Question

If traveling at equal speeds, which of the following matter waves has the longest wavelength? Explain. (a) electron; (b) proton; (c) neutron; (d) \(\alpha\) particle \(\left(\mathrm{He}^{2+}\right)\).

Short answer

The electron (a) has the longest wavelength when traveling at equal speeds since it has the smallest mass.

Step-by-step solution

Identify the Masses of the Particles

Firstly, determine the masses of the particles in question. The respective masses for the electron (e), proton(p), neutron(n) and alpha (α) particle are: (e) \( 9.11E-31 \) kg, (p) \( 1.67E-27 \) kg, (n) \( 1.67E-27 \) kg, (α) \( 6.64E-27 \) kg.

Apply the de Broglie equation

Since all the particles are traveling at equal speeds, according to the de Broglie equation \( \lambda = \frac{h}{mv} \), the particle with the least mass will have the longest wavelength.

Determine the Particle with the Smallest Mass

Comparing the masses of the particles, it is observed that the electron has the smallest mass.

Conclude

Therefore, the electron (option a) having the smallest mass, traveling at equal speeds as the rest, will have the longest wavelength according to the de Broglie equation.

Key concepts

de Broglie Wavelength

Matter waves, also known as de Broglie waves, are a fascinating concept that was introduced by physicist Louis de Broglie. The de Broglie wavelength is a fundamental concept in quantum mechanics. According to de Broglie, every particle or object in motion has an associated wavelength, known as its de Broglie wavelength.To calculate the de Broglie wavelength, the formula \( \lambda = \frac{h}{mv} \) is used, where:

• \( \lambda \) represents the wavelength,
• \( h \) is Planck's constant (approximately \( 6.63 \times 10^{-34} \ \text{m}^2 \cdot \text{kg} / \text{s} \)),
• \( m \) is the mass of the particle, and
• \( v \) is the velocity of the particle.

Since the velocity \( v \) and Planck's constant \( h \) are constants in many comparative analyses, the wavelength is inversely proportional to the mass of the particle. Thus, particles with smaller masses will have longer wavelengths when moving at the same speed.

Particle Mass

Particle mass plays a crucial role in determining the properties of matter waves. Mass influences the wavelength of a particle's wave-like nature. The smaller the mass, the longer the wavelength at a given speed.
In the exercise, four particles are considered: electrons, protons, neutrons, and alpha particles. Their respective masses are:

• Electron: \( 9.11 \times 10^{-31} \) kg
• Proton: \( 1.67 \times 10^{-27} \) kg
• Neutron: \( 1.67 \times 10^{-27} \) kg
• Alpha particle: \( 6.64 \times 10^{-27} \) kg

Evidently, the electron has a significantly smaller mass compared to the other particles. This smaller mass means that, at the same speed, the electron exhibits the longest wavelength based on de Broglie's equation.

Wave-Particle Duality

Wave-particle duality is a central concept in quantum mechanics that reveals the dual nature of matter and energy. It asserts that every particle or quantum entity can be described as both a wave and a particle.
In the context of the given exercise, this principle helps explain how subatomic particles, like electrons, can behave as waves under certain conditions.

Characteristics of Duality

This duality is not only a theoretical notion but has practical implications:

• In certain experiments, particles such as electrons can exhibit wave-like behaviors such as interference and diffraction.
• Conversely, the same electrons can behave like discrete particles, noticeable when colliding with detectors or surfaces.

Understanding wave-particle duality enhances comprehension of how subatomic particles interact with the rest of the universe, allowing scientists to predict and explore behaviors at the quantum level.

Subatomic Particles

Subatomic particles are the building blocks of atoms, consisting primarily of electrons, protons, neutrons, and sometimes alpha particles. These tiny constituents are essential for the composition and structure of matter.

Main Types of Subatomic Particles

• Electrons: Negatively charged particles that orbit the nucleus. They have a very small mass compared to protons and neutrons, making their behavior more wave-like.
• Protons: Positively charged particles found within the nucleus. They have a much larger mass than electrons.
• Neutrons: Neutral particles, also located in the nucleus, with a mass almost equivalent to that of protons.
• Alpha Particles: Composed of 2 protons and 2 neutrons, they are positively charged and more massive than a single proton or neutron.

The behavior of these particles, particularly at the quantum level, can be explained by principles such as de Broglie wavelength and wave-particle duality, demonstrating the touching complexity in the interplay between fundamental forces and matter.