Problem 48 - Chemistry: The Central Science in SI Units, Global Edition

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 \(8.85 \times 10^{5}\,\text{cm/s}\) can be calculated using the following steps: 1. Determine the mass of the muon: \(m_μ = m_e × 206.8\), where \(m_e = 9.109 \times 10^{-28}\,\text{g}\). 2. Calculate the momentum of the muon: \(p = m_μ × v\), where \(v = 8.85 \times 10^{5}\,\text{cm/s}\). 3. Calculate the de Broglie wavelength: \(λ = \frac{h}{p}\), where \(h = 6.626 \times 10^{-34}\,\text{J s}\) (converted to appropriate units). After performing the calculations, we obtain the de Broglie wavelength, λ, having appropriate units and a valid result.

Step 1: Find the mass of muon

First, we need to find the mass of the muon using the given information that its rest mass is 206.8 times the mass of an electron. The mass of an electron is approximately: \[m_e = 9.109 \times 10^{-28}\,\text{g}\] So the mass of the muon can be calculated as: \[m_μ = m_e × 206.8\]

Step 2: Calculate the momentum of the muon

Now, we can calculate the momentum of the muon using the formula: \[p = m_μ × v\] where p is the momentum, m_μ is the mass of the muon calculated in step 1, and v is the given velocity of the muon: \(8.85 \times 10^5 \, \text{cm/s}\).

Step 3: Calculate the de Broglie wavelength

Using the formula for de Broglie wavelength, we can calculate it using the values found in the previous steps: \[λ = \frac{h}{p}\] The Planck constant, h, is approximately: \[h = 6.626 \times 10^{-34} \, \text{J s}\] Remember to convert h to appropriate units (erg s) and convert the mass of muon and velocity to appropriate units before plugging the values into the formula. Then, calculate λ.

Step 4: Check the result

After solving the equation, you should check the result to ensure it is a valid wavelength, and that it has the appropriate units for a de Broglie wavelength.

Key Concepts

Muon Rest Mass

A muon is an elementary particle, much like an electron, but it is significantly heavier. The concept of the muon's rest mass is pivotal when understanding its physical properties. The rest mass of a muon is 206.8 times the rest mass of an electron. This means it has greater inertia and gravity interaction in a resting state.
To find the muon's actual mass when stationary, consider the known mass of an electron. The electron has a mass of about \(9.109 \times 10^{-28}\,\text{g}\). By multiplying this mass by 206.8, one can determine the muon's rest mass. This value is important in various calculations involving muons, such as determining their momentum when they are in motion.

Subatomic Particles

Subatomic particles include protons, neutrons, electrons, and other less common particles like muons. These particles are the building blocks of atoms. While protons and neutrons form the atom's nucleus, electrons and other particles exist outside this nucleus.
Muons fall into the category of leptons, which are a type of elementary particle. They are similar to electrons but heavier and less stable. Muons don't last long—they decay quickly, but their interactions and properties make them an interesting subject of study in the physics of subatomic particles.

• Leptons: Include electrons and muons
• Quarks: Combine to form protons and neutrons
• Bosons: Facilitate forces between particles

Understanding these categories helps us comprehend how they build the universe at a microscopic level and how different particles influence each other.

Electron Mass

Electrons are one of the fundamental components of matter. Their mass, though tiny, is a crucial reference point for understanding other subatomic particles. Electrons have a rest mass of approximately \(9.109 \times 10^{-28}\,\text{g}\). This value is used to compare and calculate the mass of other particles, such as muons, as seen earlier.
The small mass of electrons plays a significant role in their energy levels and movement within atoms. They are responsible for electrical conductivity and chemical interactions between atoms. Electrons are vital in shaping the element's atomic structure and the atom's overall identity.

Momentum Calculation

Momentum is a fundamental concept in physics that describes the motion of an object and is given by the product of its mass and velocity. For a muon, calculating its momentum involves using its rest mass and its velocity when in motion.
The formula used is:\[ p = m_μ \times v \]Where \(p\) represents momentum, \(m_μ\) is the muon's rest mass, and \(v\) is its velocity. In this exercise, the muon's velocity is provided as \(8.85 \times 10^5 \, \text{cm/s}\).
Momentum calculation is essential for further calculations, such as finding the de Broglie wavelength. This wavelength relates to the particle nature of matter and gives insight into the wave-like behavior of particles like muons. Understanding momentum helps bridge classical concepts and quantum phenomena, highlighting how mass and velocity intertwine in subatomic physics.