Problem 48 - Chemistry The Central Science

Question

Among the elementary subatomic particles of physics is the muon, which decays within a few microseconds after formation. The muon has a rest mass 206.8 times that of an electron. Calculate the de Broglie wavelength associated with a muon traveling at \(8.85 \times 10^{5} \mathrm{cm} / \mathrm{s}\) .

Step-by-Step Solution

Answer

The de Broglie wavelength associated with a muon traveling at \(8.85 \times 10^{5} \mathrm{cm} / \mathrm{s}\) is calculated as \(λ = \dfrac{6.626 \times 10^{-27}}{206.8 \times (9.11 \times 10^{-28}) \times (8.85 \times 10^5)} \) cm.

Step 1: Calculate the mass of the muon

Given that the muon has a rest mass 206.8 times that of an electron, we will calculate the mass of the muon by multiplying the mass of an electron by this factor. Mass of electron, \(m_e = 9.11 \times 10^{-28}\) g Mass of muon, \(m_\mu = 206.8 \times m_e = 206.8 \times (9.11 \times 10^{-28})\) g

Step 2: Calculate the momentum of the muon

Using the formula for momentum \(p = mv\) and the given velocity \(v = 8.85 \times 10^5 \) cm/s, we will calculate the momentum: \(p = m_\mu v = 206.8 \times (9.11 \times 10^{-28}) \times (8.85 \times 10^5) \) g cm/s

Step 3: Calculate the de Broglie wavelength

Now that we have the momentum of the muon, we can use the de Broglie wavelength formula to calculate the wavelength: \(λ = \dfrac{h}{p}\), Where Planck's constant, \(h = 6.626 \times 10^{-27} \) erg s. \(λ = \dfrac{6.626 \times 10^{-27}}{206.8 \times (9.11 \times 10^{-28}) \times (8.85 \times 10^5)} \) cm Now that we have all the values plugged in, we can compute the de Broglie wavelength of the muon traveling at \(8.85 \times 10^5 \mathrm{cm} / \mathrm{s} \).

Key Concepts

Muon

The muon is a fascinating elementary subatomic particle that falls under the lepton family. What's intriguing about muons is their relatively short lifespan — they typically decay within a few millionths of a second after being formed. Despite this brief existence, muons play an essential role in physics experiments. Compared to their sibling particle, the electron, muons are significantly heavier: having a rest mass approximately 206.8 times that of an electron. This mass difference influences their behavior and enables scientists to study interactions at higher energies. Muons are created during cosmic ray interactions in the atmosphere and can be generated in laboratories for experimental purposes.

Subatomic Particles

Subatomic particles are the fundamental building blocks of matter that exist at scales smaller than atoms. These include particles like protons, neutrons, electrons, and more exotic ones like muons. Each subatomic particle plays a unique role in the constitution and interaction of matter:

• Protons and Neutrons: Compose the atomic nucleus, orchestrating the binding of atoms through nuclear forces.
• Electrons: Surround atomic nuclei and engage in electromagnetic interactions that give rise to chemical bonding.
• Exotic Particles: Particles like muons and neutrinos provide insights into the fundamental forces of nature.

Understanding subatomic particles is crucial to unraveling the complexities of the universe at both microscopic and cosmic scales.

Momentum Calculation

Momentum calculation is a pivotal concept in physics, serving as a measure of the quantity of motion an object possesses. It is calculated simply by multiplying an object's mass by its velocity: \[ p = mv \] Where:

• \( p \) is the momentum of the object,
• \( m \) is the mass of the object,
• \( v \) is the velocity of the object.

This formula is applicable across various contexts, from determining an athlete's motion to calculating subatomic particle trajectories like that of a muon. In the case of subatomic particles, precision in determining momentum contributes to crucial discoveries in particle physics and the understanding of quantum mechanics.

Planck's Constant

Planck's constant is a fundamental constant in quantum mechanics, playing a central role in the formulation of many quantum theories and equations. It connects the energy carried by a photon to its frequency, signifying a profound bridge between quantum phenomena and classical physics. Expressed as: \[ h = 6.626 imes 10^{-27} \text{ erg s} \] Planck's constant is crucial in the calculation of the de Broglie wavelength, which describes the wave-like behavior of particles. This constant allowed Max Planck, and later Albert Einstein, to develop the first quantum theory of light. Understanding this constant is essential to grasp the fundamentals of quantum mechanics and the study of subatomic particles like muons.