Book cover for Chemistry: The Molecular Nature of Matter

Chemistry: The Molecular Nature of Matter

Neil D. Jespersen, James E. Brady, Alison Hyslop

ISBN #9781118413920

7th Edition

3,064 Questions

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53,557 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter bridges classical and quantum theories to explain the behavior of light and electrons, forming the foundation of the quantum mechanical model of the atom. It emphasizes the dual nature of light, the quantization of electron energies, and introduces crucial quantum numbers that determine electron configurations. Through models such as Bohr's theory and the wave mechanical model, it connects atomic structure to periodic properties and provides tools for solving related problems, thereby offering a comprehensive understanding of atomic behavior and periodic trends.

Learning Objectives

1

Explain the dual nature of light (wave and particle) and its role in the quantum mechanical model of the atom.

2

Describe how electron energies are quantized as demonstrated by the hydrogen line spectrum and Bohr’s model.

3

Identify and define the quantum numbers (n, ?, m?, ms) used to describe electron configurations.

4

Analyze the relationship between atomic-level descriptions (electron configurations, orbital shapes) and periodic trends such as atomic size, ionization energy, and electron affinity.

5

Apply problem-solving tools, including the Rydberg equation and orbital diagrams, to predict electronic transitions and ground state configurations.

Key Concepts

CONCEPT

DEFINITION

Dual Nature of Light

The concept that light exhibits both wave-like and particle-like properties, essential for explaining phenomena such as diffraction and photoelectric effect.

Quantized Electron Energy

The idea that electrons can only occupy specific energy levels within an atom, rather than any arbitrary energy value.

Hydrogen Line Spectrum

The set of discrete wavelengths emitted by hydrogen atoms, which provided early evidence for quantized electron energy levels.

Bohr Model

A model of the atom where electrons travel in fixed orbits around the nucleus with quantized energy levels, explaining the hydrogen spectrum.

Quantum Numbers

A set of numbers (n, ℓ, mℓ, ms) that describe the properties and arrangement of electrons in an atom, including energy level, orbital shape, orbital orientation, and spin.

Orbital Diagrams

Visual representations of the electron configuration in an atom, showing the arrangement of electrons in orbitals and their spins.

Rydberg Equation

An equation that relates the wavelengths of spectral lines of hydrogen to the difference in energy levels of the electron transitions.

Periodic Trends

Variations in atomic properties such as atomic size, ionization energy, and electron affinity across the periodic table, explained by electron configurations and quantum mechanics.

Example Problems

Example 1

In general terms, why do we call light electromagnetic radiation?

Example 2

In general, what does the term frequency imply? What is meant by the term frequency of light? What symbol is used for it, and what is the SI unit (and symbol) for frequency?

Example 3

What is meant by the term wavelength of light? What symbol is used for it?

Example 4

Sketch a picture of a wave and label its wavelength and its amplitude. Indicate how frequency could be measured.

Example 5

Which property of light waves is a measure of the brightness of the light? Which specifies the color of the light? Which is related to the energy of the light?

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Step-by-Step Explanations

QUESTION

How do you calculate the wavelength of light emitted when an electron in a hydrogen atom transitions from n = 3 to n = 2?

STEP-BY-STEP ANSWER:

Step 1: Write the Rydberg equation: 1/λ = R (1/n1² - 1/n2²), where R is the Rydberg constant (≈1.097 x 10^7 m⁻¹).
Step 2: Assign n1 = 2 and n2 = 3 for the given transition.
Step 3: Substitute these values into the equation: 1/λ = 1.097 x 10^7 m⁻¹ (1/2² - 1/3²).
Step 4: Calculate the fractions: 1/2² = 1/4 = 0.25 and 1/3² = 1/9 ≈ 0.111.
Step 5: Compute the difference: 0.25 - 0.111 = 0.139.
Step 6: Multiply by R: 1/λ = 1.097 x 10^7 m⁻¹ x 0.139 ≈ 1.525 x 10^6 m⁻¹.
Step 7: Invert the result to find the wavelength: λ ≈ 1/(1.525 x 10^6 m⁻¹) ≈ 6.56 x 10⁻⁷ m, or about 656 nm.
Final Answer: The wavelength of the emitted light is approximately 656 nm.

Using the Rydberg Equation

QUESTION

How can you use quantum numbers to determine the electron configuration for an element?

STEP-BY-STEP ANSWER:

Step 1: Identify the total number of electrons present in the atom.
Step 2: Begin filling electrons into orbitals starting from the lowest energy level, following the Aufbau principle.
Step 3: Use the principal quantum number (n) to determine the energy level and the angular quantum number (ℓ) for the type of orbital (s, p, d, f).
Step 4: Apply the Pauli Exclusion Principle, assigning a unique set of quantum numbers (including mℓ and ms) for each electron.
Step 5: Represent the configuration using shorthand notation (e.g., [Ne]3s²3p⁵) if applicable.
Final Answer: The electron configuration is determined by sequentially filling orbitals, ensuring each electron has a unique set of quantum numbers.

Determining Electron Configuration Using Quantum Numbers

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Common Mistakes

  • Confusing the dual nature of light by applying only classical or only quantum theories.
  • Misidentifying or overlooking one of the quantum numbers when determining electron configurations.
  • Incorrectly applying the Rydberg equation, especially with errors in substituting n values or calculating differences in reciprocals of squared quantum numbers.
  • Overgeneralizing the Bohr model beyond hydrogen or misinterpreting its limitations compared to the modern quantum mechanical model.