Question

Calculate the velocities of electrons with de Broglie wavelengths of \(1.0 \times 10^{2} \mathrm{nm}\) and \(1.0 \mathrm{nm} .\)

Short answer

The velocities of electrons with de Broglie wavelengths of \(1.0 \times 10^2 \,\text{nm}\) and \(1.0\,\text{nm}\) are \(v_1 = 7.272 \times 10^{5} \,\text{m/s}\) and \(v_2 = 7.272 \times 10^{6} \,\text{m/s}\), respectively.

Step-by-step solution

Recall de Broglie wavelength formula

The de Broglie wavelength is given by the formula: \[\lambda = \frac{h}{p}\] where \(\lambda\) is the de Broglie wavelength, \(h\) is the Planck constant (\(6.626 \times 10^{-34} \,\text{J}\cdot\text{s}\)), and \(p\) is the momentum of the particle.

Calculate the momentum for both wavelengths

First, we need to find the momentum (\(p\)) for both wavelengths. Using de Broglie wavelength formula and solving for momentum, we get: \[p = \frac{h}{\lambda}\] For \(\lambda_1 = 1.0 \times 10^2 \,\text{nm}\): \[p_1 = \frac{6.626 \times 10^{-34} \,\text{J}\cdot\text{s}}{1.0 \times 10^{-9} \,\text{m}} = 6.626 \times 10^{-25} \,\text{kg}\cdot\text{m/s}\] For \(\lambda_2 = 1.0\,\text{nm}\): \[p_2 = \frac{6.626 \times 10^{-34} \,\text{J}\cdot\text{s}}{1.0 \times 10^{-10} \,\text{m}} = 6.626 \times 10^{-24} \,\text{kg}\cdot\text{m/s}\]

Calculate the velocities using momentum and electron mass

Momentum and velocity are related by the formula: \[p = mv\] where \(m\) is the mass of the electron (\(9.109 \times 10^{-31} \,\text{kg}\)) and \(v\) is the velocity. We solve for \(v\) to get: \[v = \frac{p}{m}\] For \(\lambda_1 = 1.0 \times 10^2 \,\text{nm}\): \[v_1 = \frac{6.626 \times 10^{-25} \,\text{kg}\cdot\text{m/s}}{9.109 \times 10^{-31} \,\text{kg}} = 7.272 \times 10^{5} \,\text{m/s}\] For \(\lambda_2 = 1.0\,\text{nm}\): \[v_2 = \frac{6.626 \times 10^{-24} \,\text{kg}\cdot\text{m/s}}{9.109 \times 10^{-31} \,\text{kg}} = 7.272 \times 10^{6} \,\text{m/s}\]

Present the results

The velocities of electrons with de Broglie wavelengths of \(1.0 \times 10^2 \,\text{nm}\) and \(1.0\,\text{nm}\) are: \(v_1 = 7.272 \times 10^{5} \,\text{m/s}\) and \(v_2 = 7.272 \times 10^{6} \,\text{m/s}\), respectively.

Key concepts

de Broglie Equation

The de Broglie equation links the classical and quantum worlds by providing a connection between the properties of particles, like electrons, and waves. This foundational concept in quantum mechanics allows us to calculate the de Broglie wavelength, \( \lambda \), of a moving particle, which is defined as the wavelength associated with a particle's wave-like behavior. According to the de Broglie hypothesis, every moving particle has wave-like characteristics.

The equation itself is beautifully simple: \[ \lambda = \frac{h}{p} \], where \( h \) is the Planck constant and \( p \) is the momentum of the particle. Understanding this relationship allows us to determine how particles like electrons will behave on small, typically atomic scales. It is particularly important in the domain of subatomic particles, where objects demonstrate both wave-like and particle-like characteristics.

Momentum of Electrons

Momentum, in classical mechanics, is simply the product of an object's mass and its velocity: \( p = mv \).

However, when calculating the momentum of subatomic particles like electrons, we enter the realm of quantum mechanics, where properties integrate seamlessly with wave-like behavior, described by the de Broglie equation. The momentum here is essential not only because it figures into kinetic energy calculations but also because it enables us to connect the macroscopic property of motion to the microscopic phenomenon of wave propagation.

For electrons, which have a rest mass \( m \) of approximately \( 9.109 \times 10^{-31} \) kg, the momentum becomes a crucial variable in determining the aforementioned de Broglie wavelength, illustrating the dual nature of electrons displaying both wave and particle characteristics.

Planck Constant

The Planck constant, \( h \), is a fundamental constant in quantum mechanics, appearing in many formulas and principles, including the de Broglie equation. It has a value of approximately \( 6.626 \times 10^{-34} \) J\(\cdot\)s and represents the smallest action in quantum mechanics, linking the amount of energy a photon carries with the frequency of its electromagnetic wave.

The Planck constant is not just a conversion factor but an indication of the inherently quantized nature of the universe at the smallest scales. It sets the scale for the quantum of action, giving quantum phenomena their characteristic discrete nature and thereby affecting how particles like electrons interact with energy and matter.

Electron Velocity

The velocity of an electron, \( v \), is a significant term in many quantum mechanical expressions, including the de Broglie equation when solving for momentum. The velocity of an electron in an atom can vary greatly depending on the electron's energy and the attractive forces exerted by the nucleus.

High velocities suggest high kinetic energy and, by extension, smaller wavelengths according to the de Broglie equation. This is why electrons that are accelerated, as in the exercise, have different de Broglie wavelengths: their velocities are substantial, reflecting their wave-particle duality. Understanding electron velocity is critical in fields ranging from electron microscopy to semiconductor physics, where the behavior of electrons determines the operation of the technology.