Problem 92 - Chemistry: An Atoms-Focused Approach

Question

Identify the subshells with the following combinations of quantum numbers and arrange them in order of increasine energy in an atom of gold: a. \(n=2, \ell=1\) b. \(n=5, \ell=0\) c. \(n=3, e=2\) d. \(n=4, \ell=3\)

Step-by-Step Solution

Answer

Question: Arrange the subshells 2p, 5s, 3d, and 4f in order of increasing energy in an atom of gold. Answer: The order of increasing energy for the subshells in a gold atom is 2p < 3d < 5s < 4f.

Step 1: Identify the subshells with the given quantum numbers

First, let's identify the subshells using the quantum numbers provided in the exercise: a. For \(n=2, \ell=1\), the subshell is 2p b. For \(n=5, \ell=0\), the subshell is 5s c. For \(n=3, \ell=2\), the subshell is 3d d. For \(n=4, \ell=3\), the subshell is 4f

Step 2: Refer to the Aufbau Principle

According to the Aufbau Principle, electrons occupy the lowest energy subshells first. Therefore, we will order the subshells in increasing energy according to the (n + ℓ) rule. This means that the subshell with the lowest sum of the values (n + ℓ) will be the subshell with the lowest energy. a. 2p: \(2+1=3\) b. 5s: \(5+0=5\) c. 3d: \(3+2=5\) d. 4f: \(4+3=7\)

Step 3: Comparing subshells with the same (n + ℓ) value

Subshells b and c have the same (n + ℓ) value, so we need to determine which one has lower energy. In cases where the (n + ℓ) values are equal, the subshell with the lower principal quantum number n will have lower energy. Ordering the subshells with this information: 1. 2p: \(n=2, \ell=1, (n+\ell)=3\) 2. 3d: \(n=3, \ell=2, (n+\ell)=5\) 3. 5s: \(n=5, \ell=0, (n+\ell)=5\) 4. 4f: \(n=4, \ell=3, (n+\ell)=7\)

Step 4: Final order of increasing energy

Based on the steps above, the subshells in gold arranged in order of increasing energy are as follows: 2p < 3d < 5s < 4f

Key Concepts

Aufbau Principle

The Aufbau Principle is a fundamental guideline used in chemistry and physics to predict the arrangement of electrons in an atom. According to this principle, electrons fill atomic orbitals starting with the lowest energy levels before moving to higher ones. This is often articulated as "build-up," where electrons are added progressively. The energy levels are determined based on the quantum numbers, primarily the principal quantum number \( n \) and the azimuthal quantum number \( \, \ell \).

• As electrons occupy lower energy orbitals first, elements display a predictable electron configuration.
• One common technique involves the "(n + \ell)" rule. This rule helps compare orbitals where the lowest sum indicates lower energy.
• If two orbitals have the same \((n + \ell)\) value, the one with the lower \(n\) value is filled first.

Using the Aufbau Principle is crucial for determining the arrangement and behavior of electrons, which further influences the chemical properties of elements.

Subshell identification

Subshells are specific types of orbitals within energy levels of an atom. Understanding subshells is essential for identifying where electrons are located in an atom. The type of subshell is indicated by the azimuthal quantum number \( \ell \), which can take integer values from 0 to \( n-1 \), where \( n \) is the principal quantum number. Each \( \ell \) value corresponds to a particular type of orbital:

• \( \ell = 0 \): s orbital (spherical in shape)
• \( \ell = 1 \): p orbitals (dumbbell-shaped)
• \( \ell = 2 \): d orbitals (more complex shapes)
• \( \ell = 3 \): f orbitals (even more complex shapes)

For example, for a given quantum number of \( n = 3 \), the possible subshells include \(3s\), \(3p\), and \(3d\). Recognizing these subshells helps in categorizing elements based on their electron configuration, thereby aiding in predicting their chemical behavior.

Electron configuration

Electron configuration is the distribution of electrons in an atom's orbitals. It is usually represented using numbers and letters correlated to quantum numbers, where the numbers reflect the principal quantum number \( n \) and the letters depict the type of subshell \( \ell \). Writing the electron configuration involves arranging electrons according to certain rules:

• The Pauli Exclusion Principle states that no two electrons can have the same set of quantum numbers, so each orbital can hold a maximum of two electrons with opposite spins.
• Hund's Rule suggests that electrons fill every orbital in a subshell singly before any orbital gets a second electron, minimizing repulsion.

For instance, the electron configuration of gold (Au) begins as follows: 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d¹⁰ 4p⁶ 5s² 4d¹⁰ 5p⁶ 6s¹ 4f¹⁴ 5d¹⁰ 6p². Each number and letter denote specific orbitals filled with electrons. Understanding these configurations allows us to infer an element’s chemical properties and how they might interact with other elements.