Problem 37 - Chemistry The Molecular Science

Question

Arrange these items from smallest to largest de Broglie wavelength: baseball; bowling ball; electron moving at the velocity of light; the Moon; neon atom.

Step-by-Step Solution

Answer

The order is: Moon, bowling ball, baseball, neon atom, electron.

Step 1: Identify the de Broglie Wavelength Formula

To rank items by de Broglie wavelength, use the formula: \( \lambda = \frac{h}{mv} \), where \( \lambda \) is the wavelength, \( h \) is Planck's constant, \( m \) is the mass, and \( v \) is the velocity.

Step 2: Determine Mass and Velocity

List the mass and typical velocities for each object: baseball (~145g, ~40m/s), bowling ball (~7kg, ~5m/s), electron (~9.11x10^-31 kg, close to light speed), Moon (~7.35x10^22 kg, orbital velocity ~1022m/s), neon atom (~20.18 amu, thermal speed ~500m/s).

Step 3: Calculate Their Momentum

Calculate momentum \( mv \) for each object: baseball (~5.8 kg·m/s), bowling ball (~35 kg·m/s), electron (negligible), Moon (~7.51x10^25 kg·m/s), neon atom (~1.67x10^-23 kg·m/s).

Step 4: Compute de Broglie Wavelengths

Using \( \lambda = \frac{h}{mv} \), calculate the de Broglie wavelengths: electron has the largest (close to one angstrom), while massive, slow-moving objects like the baseball, bowling ball, and Moon have tiny wavelengths.

Step 5: Arrange by Wavelength

Order from smallest to largest wavelength: Moon, bowling ball, baseball, neon atom, electron.

Key Concepts

Quantum Mechanics

Quantum mechanics is a branch of physics that explores the behavior of matter and energy at the smallest scales, such as atoms and subatomic particles. Unlike classical physics, which describes a largely deterministic universe, quantum mechanics allows for probabilistic outcomes.
It reveals a fascinating world where particles can exist in multiple states at once, a phenomenon known as superposition. Observations on this scale are often counter-intuitive, leading to a better understanding of how even large systems can act under certain quantum rules.

• Quantum mechanics explains phenomena like electron orbits in atoms and the behavior of semiconductors, crucial for modern electronics.
• It introduces the concept of quantization, which means energy is discrete rather than continuous.

This branch is essential for understanding the de Broglie wavelength, a key principle that connects particles with their wave nature.

Wave-Particle Duality

Wave-particle duality is the concept that every particle or quantum entity can exhibit both wave and particle properties. This dual nature becomes significant at the atomic and subatomic levels.
In classical mechanics, particles like billiard balls follow well-defined paths and waves like sound or light spread out. However, in the quantum realm, entities can display characteristics of both.

• Louis de Broglie suggested that particles such as electrons have wave-like properties, characterized by the de Broglie wavelength.
• This is demonstrated by experiments like electron diffraction, where electrons exhibit interference patterns typical of waves.

The de Broglie wavelength is calculated using the particle's momentum, highlighting the dual nature of matter.

Planck's Constant

Planck's constant (\( h \)) is a fundamental quantity in quantum mechanics that represents the scale at which quantum effects become significant. Its value is approximately \( 6.626 imes 10^{-34} ext{Js} \) and is pivotal in the equation for the de Broglie wavelength \( \lambda = \frac{h}{mv} \).
It sets the foundation for the concept of quantization, indicating the smallest possible energy units involved in interactions at the quantum level.

• Introduced by Max Planck, this constant was originally used to explain black-body radiation, a problem classical physics couldn't solve.
• Planck's constant appears in several fundamental quantum mechanics equations, such as the Schrödinger equation and the Planck-Einstein relation.

It provides the link between energy and frequency, making it an essential component in understanding the dual nature of particles.

Momentum

Momentum (\( p \)) in physics is the product of an object's mass and its velocity (\( p = mv \)). It is a vector quantity, having both magnitude and direction. In the context of quantum mechanics and de Broglie wavelength, momentum helps in determining the wave nature of particles.
For objects moving at or near the speed of light, the mass and the relativistic velocity come into play, making calculations more complex.

• As particles move faster or have greater mass, their momentum increases, resulting in shorter de Broglie wavelengths.
• Conversely, slower or less massive particles have longer wavelengths, illustrating the particle's wave-like characteristics.

Understanding momentum allows us to differentiate how differently sized objects exhibit their dual nature, as seen when comparing particles like electrons to cosmic bodies like the Moon.