Atomic Emission Spectrum and Bohr Models: Structured Study Notes

Atomic Emission Spectrum and Bohr Models

Introduction

The atomic emission spectrum and Bohr model are foundational concepts in general chemistry, explaining how atoms absorb and emit light and how electrons are arranged in atoms. These topics are essential for understanding atomic structure, quantum mechanics, and the behavior of elements.

The Quantization of Energy

Energy Quantization in Atoms

Energy in atoms is quantized, meaning electrons can only occupy specific energy levels. Energy can be absorbed or emitted only in discrete amounts called quanta.

• Quantum: The smallest possible unit of energy, given by $E = h\nu$, where $h$ is Planck's constant and $\nu$ is the frequency.

• Allowed Transitions: Only whole number multiples of $h\nu$ are permitted (e.g., $1 \text{ quanta} = h\nu$, $2 \text{ quanta} = 2h\nu$).

• Not Allowed: Fractional multiples (e.g., $\frac{1}{2}h\nu$, $\frac{2}{3}h\nu$) are not permitted.

Allowed	Not Allowed
1 quanta = $h\nu$	$\frac{1}{2}$ quanta = $\frac{1}{2}h\nu$
2 quanta = $2h\nu$	$\frac{2}{3}$ quanta = $\frac{2}{3}h\nu$
3 quanta = $3h\nu$	$\frac{7}{3}$ quanta = $\frac{7}{3}h\nu$

Analogy: Quantized energy levels are like climbing a ladder (discrete steps), while continuous energy is like walking up a ramp (any value possible).

Absorption and Emission Spectrum

Types of Spectra

• Absorption Spectrum: Produced when atoms or molecules absorb specific wavelengths of light, resulting in a spectrum with dark lines (missing colors).

• Emission Spectrum: Produced when excited atoms emit light at specific wavelengths, resulting in bright lines on a dark background.

• Spectral Signature: Each element has a unique pattern of absorption and emission lines, which can be used to identify unknown samples.

Comparison Table: Absorption vs. Emission Spectra

Element	Absorption	Emission
Sodium	Dark lines at specific wavelengths	Bright lines at same wavelengths
Nitrogen	Dark lines at specific wavelengths	Bright lines at same wavelengths
Hydrogen	Dark lines at specific wavelengths	Bright lines at same wavelengths
Oxygen	Dark lines at specific wavelengths	Bright lines at same wavelengths

Application: Identifying Unknown Samples

• By comparing the emission or absorption spectrum of an unknown sample to known spectra, the element can be identified.

Atomic Spectrum of Hydrogen

Discrete Energy Levels and Spectral Lines

The hydrogen atom emits light at specific wavelengths, corresponding to electron transitions between energy levels.

• Rydberg Formula: $\frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$, where $R$ is the Rydberg constant, $n_1$ and $n_2$ are integers ($n_2 > n_1$).

• Energy of Photon: $E = h\nu = \Delta E = E_{\text{final}} - E_{\text{initial}}$

• When an electron drops from a higher to a lower energy level, a photon is emitted with energy equal to the difference between the two levels.

Example Calculation

• Transition from $n=4$ to $n=1$ in hydrogen:

• $E = -2.18 \times 10^{-18} \left( \frac{1}{n_{\text{final}}^2} - \frac{1}{n_{\text{initial}}^2} \right)$

• Calculate the energy and wavelength of the emitted photon using the above formula.

Ground State and Excited State

Definitions

• Ground State: The lowest energy state of an atom, where electrons occupy the lowest possible energy levels.

• Excited State: Any energy state higher than the ground state, where electrons have absorbed energy and moved to higher levels.

Absorption and Emission Processes

Mechanism

• Absorption: Electron absorbs a photon and moves to a higher energy level ($\Delta E = h\nu$).

• Emission: Electron drops to a lower energy level and emits a photon ($\Delta E = h\nu$).

Photoelectric Effect vs. Atomic Spectra

Comparison

• Photoelectric Effect: Electrons are ejected from a material when it absorbs photons with energy greater than the binding energy. Excess energy becomes kinetic energy of the electron.

• Atomic Absorption/Emission: Electrons move between quantized energy levels only if the photon energy matches the energy difference exactly.

Bohr Model of the Atom

Key Features

• Electrons move in fixed circular orbits around the nucleus, each with a specific energy.

• Electrons can transition between orbits by absorbing or emitting a quantum of energy.

• Bohr model successfully explains the emission spectrum of hydrogen and hydrogen-like atoms.

Bohr Model Equation

• Energy of electron in orbit $n$: $E_n = -2.18 \times 10^{-18} \frac{1}{n^2}$

• Energy change for transition: $\Delta E = E_{\text{final}} - E_{\text{initial}}$

Strengths and Limitations

• Strengths: Explains hydrogen spectrum, visualizes energy levels, simple model.

• Limitations: Does not account for wave properties of electrons, cannot explain spectra of multi-electron atoms, does not predict intensities of spectral lines.

Bohr Models and Wave Mechanics

Comparison of Models

• Bohr Model: Electrons in fixed orbits (circular paths).

• Wave Mechanics Model: Electrons described by probability distributions (charge density clouds), not fixed paths.

Summary Table: Bohr vs. Wave Mechanics

Model	Description
Bohr Model	Electrons in circular orbits, fixed energy levels
Wave Mechanics	Electrons as probability clouds, energy levels from quantum numbers

Conclusion

The atomic emission spectrum and Bohr model provide a framework for understanding atomic structure and the quantization of energy. These concepts are essential for interpreting spectral data, identifying elements, and appreciating the development of quantum theory in chemistry.

Additional info: Some context and equations have been expanded for clarity and completeness.