Question

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

Classically, the electron would radiate significant energy quickly, indicating classical physics' inadequacy for atomic descriptions.

Step-by-step solution

Identify the Problem

To find out how much energy an electron in a hydrogen atom would radiate per second if treated classically, we apply the classical electromagnetism concept which states that an accelerating charge radiates energy.

Use Larmor Formula

The rate of energy radiation by an accelerating charge is given by the Larmor formula: \[ P = \frac{2}{3} \frac{e^2 a^2}{c^3} \]where \( e \) is the elementary charge, \( a \) is the centripetal acceleration, and \( c \) is the speed of light.

Calculate Centripetal Acceleration

For an electron in circular motion, the centripetal acceleration \( a \) is given by \( a = \frac{v^2}{r} \), where \( v \) is the velocity of the electron and \( r = 0.0529 \times 10^{-9} \) m is the radius of the orbit.

Find Electron Velocity

The kinetic energy \( KE \) of the electron is given as 13.6 eV. Convert it to Joules: \[ KE = \frac{1}{2}mv^2 \rightarrow v^2 = \frac{2 \times KE}{m} \] Use \( 1 \, \text{eV} = 1.602 \times 10^{-19} \) Joules and \( m = 9.109 \times 10^{-31} \) kg for the mass of an electron.

Calculate Acceleration Value

Insert the calculated velocity \( v \) into \( a = \frac{v^2}{r} \) to find the centripetal acceleration \( a \).

Calculate Power Radiated

Insert the value of \( a \) back into the Larmor formula to find the power \( P \), which is the energy radiated per second.

Interpretation of the Result

Notably, classical physics predicts that the electron would radiate a large amount of energy, quickly spiraling into the nucleus. This discrepancy suggests the necessity of quantum mechanics to accurately describe atomic structure.

Key concepts

Classical Electromagnetism

Classical electromagnetism is a branch of physics that describes how electric charges and currents interact. At its core, it’s about forces and interactions that are transmitted by electric and magnetic fields. These fields manifest the power to both attract and repel charged particles.

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

The Larmor formula is an essential tool in understanding how accelerating charges like electrons radiate energy. It provides a mathematical framework for calculating this energy emission.

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:

• \( P = \frac{2}{3} \frac{e^2 a^2}{c^3} \)

In this equation, \( e \) represents the elementary charge of the electron, \( a \) is the acceleration, and \( c \) is the speed of light.

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

Centripetal acceleration, a key concept in the study of circular motion, refers to the acceleration experienced by an object moving in a circle at constant speed. It is always directed towards the center of the circle.

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:

• \( a = \frac{v^2}{r} \)

Here, \( v \) is the velocity, and \( r \) is the radius of the circular path. Understanding this acceleration helps in deriving how much kinetic energy is involved and in applying formulas like Larmor’s to possibly determine radiative power loss.

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

Kinetic energy is the energy that a body possesses due to its motion. For an electron, which is a subatomic particle with negligible mass compared to everyday objects, kinetic energy is significant in maintaining it in motion around a nucleus rather than collapsing into it.

The kinetic energy (KE) of an electron, especially in context to an atomic orbit, can be calculated using the formula:

• \( KE = \frac{1}{2} mv^2 \)

Where \( m \) is the mass of the electron and \( v \) is its velocity in orbit. In atomic scale energies, units like electron volts (eV) are often used because they are more convenient than joules.

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.