Emission and Absorption Spectra - AP Physics 2: Algebra-Based Study Guide

Getting Started

We will investigate the interaction between light and individual atoms. At this subatomic scale, the system consists of an atom's nucleus and its electrons. The core question we explore is: How do atoms absorb and release energy in the form of light, and why does this process reveal a unique "fingerprint" for every chemical element?

What You Should Be Able to Do

After working through this section, you should be able to:

• Describe the conditions under which an atom can absorb a photon of light.

• Explain the process of spontaneous emission, where an atom releases a photon of light.

• Use an energy level diagram to calculate the specific energy, frequency, and wavelength of a photon that can be absorbed or emitted by a particular atom.

• Relate an element's unique emission and absorption spectra to its unique set of atomic energy levels.

Key Concepts & Mechanisms

This topic is best understood through the lens of Interactions and Conservation. The central process is an energy exchange between a photon and an atom, governed by the law of conservation of energy.

System & Preconditions

• System: Our system is a single, isolated atom, which we model as a central nucleus with one or more electrons bound to it by the electric force.

• Idealizations: We assume the atom's electrons can only exist in a set of discrete, well-defined energy states, known as energy levels. We ignore smaller effects like interactions between atoms or the slight blurring of these energy levels.

• Preconditions for Interaction:

  1. Quantized Energy States: An electron cannot have just any amount of energy. It must occupy one of the specific energy levels ($E_n$) allowed for that atom. The lowest energy level is called the ground state, and all higher levels are called excited states. Atomic energy is typically measured in electron-volts (eV), where $1\text{ eV}=1.602\times 10^{-19}\text{ J}$.

  2. Energy Matching: For an interaction to occur, the energy of an incoming or outgoing photon must precisely match the energy difference ($\Delta E$) between two of the atom's allowed energy levels.

Key Steps / Relations

1. The Photon: Light is composed of discrete packets of energy called photons. The energy of a single photon ($E_{photon}$) is directly proportional to its frequency ($f$, in Hertz) and inversely proportional to its wavelength ($\lambda$, in meters). This relationship is described by the Planck-Einstein relation:

   $E_{photon}=hf=\frac{hc}{\lambda }$

   Here, $h$ is Planck's constant ($h\approx 6.63\times 10^{-34}\text{ J}\cdot \text{s}$ or $4.14\times 10^{-15}\text{ eV}\cdot \text{s}$), and $c$ is the speed of light in a vacuum ($c\approx 3.00\times 10^8\text{ m/s}$).

2. Interaction 1: Absorption. An atom in a lower energy state, $E_{initial}$, can absorb an incoming photon and jump to a higher energy state, $E_{final}$. This process is governed by the conservation of energy. The atom can only absorb the photon if the photon's energy is exactly equal to the energy gap.

   Energy Conservation for Absorption:$E_{photon}=E_{final}-E_{initial}=\Delta E$

   If a photon's energy does not precisely match an allowed energy transition, it will not be absorbed and will pass through the atom unaffected.

3. Interaction 2: Emission. An atom in an excited state, $E_{initial}$, is unstable and will spontaneously transition to a lower energy state, $E_{final}$. To conserve energy, the atom emits a single photon with an energy equal to the energy it lost.

   Energy Conservation for Emission:$E_{photon}=E_{initial}-E_{final}=\Delta E$

Outputs & Effects

• What Changes: The energy state of the atom changes (it becomes excited or de-excited). A photon is either removed from its path (absorption) or created and sent out in a random direction (emission).

• What Remains Constant: The total energy of the isolated atom-photon system is conserved in every interaction. The set of allowed energy levels for a given element is a fixed, fundamental property of that element.

Regulation & Limits

• Domain of Validity: This model is highly effective for single atoms or low-density gases where atoms do not significantly interact with each other. In solids or high-density gases, the energy levels can broaden into bands.

• Reading Energy Level Diagrams: These diagrams are the primary tool for visualizing this process. Energy is plotted on the vertical axis. By convention, the energy of a free, unbound electron (an ionized atom) is set to 0 eV. Therefore, the bound states (ground and excited states) have negative energy values. A "higher" energy level means "less negative." Transitions are shown as arrows pointing up for absorption and down for emission.

Key Models & Diagrams

The relationship between the atomic process, the energy level diagram, and the resulting spectrum is summarized below.

| Absorption | An incoming photon's energy matches the upward energy transition from a lower state ($E_i$) to a higher state ($E_f$).