Problem 24 - Chemistry and Chemical Reactivity

Question

A beam of electrons \(\left(m=9.11 \times 10^{-31} \mathrm{kg} / \text { electron }\right)\) has an average speed of \(1.3 \times 10^{8} \mathrm{m} / \mathrm{s}\). What is the wavelength of electrons having this average speed?

Step-by-Step Solution

Answer

The wavelength of the electrons is approximately \(5.596 \times 10^{-12} \mathrm{m}\).

Step 1: Understanding the Problem Requirements

To find the wavelength of an electron beam, we need to use the de Broglie wavelength formula, which relates a moving particle's momentum to its wavelength. The formula is given by \( \lambda = \frac{h}{p} \), where \( \lambda \) is the wavelength, \( h \) is the Planck's constant, and \( p \) is the momentum of the particle.

Step 2: Calculating the Momentum

The momentum \( p \) of a particle is calculated as \( p = mv \), where \( m \) is the mass and \( v \) is the velocity of the particle. Given the mass of an electron as \( 9.11 \times 10^{-31} \mathrm{kg} \) and the speed as \( 1.3 \times 10^8 \mathrm{m/s} \), we calculate: \[ p = (9.11 \times 10^{-31} \mathrm{kg})(1.3 \times 10^8 \mathrm{m/s}) = 1.1843 \times 10^{-22} \mathrm{kg \, m/s}. \]

Step 3: Using the de Broglie Wavelength Formula

Substitute the calculated momentum into the de Broglie wavelength formula. The Planck's constant is \( h = 6.626 \times 10^{-34} \mathrm{Js} \). Using \( \lambda = \frac{h}{p} \): \[ \lambda = \frac{6.626 \times 10^{-34} \mathrm{Js}}{1.1843 \times 10^{-22} \mathrm{kg \, m/s}} = 5.596 \times 10^{-12} \mathrm{m}. \]

Key Concepts

Electron Momentum

The momentum of an electron is all about its mass and speed. In physics, momentum (often symbolized by \( p \)) plays a crucial role in understanding how objects move. For any particle, like an electron, its momentum can be simply calculated using the formula:

• \( p = mv \)

where \( m \) is the mass of the electron and \( v \) is its velocity.
Electrons are tiny, with a mass of approximately \( 9.11 \times 10^{-31} \text{ kg} \).
Even with such little mass, electrons can achieve significant momentum because they move at high speeds, often close to a fraction of the speed of light!
In the problem we're examining here, the electron speed is given as \( 1.3 \times 10^{8} \text{ m/s} \). By multiplying this velocity with the electron's mass, we compute the momentum:

• \( p = 9.11 \times 10^{-31} \times 1.3 \times 10^{8} = 1.1843 \times 10^{-22} \, \text{kg m/s} \)

This result gives us a tangible idea of the electron's movement energy, essential for determining its wavelength.

Planck's Constant

Planck's constant is a fundamental value in quantum mechanics. It forms part of the bridge between the particle and wave descriptions of matter. This constant is denoted by \( h \) and has a value of \( 6.626 \times 10^{-34} \text{ Js} \).
The small value of Planck's constant reveals why quantum effects are generally only observable at extremely small scales.
In our exercise, it is a key ingredient in the de Broglie wavelength formula:

• \( \lambda = \frac{h}{p} \)

Using this special constant, we can calculate the wavelength of a particle given its momentum.
This constant embodies the essence of directional quantum physics by correlating energy levels to their respective frequencies or momenta.
As a student, you'll often see Planck's constant applied in situations where energy, frequency, and wavelength interrelate.

Particle Wave Properties

One of the most fascinating concepts in modern physics is that particles, like electrons, exhibit wave-like properties. This dual nature is at the heart of quantum mechanics. It's why we can describe electrons in terms of their wavelength.
A key figure in this exploration was the physicist Louis de Broglie, who proposed that every moving particle or object has an associated wavelength. "De Broglie wavelength" is a term used to describe this idea. The de Broglie wavelength is calculated using the formula:

• \( \lambda = \frac{h}{p} \)

Where \( \lambda \) is the wavelength, \( h \) is Planck's constant, and \( p \) is the momentum.
When you calculate a wavelength for particles like electrons, you're diving into their wave properties.
This understanding helps in fields like electron microscopy, where we exploit these wave-like properties to visualize objects smaller than light's wavelength.
By getting a grip on particle wave properties, you'll appreciate the nuances of how our universe operates at a fundamental level, intertwining the notions of particles and waves.