A 0.22-caliber handgun fires a 1.9-g bullet at a velocity of 745 m/s. Calculate the de Broglie wavelength of the bullet. Is the wave nature of matter significant for bullets?

Name: Chemistry: A Molecular Approach
Author: Tro
ISBN: 9780137832217

Verified step by step guidance

Identify the formula for the de Broglie wavelength: \( \lambda = \frac{h}{mv} \), where \( \lambda \) is the wavelength, \( h \) is Planck's constant \( 6.626 \times 10^{-34} \text{ m}^2 \text{ kg/s} \), \( m \) is the mass in kilograms, and \( v \) is the velocity in meters per second.

Convert the mass of the bullet from grams to kilograms. Since 1 gram = 0.001 kilograms, multiply the mass of the bullet (1.9 g) by 0.001 to get the mass in kilograms.

Substitute the values for \( h \), \( m \), and \( v \) into the de Broglie wavelength formula. Use the converted mass in kilograms and the given velocity (745 m/s).

Calculate the de Broglie wavelength using the substituted values. This will give you the wavelength in meters.

Discuss the significance of the wave nature of matter for bullets. Consider the size of the calculated wavelength compared to the size of the bullet and typical wavelengths of visible light to determine if the wave nature is significant.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

de Broglie Wavelength

The de Broglie wavelength is a concept in quantum mechanics that describes the wave-like behavior of particles. It is calculated using the formula λ = h/p, where λ is the wavelength, h is Planck's constant, and p is the momentum of the particle. For a bullet, its momentum is the product of its mass and velocity, which allows us to determine its associated wavelength.

Wave-Particle Duality

Wave-particle duality is a fundamental principle of quantum mechanics stating that every particle exhibits both wave and particle properties. This concept is crucial for understanding phenomena at the atomic and subatomic levels, but for macroscopic objects like bullets, the wave nature is negligible compared to their particle characteristics, making classical mechanics more applicable.

Significance of Wave Nature

The significance of the wave nature of matter is context-dependent. For microscopic particles, such as electrons, wave properties are essential for explaining behaviors like interference and diffraction. However, for macroscopic objects like bullets, the de Broglie wavelength is extremely small, rendering wave effects insignificant in practical terms, thus classical physics suffices for their analysis.