The Doppler effect is a fundamental principle in atomic spectroscopy that allows us to determine the velocity of atoms by analyzing the shift in the frequency of the light they emit or absorb. This comprehensive guide will walk you through the step-by-step process of computing velocity in atomic spectra, providing you with the necessary theoretical background, formulas, examples, and practical applications.
Understanding the Doppler Effect
The Doppler effect describes the change in the observed frequency of a wave when the source or the observer is in motion relative to each other. In the context of atomic spectra, the Doppler effect arises due to the motion of the atoms emitting or absorbing the light.
Theorem: Doppler Effect
The Doppler shift in frequency, Δν, is related to the velocity of the atom, v, by the following equation:
Δν = (v/c) × ν₀
where c is the speed of light, and ν₀ is the rest frequency of the atomic transition.
Measuring the Doppler Shift
To compute the velocity of an atom using the Doppler effect, you first need to measure the shift in the frequency of the emitted or absorbed light. This can be done using various spectroscopic techniques, such as:
- Spectroscopy: Analyzing the spectrum of the light emitted or absorbed by the atom to identify the shift in the spectral lines.
- Interferometry: Using interferometric techniques to measure the change in the wavelength of the light, which is inversely proportional to the change in frequency.
Once the shift in frequency, Δν, has been measured, you can use the Doppler equation to calculate the velocity of the atom.
Calculating the Velocity
The Doppler equation can be rearranged to solve for the velocity, v:
v = (Δν / ν₀) × c
Here’s an example of how to use this equation:
Physics Example:
Suppose an atom emits light with a rest frequency of 5.0 × 10^14 Hz, and the measured frequency of the emitted light is 5.1 × 10^14 Hz. Using the Doppler equation, we can calculate the velocity of the atom as follows:
v = (Δν / ν₀) × c
v = ((5.1 × 10^14 Hz – 5.0 × 10^14 Hz) / 5.0 × 10^14 Hz) × 3.0 × 10^8 m/s
v ≈ 3.0 × 10^6 m/s
Therefore, the velocity of the atom is approximately 3.0 × 10^6 m/s.
Radial Velocity and Full Velocity Vector
It’s important to note that the Doppler effect only provides a measure of the radial velocity of the atom, which is the component of the velocity directed along the line of sight between the atom and the observer. To determine the full velocity vector of the atom, you would need to measure the Doppler shift in multiple directions and use trigonometry to calculate the other components of the velocity.
Physics Numerical Problem:
A star is observed to have a redshift in its spectral lines, indicating that it is moving away from the observer. If the rest frequency of a particular spectral line is 6.0 × 10^14 Hz, and the measured frequency of the line is 5.8 × 10^14 Hz, calculate the radial velocity of the star.
To solve this problem, we can use the Doppler equation:
v = (Δν / ν₀) × c
v = ((5.8 × 10^14 Hz – 6.0 × 10^14 Hz) / 6.0 × 10^14 Hz) × 3.0 × 10^8 m/s
v ≈ -3.3 × 10^6 m/s
The negative sign indicates that the star is moving away from the observer.
Graphical Representation
The relationship between the Doppler shift in frequency and the velocity of the source can be visualized using a graph. The graph below shows the linear relationship between Δν and v, as described by the Doppler equation.
Figure: A graph showing the relationship between the Doppler shift in frequency and the velocity of the source, based on the equation Δν = (v/c) × ν₀.
Data Points, Values, and Measurements
To compute the velocity in atomic spectra, you’ll need to know the following data points, values, and measurements:
- Rest frequency of the atomic transition, ν₀: This is the frequency of the light emitted or absorbed by the atom when it is at rest.
- Measured frequency of the emitted or absorbed light, Δν: This is the frequency of the light as measured by the observer, which is shifted due to the Doppler effect.
- Speed of light, c: The speed of light in a vacuum, which is approximately 3.0 × 10^8 m/s.
- Radial velocity of the atom or object, v: The component of the velocity directed along the line of sight between the atom and the observer.
Units:
– Frequency: Hz (Hertz)
– Velocity: m/s (meters per second)
– Speed of light, c: 3.0 × 10^8 m/s
References
- Kinetic Energy and Velocity Lab – Arbor Scientific
- Measurable Quantity – an overview | ScienceDirect Topics
- Understanding Atomic Spectra – Chemistry LibreTexts
- 26.2 Borhr Model of the Atom and Atomic Spectra | Quantum Physics
- 14 Shape of Spectral Lines – Astrophysics Data System
By following the steps outlined in this comprehensive guide, you’ll be able to effectively compute the velocity of atoms in atomic spectra using the Doppler effect. Remember to always consider the limitations of the Doppler effect in measuring the full velocity vector, and be mindful of the various spectroscopic techniques available for measuring the frequency shift.
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