The Magnetic Quantum Number, symbolized by \( m_e \), defines the orientation of the orbital in space. This quantum number is crucial for understanding how orbitals are positioned around the nucleus once the number of subshells has been determined by \( \ell \).

• \( m_e \) ranges from \(-\ell \) to \(+\ell \), including zero.
• For instance, if \( \ell = 1 \), then \( m_e \) can be \(-1, 0, \) or \(+1 \).
• In the case where \( n=3 \), we find a total of 9 possible \( m_e \) values by summing allowed values for each \( \ell \): \( 1 + 3 + 5 = 9 \).

These values show how the subshells are oriented differently, which is important in understanding how atoms form bonds and interact with magnetic fields.

Electron Spin Quantum Number

Notated as \( m_s \), the Electron Spin Quantum Number describes an interesting and unique property of electrons: their intrinsic angular momentum or 'spin'. Despite being a fundamental property, it only has two possible values.

• The values are \(+\frac{1}{2} \) and \(-\frac{1}{2} \), often thought of as "up" and "down" spins.
• Spin is a critical aspect of the Pauli Exclusion Principle, which states that no two electrons in an atom can have the same set of quantum numbers.

This quantum number adds a layer of complexity, helping explain electron pairing within an orbital. In other words, while many electrons can be in the same space, each must have a unique spin.

Atomic Structure

Understanding atomic structure involves piecing together all the quantum numbers, which effectively map out where electrons reside around an atom. The principal, azimuthal, and magnetic quantum numbers define an electron's "address" within an atom.

• A complete set gives rise to specific "states" within an atom, each electron having a precise location in terms of energy level and spatial distribution.
• For \( n=3 \), combining these quantum numbers gives us 18 distinct states (combinations of \( \ell, m_e, \) and \( m_s \)), and allows the existence of 3 subshells: \( s, p, \) and \( d \).

Recognizing how these quantum numbers interact provides insight not just into a single atom's structure, but into how bonds form and chemical reactions occur. This knowledge is critical for chemistry and understanding the natural world at the atomic level.