Problem 39 - General Chemistry: Principles and Modern Applications

Question

Which must possess a greater velocity to produce matter waves of the same wavelength (such as \(1 \mathrm{nm}\) ), protons or electrons? Explain your reasoning.

Step-by-Step Solution

Answer

Electrons must possess a greater velocity compared to protons to produce matter waves of the same wavelength, as electrons have a smaller mass than protons.

Step 1: Recall the formula between de Broglie wavelength and momentum

The fundamental principle underlying this exercise is the de Broglie's wavelength formula, \(\lambda = \frac{h}{p}\), where \(\lambda\) is the de Broglie wavelength, \(h\) is the Planck's constant and \(p\) is the momentum of the particle.

Step 2: Understand the relationship between momentum and velocity

Momentum can be determined using the formula \(p = mv\), where m is the mass of the particle and v is its velocity. Thus, the velocity can be found using the formula:\(v = \frac{p}{m}\).

Step 3: Determine the velocity needed for each particle to have the same wavelength

The same wavelength can be achieved by having the same momentum, according to the de Broglie's wavelength formula. However, having the same momentum means having a different velocity, as the electrons and the protons have different masses. Now, since the mass of a proton is much larger than the mass of an electron, the smaller mass of the electron implies that it must move faster than the proton.

Key Concepts

Momentum and Velocity Relationship

In the world of physics, understanding the relationship between momentum and velocity is crucial, especially when discussing matter waves. Momentum, represented by the symbol \(p\), is the product of a particle's mass \(m\) and its velocity \(v\). The formula for momentum is \(p = mv\). This equation tells us that momentum depends directly on both mass and velocity.

When considering how velocity relates to momentum, you can rearrange the formula to find that \(v = \frac{p}{m}\). This equation states that the velocity of an object is its momentum divided by its mass. Thus, for two objects with the same momentum, the one with the smaller mass must have a greater velocity. This concept is especially important when determining the behavior of tiny particles like electrons and protons in terms of their speed.

Understanding this relationship helps us analyze how different particles need to move to achieve the same de Broglie wavelength.

Electron vs Proton Velocity

When attempting to determine which particle, an electron or a proton, must possess greater velocity to produce the same matter wave wavelength, it's vital to consider their masses.

- An electron has significantly less mass compared to a proton. Thus, to achieve the same momentum as a proton, the electron must travel faster.- Protons, having a larger mass, can achieve the same momentum at a lower speed compared to electrons.

Since the de Broglie wavelength formula \(\lambda = \frac{h}{p}\) indicates that a particle's wavelength is inversely proportional to its momentum, particles with the same momentum have identical wavelengths. Therefore, due to their smaller mass, electrons must attain higher velocities than protons to match their wavelengths. This is a fascinating example of how mass differences between particles affect their velocities and wave properties.

Matter Waves

Matter waves, also known as de Broglie waves, describe the wave-like behavior of particles. This concept originates from the de Broglie hypothesis, which proposes that not just light, but all particles also exhibit wave properties. This idea challenges the traditional view of particles as point-like objects. Instead, matter waves describe particles as having a wavelength, which can be calculated using the de Broglie wavelength formula \(\lambda = \frac{h}{p}\). - Here, \(\lambda\) is the wavelength of the particle, \(h\) is Planck's constant, a fundamental constant in nature, and \(p\) represents the particle's momentum.- The notion that particles, like electrons or protons, can have wave properties is fundamental to quantum mechanics. It forms the basis for understanding phenomena like electron diffraction and interference patterns.

The concept of matter waves bridges classical and quantum physics, illustrating that particles can behave like waves, influencing how we understand the micro-world. Therefore, whether discussing electrons, protons or any particle, their wave characteristics are crucial in predicting and explaining their behaviors at quantum scales.