Calculate the de Broglie wavelength of a 143-g baseball traveling at 85 mph. Is the wave nature of matter significant for a baseball?

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

Verified step by step guidance

Convert the mass of the baseball from grams to kilograms by dividing by 1000.

Convert the speed of the baseball from miles per hour (mph) to meters per second (m/s) using the conversion factor: 1 mph = 0.44704 m/s.

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

Substitute the values of \( h \), \( m \), and \( v \) into the de Broglie wavelength formula to calculate \( \lambda \).

Discuss whether the calculated wavelength is significant by comparing it to the size of the baseball, noting that the wave nature of matter is generally not significant for macroscopic objects like a baseball.

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. This concept illustrates that all matter exhibits both particle and wave characteristics, with the wavelength being inversely proportional to the momentum.

Momentum

Momentum is a physical quantity defined as the product of an object's mass and its velocity (p = mv). In the context of the de Broglie wavelength, momentum is crucial because it determines the wavelength of a particle. For macroscopic objects like a baseball, the momentum is typically large, resulting in a very small de Broglie wavelength, which makes wave properties negligible.

Wave-Particle Duality

Wave-particle duality is a fundamental principle of quantum mechanics stating that every particle or quantum entity exhibits both wave and particle properties. While this duality is significant for microscopic particles like electrons, it becomes less relevant for larger objects, such as a baseball, where the wave-like behavior is imperceptible due to their relatively large mass and momentum.

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