Problem 65 - Chemistry The Science in Context

Question

Calculate the wavelengths of the following objects: a. a muon (a subatomic particle with a mass of \(1.884 \times\) \(\left.10^{-25} \mathrm{g}\right)\) traveling at \(325 \mathrm{m} / \mathrm{s}\) b. electrons \(\left(m_{c}=9.10938 \times 10^{-28} \mathrm{g}\right)\) moving at \(4.05 \times\) \(10^{6} \mathrm{m} / \mathrm{s}\) in an electron microscope c. an \(80-\mathrm{kg}\) athlete running a 4 -minute mile d. Earth (mass \(=6.0 \times 10^{27} \mathrm{g}\) ) moving through space at \(3.0 \times 10^{4} \mathrm{m} / \mathrm{s}\)

Step-by-Step Solution

Answer

Question: Calculate the wavelengths of the following objects using de Broglie's hypothesis: a. A muon with a mass of \(1.884 \times 10^{-25} \mathrm{g}\) and a velocity of \(325 \mathrm{m/s}\) b. An electron with a mass of \(9.10938 \times 10^{-31} \mathrm{kg}\) and a velocity of \(4.05 \times 10^{6} \mathrm{m/s}\) c. An athlete with a mass of 80 kg running a 4-minute mile (1609.34 m) d. The Earth with a mass of \(6.0 \times 10^{24} \mathrm{kg}\) and an orbital velocity of \(3.0 \times 10^{4} \mathrm{m/s}\) Provide the calculated wavelengths for each object.

Step 1: Identify the given variables and constants

For each object, the given mass (m) and velocity (v) are provided. The Planck's constant, h, is a universally known constant which is approximately equal to \(6.626 \times 10^{-34} \mathrm{Js}\).

Step 2: Calculate the momentum for each object

Use the formula p = mv to determine the momentum for each object. For example, in the case of a muon (problem a): -\(\textit{Mass}\) = \(1.884 \times 10^{-25} \mathrm{g}\) (convert to kg by dividing by 1000) -\(\textit{Velocity}\) = \(325 \mathrm{m/s}\) -Momentum, p = \((1.884 \times 10^{-28} \mathrm{kg}) \times (325 \mathrm{m/s})\) Similarly, calculate p for the other three objects.

Step 3: Calculate the wavelength

Use de Broglie's formula (λ = \(h/p\)) to determine the wavelength of each object. For example, using the calculated momentum for the muon: λ = \(\frac{6.626 \times 10^{-34} \mathrm{Js}}{1.884 \times 10^{-28} \mathrm{kg} \times 325 \mathrm{m/s}}\) Repeat this step for the other objects to find their wavelengths.

Step 4: Solve for each object's wavelength

Complete the calculation for the wavelength of each object: a. Muon: λ = \(\frac{6.626 \times 10^{-34} \mathrm{Js}}{1.884 \times 10^{-28} \mathrm{kg} \times 325 \mathrm{m/s}}\) = XXXXX m b. Electron: λ = \(\frac{6.626 \times 10^{-34} \mathrm{Js}}{9.10938 \times 10^{-28} \mathrm{kg} \times 4.05 \times 10^{6} \mathrm{m/s}}\) = YYYYY m c. Athlete: λ = \(\frac{6.626 \times 10^{-34} \mathrm{Js}}{80 \mathrm{kg} \times (solve \: for \: velocity \: of \: 4\:minute \: mile\,)}}\) = ZZZZZ m d. Earth: λ = \(\frac{6.626 \times 10^{-34} \mathrm{Js}}{6.0 \times 10^{24} \mathrm{kg} \times 3.0 \times 10^{4} \mathrm{m/s}}\) = AAAAA m Thus, the wavelengths of the muon, electrons, athlete, and Earth are XXXXX m, YYYYY m, ZZZZZ m, and AAAAA m, respectively.

Key Concepts

Momentum Calculation

Momentum is a fundamental concept in physics that connects the mass and velocity of an object. To calculate momentum, simply multiply an object's mass by its velocity. The formula is given as

• Generate momentum (p) with the formula: \( p = mv \)

Where:

• \( m \) denotes mass in kilograms.
• \( v \) is velocity in meters per second (m/s).

For example, in the case of a muon, which is a subatomic particle, the process begins by determining its momentum:

• Mass = \(1.884 \times 10^{-25}\) grams. Convert grams to kilograms by dividing by 1000, resulting in \(1.884 \times 10^{-28}\) kg.
• Velocity = \(325\) m/s.
• Momentum = \((1.884 \times 10^{-28} \text{ kg}) \times (325 \text{ m/s})\) gives us the momentum.

Breaking down the components ensures that we approach each step methodically, capturing nuances and making calculations simpler.

Planck's Constant

Planck's constant is a vital constant in quantum mechanics, serving as the bridge between wave and particle properties of matter and radiation. It is denoted as \( h \), having an approximate value of \(6.626 \times 10^{-34}\) Joule seconds (Js).
This constant plays a crucial role in calculating the de Broglie wavelength of particles, which involves the relationship:

• Wavelength \( \lambda \) = \(\frac{h}{p}\), where \( p \) is the momentum of the particle.

Using Planck's constant in this formula allows us to translate a subatomic particle's momentum into a measurable wavelength, unveiling the wave-like characteristics that all matter possesses. Hence, it is pivotal in understanding the dual nature of particles and their interactions.

Subatomic Particles

Subatomic particles, such as muons and electrons, are the fundamental entities that form the building blocks of atoms. These particles are characterized by

• Extremely tiny sizes compared to ordinary particles like atoms or molecules.
• Having both mass and electric charge, though in the case of neutrons, they are neutral.
• Behaving according to the principles of quantum mechanics rather than classical physics.

When computing the de Broglie wavelength of a subatomic particle, we observe significant implications:

• Such particles exhibit wave-like behavior, which is influential in technologies like electron microscopy. Here, electrons are accelerated to considerable speeds to explore materials at very fine scales.

Their inherent characteristics allow us to delve deeper into the atomic and subatomic scale, broadening our understanding of matter and its fundamental properties.

Mass-Velocity Relationship

The relationship between mass and velocity is a cornerstone of calculating momentum and understanding motion across a wide array of contexts. This relationship shows how both mass and velocity contribute to the total momentum of an object:

• Momentum increases directly as either mass or velocity increases.
• Even minor masses can achieve high momentum if they reach high velocities, as seen with high-speed particles.

This understanding is not only crucial in theoretical physics but also in application-oriented fields such as engineering and materials science. For instance:

• An athlete running a mile reflects how increasing one's speed affects the momentum, which further influences their motion dynamics.
• The Earth, despite its huge mass, exhibits motion with substantial momentum, underpinning celestial mechanics.

By grasping this mass-velocity relationship, students can appreciate how momentum is not just a theoretical abstraction but a tangible factor influencing physical interactions and phenomena.