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 \(1.3 \times 10^{8} \mathrm{m} \cdot \mathrm{s}^{-1} .\) What is the wavelength of electrons having this average speed?

Step-by-Step Solution

Answer

The wavelength of electrons is approximately \( 5.60 \times 10^{-12} \text{ m} \).

Step 1: Identify Known Values

We start by noting down the values we have. The mass of an electron is \( m = 9.11 \times 10^{-31} \text{ kg} \) and the average speed \( v = 1.3 \times 10^8 \text{ m/s} \).

Step 2: Determine Formula to Use

Given these values, we need to find the wavelength of the electrons. This can be achieved using the de Broglie wavelength formula, defined as \( \lambda = \frac{h}{mv} \), where \( h \) is the Planck's constant \( h = 6.63 \times 10^{-34} \text{ J·s} \).

Step 3: Substitute Values into the Formula

Substitute the known values into the de Broglie wavelength formula: \[ \lambda = \frac{6.63 \times 10^{-34} \text{ J·s}}{9.11 \times 10^{-31} \text{ kg} \times 1.3 \times 10^8 \text{ m/s}}. \]

Step 4: Calculate the Wavelength

Now, perform the calculation: \[ \lambda = \frac{6.63 \times 10^{-34}}{(9.11 \times 10^{-31}) \times (1.3 \times 10^8)} = \frac{6.63 \times 10^{-34}}{1.1843 \times 10^{-22}}. \] \[ \lambda \approx 5.60 \times 10^{-12} \text{ m}. \]

Step 5: Result Interpretation

The calculated wavelength \( \lambda \approx 5.60 \times 10^{-12} \text{ m} \) represents the wavelength of electrons moving at an average speed of \( 1.3 \times 10^8 \text{ m/s} \). This wavelength is in the picometer range, typical for particles like electrons.

Key Concepts

Electron Speed

In this scenario, we are dealing with electrons moving at a significant speed. Understanding electron speed is crucial in particle physics, as the behavior of particles at different velocities can vary greatly.

• Electrons have mass: \(9.11 \times 10^{-31} \text{ kg}\).
• In the given problem, the electrons' speed is \(1.3 \times 10^8 \text{ m/s}\).

This speed suggests that electrons are traveling at almost half the speed of light! When electrons move at such high speeds, we often need to consider relativistic effects. However, for this basic calculation, we are using classical speed to ensure simplicity in our findings.

Planck Constant

The Planck constant is a fundamental figure in quantum mechanics. It is essential for calculations involving particle wavelengths, notably in the realm of wave-particle duality.The value of the Planck constant is \(6.63 \times 10^{-34} \text{ J·s}\).

• This constant relates the energy of a photon to its frequency: \(E = h u\).
• It is critical when calculating the de Broglie wavelength.

Since electrons, like other particles, exhibit wave-like properties, the Planck constant provides a bridge between classical and quantum physics with its unique ability to quantify the size of quantum effects.

Wavelength Calculation

To compute the wavelength of the electrons, we apply the de Broglie wavelength formula:\[ \lambda = \frac{h}{mv} \]Here, \(\lambda\) represents the wavelength, \(h\) is the Planck constant, \(m\) refers to the mass of the electron, and \(v\) is the speed of the electron.

• The aim is to find the wavelength at a given speed.
• Plugging in the values: \( \lambda = \frac{6.63 \times 10^{-34}}{9.11 \times 10^{-31} \times 1.3 \times 10^8} \).

The calculation yields a wavelength of approximately \(5.60 \times 10^{-12} \text{ m}\), which indicates the electron's wave feature at a given speed. This wavelength lies in the picometer range, demonstrating the particle's dual nature.

Particle Physics

Particle physics explores the nature of subatomic particles, like electrons. It explains how these particles behave and interact at high velocities, as well as incorporating both particle and wave-like traits. The de Broglie wavelength is central in understanding wave-particle duality. In essence, subatomic particles can exhibit characteristics of both waves and particles.

• Particles have measurable wavelengths depending on their momentum.
• This duality supports quantum theories, enriching our understanding of atomic scales.

By comprehending these concepts, scientists can predict behaviors and interactions of particles, influencing fields from electronics to quantum computing.