Chapter 32: Problem 52
The electron in a hydrogen atom can be considered to be in a circular orbit with a radius of 0.0529 nm and a kinetic energy of 13.6 eV. If the electron behaved classically, how much energy would it radiate per second (see Challenge Problem 32.51)? What does this tell you about the use of classical physics in describing the atom?
Short Answer
Step by step solution
Identify the Problem
Use Larmor Formula
Calculate Centripetal Acceleration
Find Electron Velocity
Calculate Acceleration Value
Calculate Power Radiated
Interpretation of the Result
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Classical Electromagnetism
One critical element of classical electromagnetism is the idea that accelerating charges radiate energy. This means when a charged particle, like an electron, is in motion and changes direction or speed, it emits electromagnetic radiation. This principle is central to various phenomena and technologies. For example, the radiation emitted by electrons is the principle behind radio and microwaves.
Despite its wide applicability, classical electromagnetism faces challenges when applied to atomic structures. At the atomic scale, especially with electrons in an atom, classical predictions can markedly diverge from observed reality. This is largely because classical models don't incorporate quantum effects that are crucial at this scale.
Larmor Formula
According to the Larmor formula, the power radiated by an accelerating charge (such as an electron) is proportional to the square of its acceleration. Mathematically, this is expressed as:
- \( P = \frac{2}{3} \frac{e^2 a^2}{c^3} \)
This formula plays a pivotal role, especially in classical mechanics, in predicting how much energy is lost and hence cannot describe how atoms remain stable. Classical physics, using the Larmor formula, predicts that electrons in atoms should rapidly lose energy and spiral into the nucleus—a scenario not observed in reality.
Centripetal Acceleration
For an electron moving in a circular path around a nucleus, like in a simplified model of a hydrogen atom, centripetal acceleration is crucial. It maintains the electron in its circular orbit. The mathematical formula for this type of acceleration is:
- \( a = \frac{v^2}{r} \)
Given that the classical electron orbit would radiate energy due to this centripetal acceleration, classical electromagnetism would expect a rapid decay of the electron’s energy—something that requires quantum mechanics for a more accurate explanation.
Electron Kinetic Energy
The kinetic energy (KE) of an electron, especially in context to an atomic orbit, can be calculated using the formula:
- \( KE = \frac{1}{2} mv^2 \)
In the realm of classical physics applied to the atomic level, the consideration of kinetic energy leads to predictions of an unstable atom due to continuous energy radiation. However, this is contrary to the stable nature of atoms, leading to the adoption of quantum mechanics where kinetic energy is quantized and leads to discrete energy levels that maintain electrons in stable orbits.