Problem 26 - University Physics with Modern Physics

Question

For germanium \((\mathrm{Ge}, \mathrm{Z}=32),\) make a list of the number of electrons in each subshell \((1 s, 2 s, 2 p, \ldots) .\) Use the allowed values of the quantum numbers along with the exclusion principle; do not refer to Table \(41.3 .\)

Step-by-Step Solution

Answer

Germanium's electron configuration is: \(1s^2, 2s^2, 2p^6, 3s^2, 3p^6, 4s^2, 3d^{10}, 4p^2\).

Step 1: Understand Quantum Numbers and the Aufbau Principle

Electrons fill subshells in an atom based on quantum numbers. The principal quantum number \(n\) determines the energy level, the azimuthal quantum number \(l\) defines the shape of the orbital, and subshells are designated \(s, p, d, f\). Electrons fill the lowest energy subshells first, following the Aufbau principle.

Step 2: List Subshell Order

The order in which subshells are filled according to increasing energy is: \(1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, \ldots\). For elements like germanium with \(Z = 32\), we will follow this order until we have accounted for all 32 electrons.

Step 3: Filling Electrons into Subshells

Assign electrons to each subshell in order, according to their maximum electron capacity: \(1s\) can hold 2, \(2s\) holds 2, \(2p\) holds 6, \(3s\) holds 2, \(3p\) holds 6, \(4s\) holds 2, \(3d\) holds 10, \(4p\) holds 6. Continue until all 32 electrons are assigned.

Step 4: List the Electron Distribution

Distribute the 32 electrons in the subshells as follows:- \(1s^2\)- \(2s^2\)- \(2p^6\)- \(3s^2\)- \(3p^6\)- \(4s^2\)- \(3d^{10}\)- \(4p^2\)This represents the electron configuration of germanium at \(Z = 32\).

Key Concepts

Quantum Numbers

Quantum numbers are vital for understanding the distribution of electrons in an atom. They describe various attributes of electrons that orbit around the nucleus. Four quantum numbers define the position and energy of an electron:

• Principal Quantum Number ( , n): Indicates the energy level of the electron. Higher values of n mean the electron is further from the nucleus and has more energy.
• Azimuthal Quantum Number ( , l): Describes the shape of the orbital and subshell. It takes integer values ranging from 0 to n-1. Here, l=0 represents an s orbital, l=1 a p orbital, l=2 a d orbital, and l=3 an f orbital.
• Magnetic Quantum Number (m_l): Specifies the orientation of the orbital in space. It ranges from -l to +l.
• Spin Quantum Number (m_s): Describes the electron's spin, with possible values of -1/2 or +1/2, indicating the two possible spin states of an electron.

These quantum numbers help us to organize where and how electrons are arranged in atoms, translating into the atom's electron configuration.

Aufbau Principle

The Aufbau Principle is a fundamental concept in chemistry that explains how electrons fill available atomic orbitals. The term 'Aufbau' is German for 'building up' or 'construction.' This principle sets the rule that electrons fill orbitals starting with the lowest energy levels first:

• Electrons will occupy the lowest energy orbital available.
• Only when lower orbitals are filled will electrons start to fill higher orbitals.
• This approach helps in predicting the electron configuration of elements systematically.

By following the Aufbau Principle, we can deduce the ground state electron configuration of atoms, which is essential in predicting chemical behavior and properties.

Subshell Order

The sequence of filling electron subshells is based on their increasing energy levels, which follows a specific order. This order is crucial for understanding the electron configurations of atoms, as it tells us the sequence in which electrons will fill their respective orbitals:

• The typical order starts with the filling of the 1s orbital, followed by 2s, 2p, 3s, and so forth.
• The mnemonic to remember this order is by following diagonal lines in the periodic table or using the Aufbau diagram.
• For example, the complement of subshells for germanium are filled in the order: 1s², 2s², 2p⁶, 3s², 3p⁶, 4s², 3d¹⁰, and finally 4p².

This structured order allows us to predict and confirm the electron configurations for various elements, giving insight into their chemical properties.

Pauli Exclusion Principle

The Pauli Exclusion Principle is a fundamental rule in quantum mechanics with profound chemical implications. Formulated by Wolfgang Pauli, it states that no two electrons in a single atom can have the same set of four quantum numbers:

• This principle means each orbital can hold a maximum of two electrons.
• When paired in an orbital, the two electrons must have opposite spins, one with spin +1/2 and the other with spin -1/2.
• This principle ensures electrons are distributed among orbitals in a way that minimizes energy levels and potential repulsion.

The Pauli Exclusion Principle is key for determining the electronic structure. It prevents electrons from occupying the same quantum state, providing detailed organization and structure to every element's electron configuration.