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Chapter 9: Problem 26

A proton and an electron have the same de Broglie wavelength. How do their speeds compare?

Short Answer

Expert verified
The proton's speed is much smaller than the electron's speed.

Step by step solution

01

Understand the de Broglie wavelength

The de Broglie wavelength \(\lambda\) is given by the de Broglie equation: \[\lambda = \frac{h}{p}\] where \(h\) is the Planck constant and \(p\) is the momentum of the particle.
02

Express momentum in terms of mass and velocity

Momentum \(p\) can be expressed as \[p = mv\] where \(m\) is the mass of the particle and \(v\) is its velocity.
03

Compare the de Broglie wavelengths of proton and electron

Since the proton and electron have the same de Broglie wavelength, their de Broglie equations can be set equal: \[\frac{h}{p_{proton}} = \frac{h}{p_{electron}}\] Simplifying this, we get: \[p_{proton} = p_{electron}\]
04

Substitute momentum expressions

Using \(p = mv\), the equality becomes: \[m_{proton} v_{proton} = m_{electron} v_{electron}\]
05

Solve for the ratio of velocities

Rearranging the equation to solve for the ratio of velocities: \[\frac{v_{proton}}{v_{electron}} = \frac{m_{electron}}{m_{proton}}\] Since the mass of the electron \(m_{electron} \) is much smaller than the mass of the proton \(m_{proton} \), \( v_{proton} \) will be much smaller than \( v_{electron} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

momentum
Momentum is a fundamental concept in physics. It measures the quantity of motion an object has. The formula for momentum is: \[ p = mv \] where \( p \) represents momentum, \( m \) stands for mass, and \( v \) is velocity. This relationship shows that momentum is directly proportional to both mass and velocity. Therefore, if either mass or velocity increases, momentum will also rise.
mass and velocity relationship
The relationship between mass and velocity is pivotal to understanding various physical phenomena. According to the formula \( p = mv \), for a given momentum, increasing the mass of an object will decrease its velocity, and vice versa. Using the de Broglie wavelength equation, \[ \lambda = \frac{h}{p} \] we can see that for a constant wavelength, an increase in mass results in a decrease in velocity to maintain the same momentum. This is crucial in differentiating how particles of different masses, like protons and electrons, behave.
proton and electron comparison
Protons and electrons, though both subatomic particles, have vastly different masses. A proton is about 1836 times heavier than an electron. Given that they have the same de Broglie wavelength, their momenta must be equal due to \( \lambda = \frac{h}{p} \). However, because \[ p = mv \] and \( m_{proton} \) is significantly larger than \( m_{electron} \), the velocity of the proton (\( v_{proton} \)) must be much smaller than the velocity of the electron (\( v_{electron} \)). Essentially, for the de Broglie wavelengths to be equal, the electron must move much faster than the proton.

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