Problem 42 - Chemistry: The Central Science

Question

Among the elementary subatomic particles of physics is the muon, which decays within a few nanoseconds 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 a velocity of \(8.85 \times 10^{5} \mathrm{~cm} / \mathrm{s}\).

Step-by-Step Solution

Answer

The de Broglie wavelength associated with a muon traveling at a velocity of \(8.85 \times 10^{5} \mathrm{cm/s}\) is approximately \(4.16 \times 10^{-11} \mathrm{cm}\).

Step 1: Find the mass of the muon

To find the mass of the muon (m), we'll first need the mass of an electron (me), which is approximately \(9.11 \times 10^{-28} \mathrm{g}\). Since the muon has a rest mass of 206.8 times that of an electron, we can find its mass by multiplying this factor by the mass of an electron: \[m = 206.8 \times m_e = 206.8 \times (9.11 \times 10^{-28} \mathrm{g})\]

Step 2: Calculate the momentum of the muon

Now that we have the mass of the muon, we can find its momentum (p) by multiplying its mass by its given velocity (v): \[p = mv = (206.8 \times (9.11 \times 10^{-28} \mathrm{g})) \times (8.85 \times 10^{5} \mathrm{cm/s})\]

Step 3: Calculate the de Broglie wavelength

Finally, we can use the de Broglie formula to find the wavelength (λ) of the muon. Recall that Planck's constant (h) is approximately \(6.63 \times 10^{-27} \mathrm{erg} \cdot \mathrm{s}\). Now substitute the momentum (p) we found in the formula: \[λ = \frac{h}{p} = \frac{6.63 \times 10^{-27} \mathrm{erg} \cdot \mathrm{s}}{(206.8 \times (9.11 \times 10^{-28} \mathrm{g})) \times (8.85 \times 10^{5} \mathrm{cm/s})}\] Now, calculate the value of λ: \[λ \approx 4.16 \times 10^{-11} \mathrm{cm}\] So, the de Broglie wavelength associated with a muon traveling at a velocity of \(8.85 \times 10^{5} \mathrm{cm/s}\) is approximately \(4.16 \times 10^{-11} \mathrm{cm}\).

Key Concepts

Muon

The muon is a fascinating subatomic particle that shares many similarities with the electron, yet it is much more massive. It is classified as a lepton, one of the fundamental particles in the universe. Unlike electrons, muons are unstable and exist only for a brief period, typically just a few microseconds, before decaying into other particles. As outlined in the exercise, the rest mass of the muon is about 206.8 times that of an electron. Given this characteristic, when calculating properties like the de Broglie wavelength, it's crucial to start with an accurate determination of its mass. Its larger mass compared to an electron significantly impacts calculations that involve its momentum and wavelength.

Subatomic Particles

Subatomic particles are the building blocks of atoms. They include particles like protons, neutrons, and electrons, each with different properties and functions within the atom. Beyond these, other subatomic particles exist, such as muons and neutrinos, which do not contribute directly to atomic structure. These particles are studied under particle physics, which delves into understanding their behavior and interactions. Identifying the properties of subatomic particles like the muon, which is involved in this exercise, helps scientists grasp fundamental forces and matter composition in the universe. They are integral to understanding quantum physics and its numerous applications.

Planck's Constant

Planck's constant, denoted by \(h\), is a pivotal constant in physics, particularly in quantum mechanics. It relates energy to frequency via the equation \(E = h u\), where \(E\) is energy and \(u\) is frequency. Similarly, it is used in the calculation of de Broglie wavelengths, linking it to particle wave properties. In the provided solution, Planck's constant has a value of approximately \(6.63 \times 10^{-27} \mathrm{erg} \cdot \mathrm{s}\). This constant enables the conversion of momentum into a wavelength, offering a bridge between the macroscopic and quantum worlds. Thanks to Planck's constant, physicists can explore dual wave-particle properties of matter.

Momentum

Momentum, in the realm of physics, is defined as the product of an object's mass and its velocity. It is a vector quantity, possessing both magnitude and direction. Calculating momentum is essential for understanding the motion of objects and particles. In quantum mechanics, like the exercise with the muon, momentum is key to determining the de Broglie wavelength. The momentum \(p\) of a particle such as the muon is calculated using \(p = mv\), where \(m\) is mass and \(v\) is velocity. This relationship helps in transitioning from traditional physics to quantum mechanics, highlighting the wavelike behavior of particles.

Mass of Electron

Electrons are subatomic particles with a fundamental role in chemistry and physics, particularly in forming atoms and participating in chemical reactions. The mass of an electron is crucial for many calculations in physics, such as determining other particle masses through comparison. It is approximately \(9.11 \times 10^{-28} \mathrm{g}\), a small figure reflecting the lightness of electrons compared to other subatomic particles. When calculating the mass of a muon, this mass serves as a reference. With the muon being 206.8 times more massive than an electron, it influences properties like speed and wavelength. Understanding electron mass aids in grasping the distinct characteristics and behaviors of other particles.