Problem 68 - Chemistry The Science in Context

Question

How rapidly would each of the following particles be moving if they all had the same wavelength as a photon of red light \((\lambda=750 \mathrm{nm}) ?\) a. an electron of mass \(9.10938 \times 10^{-28} \mathrm{g}\) b. a proton of mass \(1.67262 \times 10^{-24} \mathrm{g}\) c. a neutron of mass \(1.67493 \times 10^{-24} \mathrm{g}\) d. an \(\alpha\) particle of mass \(6.64 \times 10^{-24} \mathrm{g}\)

Step-by-Step Solution

Answer

Answer: The velocities of the particles are: - Electron: 2.439 x 10^6 m/s - Proton: 5.272 x 10^3 m/s - Neutron: 5.263 x 10^3 m/s - Alpha particle: 1.321 x 10^3 m/s

Step 1: Write down the given information

We know the following data: - Wavelength of red light, \(\lambda = 750 \, \text{nm} = 750 \times 10^{-9} \, \text{m}\) - Planck's constant, \(h = 6.626 \times 10^{-34} \, \text{Js}\) Masses of the particles: - Electron, \(m_e = 9.10938 \times 10^{-31} \, \text{kg}\) - Proton, \(m_p = 1.67262 \times 10^{-27} \, \text{kg}\) - Neutron, \(m_n = 1.67493 \times 10^{-27} \, \text{kg}\) - Alpha particle, \(m_{\alpha} = 6.64 \times 10^{-27} \, \text{kg}\) Note: We have converted the masses to SI units (kilograms).

Step 2: Calculate the velocity of the electron

Using the de Broglie wavelength equation, we get: $$v_e = \frac{h}{m_e\lambda} = \frac{6.626 \times 10^{-34}}{(9.10938 \times 10^{-31})(750 \times 10^{-9})} = 2.439 \times 10^{6} \, \text{m/s}$$ The electron's velocity is \(2.439 \times 10^{6} \, \text{m/s}\).

Step 3: Calculate the velocity of the proton

Using the de Broglie wavelength equation, we get: $$v_p = \frac{h}{m_p\lambda} = \frac{6.626 \times 10^{-34}}{(1.67262 \times 10^{-27})(750 \times 10^{-9})} = 5.272 \times 10^{3} \, \text{m/s}$$ The proton's velocity is \(5.272 \times 10^{3} \, \text{m/s}\).

Step 4: Calculate the velocity of the neutron

Using the de Broglie wavelength equation, we get: $$v_n = \frac{h}{m_n\lambda} = \frac{6.626 \times 10^{-34}}{(1.67493 \times 10^{-27})(750 \times 10^{-9})} = 5.263 \times 10^{3} \, \text{m/s}$$ The neutron's velocity is \(5.263 \times 10^{3} \, \text{m/s}\).

Step 5: Calculate the velocity of the alpha particle

Using the de Broglie wavelength equation, we get: $$v_{\alpha} = \frac{h}{m_{\alpha}\lambda} = \frac{6.626 \times 10^{-34}}{(6.64 \times 10^{-27})(750 \times 10^{-9})} = 1.321 \times 10^{3} \, \text{m/s}$$ The alpha particle's velocity is \(1.321 \times 10^{3} \, \text{m/s}\).

Step 6: Present the results

The velocities of the particles when they have the same wavelength as a photon of red light are: - Electron: \(2.439 \times 10^{6} \, \text{m/s}\) - Proton: \(5.272 \times 10^{3} \, \text{m/s}\) - Neutron: \(5.263 \times 10^{3} \, \text{m/s}\) - Alpha particle: \(1.321 \times 10^{3} \, \text{m/s}\)

Key Concepts

Particle Velocity

Particle velocity is an important concept when studying wave-particle duality, as described by de Broglie's theory. The velocity of a particle can be determined when its wavelength is known, using de Broglie's equation:

• Wavelength (\( \lambda \)) refers to the distance over which the wave's shape repeats.
• Planck's constant (\( h \)) is a fundamental constant applicable to quantum mechanics, indicating the smallest possible unit of energy.
• Mass (\( m \)) is the amount of matter present in the particle.

The de Broglie wavelength equation is:\[v = \frac{h}{m \lambda}\]This equation signifies that the particle velocity (\( v \)) depends inversely on its mass and directly on its wavelength. Thus, the smaller the mass, the higher the velocity for the same wavelength.
For example, an electron will move rapidly compared to a proton, neutron, or alpha particle when they share the same wavelength, due to its much smaller mass.

Quantum Mechanics

Quantum mechanics is the branch of physics that explores the behaviors of microscopic particles like electrons and protons. This field provides the framework for understanding how particles can behave like waves, as demonstrated by de Broglie's hypothesis of wave-particle duality.

• Wave-particle duality introduces the idea that every particle exhibits both wave-like and particle-like properties.
• This is crucial for explaining phenomena like the de Broglie wavelength, which allows us to calculate particle velocities based on wave characteristics.
• The principles of quantum mechanics aid in understanding the probability and uncertainty of particle positions and behaviors.

One cornerstone of quantum mechanics is Heisenberg's uncertainty principle: >This principle states that it is impossible to simultaneously know both the exact position and exact velocity of a particle.

Wave-Particle Duality

Wave-particle duality suggests that particles can be described not just as tiny spheres, but as waves, hence having a wavelength. So, under certain conditions, such as when calculating velocities using a given wavelength, their behavior aligns more with waves than classical particles.

Photon Wavelength

Photon wavelength refers to the characteristic distance over which identical phase points of a photon travel — typically measured in nanometers (nm). Photons are massless particles of light that exhibit both wave-like and particle-like properties:

• Red light, for instance, has a wavelength of about 750 nm, which lays in the visible spectrum.
• The photon's energy and frequency are inversely related to its wavelength: the longer the wavelength, the lower the energy.
• The photon wavelength influences how it interacts with matter, such as causing electron excitations.

De Broglie Wavelength of Particles

While photons naturally exhibit wave properties, other particles (e.g., electrons, protons) can be assigned a de Broglie wavelength. For particles like these, matching their de Broglie wavelength to that of a photon allows for interesting theoretical calculations of their potential velocities, as seen with the corresponding photon wavelength.
Thus, photons set the stage for exploring how particles with actual mass move according to the wave-like properties imparted upon them through their de Broglie wavelengths.