Problem 59 - University Physics with Modern Physics

Question

(a) Show that the total number of atomic states (including different spin states) in a shell of principal quantum number \(n\) is 2\(n^{2}\) . [Hint: The sum of the first \(N\) integers \(1+2+3+\cdots+N\) is equal to \(N(N+1) / 2 . ](\mathrm{b})\) Which shell has 50 states?

Step-by-Step Solution

Answer

The total number of atomic states in a shell is \(2n^2\). The shell with 50 states is \(n=5\).

Step 1: Understanding the Quantum Numbers

Each electron in an atom is characterized by a set of quantum numbers. The principal quantum number \( n \) determines the size of the shell, while the azimuthal quantum number \( l \) can range from \( 0 \) to \( n-1 \). Each \( l \) value corresponds to an orbital type within the shell.

Step 2: Calculate Number of Orbitals per Shell

For a fixed \( n \), the number of orbitals \( m_l \) (magnetic quantum number) ranges from \( -l \) to \( +l \), which gives \( 2l+1 \) possible \( m_l \) values for each \( l \). Therefore, for each \( n \), the total number of orbitals is \( \sum_{l=0}^{n-1} (2l+1) \).

Step 3: Compute Total Orbitals for a Given Shell

Sum up the possible \( m_l \) values from \( l=0 \) to \( n-1 \): \[ \sum_{l=0}^{n-1} (2l+1) = n^2 \]. This is derived from the fact that the sequence \( (2l+1) \) is an arithmetic sequence.

Step 4: Consider Electron Spin

Each orbital can host 2 electrons with opposite spins (spin quantum number \( m_s = +1/2 \) or \( m_s = -1/2 \)). This means that the total number of states in the shell is \( 2 \times n^2 = 2n^2 \).

Step 5: Determine the Shell with 50 States

The given expression for the number of states is \( 2n^2 \). We need to solve for \( n \) such that \( 2n^2 = 50 \). This simplifies to \( n^2 = 25 \), so \( n = 5 \).

Step 6: Final Verification

For \( n = 5 \), the total number of states is \( 2 \times 5^2 = 50 \), confirming that the shell with principal quantum number \( n = 5 \) satisfies the requirement.

Key Concepts

Quantum Numbers

In quantum mechanics, quantum numbers are essential for describing the unique quantum state of an electron in an atom. Each electron is identified by a set of four quantum numbers.

• Principal quantum number ( ) : Defines the energy level or shell. Ranges from 1 to infinity.
• Azimuthal quantum number ( ) : Indicates the orbital shape. Values range from 0 to - 1.
• Magnetic quantum number ( ) : Determines the orbital's orientation in space. Ranges from -l to +l.
• Spin quantum number ( ) : Designates the electron's spin direction. Can be +1/2 or -1/2.

These numbers help to describe the probability distribution of an electron in an atom and are fundamental to understanding atomic structure.

Atomic States

An atomic state refers to the specific configuration and energy level of an electron or electrons within an atom. Each state is characterized by a unique set of quantum numbers.

• Electrons can jump between different states by absorbing or emitting energy.
• The state of an atom is influenced by the interactions among electrons and the forces within the atom's nucleus.

Exploring atomic states is crucial for understanding phenomena like spectral lines, where electrons transition between energy levels, emitting light of specific wavelengths.

Principal Quantum Number

The principal quantum number, denoted as , is a fundamental concept in quantum mechanics. It signifies the main energy level or shell in which an electron resides.

• The value of defines the size and energy of the orbital.
• Higher values correspond to higher energy levels and generally larger orbitals.
• The total number of atomic states in a shell depends on the principal quantum number as given by the expression: 2n^2.

For example, for = 5, there are 50 possible atomic states. This includes all the orbitals and electron spins within the shell, providing insight into the electron arrangement at a more detailed level.

Electron Spin

Spin is an intrinsic form of angular momentum carried by electrons. Each electron has a spin quantum number, often denoted as .

• Only two possible values: +1/2 and -1/2, representing the two opposite spin directions.
• Electrons in the same orbital must have opposite spins to comply with the Pauli Exclusion Principle.

The concept of spin is crucial in determining how electrons fill available orbitals in an atom, affecting chemical properties and magnetism in materials. Understanding electron spin helps explain why electrons pair up in orbitals and the overall magnetic behavior of atoms.

Magnetic Quantum Number

The magnetic quantum number ( ) describes the orientation of an electron's orbital in space relative to an external magnetic field. It provides a more detailed view than the azimuthal quantum number.

• For a given orbital with azimuthal quantum number l, can take on integer values ranging from -l to +l, thus giving 2l + 1 possible values.
• This number affects how orbitals orient in an atom, determining, in part, how atoms interact with magnetic fields.

Understanding the magnetic quantum number is key to analyzing atomic and molecular structures, particularly in how electrons occupy differently oriented orbitals within a given energy level.