Question

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

Short answer

The velocities of electrons with de Broglie wavelengths of \(1.0 \times 10^2 \mathrm{~nm}\) and \(1.0 \mathrm{~nm}\) are \(7.27 \times 10^5 \mathrm{\dfrac{m}{s}}\) and \(7.27 \times 10^7 \mathrm{\dfrac{m}{s}}\) respectively.

Step-by-step solution

Rewrite the de Broglie equation in terms of velocity

We need to find the velocity, so we need to rewrite the given formula in terms of velocity. We have the momentum \(p = m \cdot v\). We can rewrite the de Broglie equation as: \( \lambda = \dfrac{h}{m \cdot v} \) Now, we can solve for the velocity: \(v = \dfrac{h}{\lambda \cdot m} \) We will use this equation to calculate the velocities.

Identify the constants and convert the wavelengths to meters

In the equation we derived, we have: - \(h\) (Planck's constant) = \(6.626 \times 10^{-34} \mathrm{Js}\) - \(m\) (mass of an electron) = \(9.109 \times 10^{-31} \mathrm{kg}\) Our given wavelengths are in nanometers. We need to convert them to meters to use them in the equation. \(1.0 \times 10^2 \mathrm{~nm} = 1.0 \times 10^{2} \times 10^{-9} \mathrm{m} = 1.0 \times 10^{-7} \mathrm{m}\) \(1.0 \mathrm{~nm} = 1.0 \times 10^{-9} \mathrm{m}\) We'll now use these values and the equation to calculate the velocities.

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

For the de Broglie wavelength of \(1.0 \times 10^2 \mathrm{~nm}\): \(v_{1} = \dfrac{6.626 \times 10^{-34} \mathrm{Js}}{(1.0 \times 10^{-7} \mathrm{m})(9.109 \times 10^{-31} \mathrm{kg})} \) \(v_{1} = 7.27 \times 10^5 \mathrm{\dfrac{m}{s}}\) For the de Broglie wavelength of \(1.0 \mathrm{~nm}\): \(v_{2} = \dfrac{6.626 \times 10^{-34} \mathrm{Js}}{(1.0 \times 10^{-9} \mathrm{m})(9.109 \times 10^{-31} \mathrm{kg})} \) \(v_{2} = 7.27 \times 10^7 \mathrm{\dfrac{m}{s}}\) So, the velocities of electrons with de Broglie wavelengths of \(1.0 \times 10^2 \mathrm{~nm}\) and \(1.0 \mathrm{~nm}\) are \(7.27 \times 10^5 \mathrm{\dfrac{m}{s}}\) and \(7.27 \times 10^7 \mathrm{\dfrac{m}{s}}\) respectively.

Key concepts

Electron Velocity

The velocity of an electron is an essential concept when discussing the de Broglie wavelength. In quantum mechanics, every particle can be associated with a wave, and the wavelength of this wave is inversely proportional to the particle's momentum. To determine the electron's velocity given a wavelength, we start by rearranging the de Broglie equation.

• Momentum, denoted as \( p \), is calculated using \( p = m \cdot v \) where \( m \) is the mass and \( v \) is the velocity.
• The de Broglie wavelength formula is \( \lambda = \dfrac{h}{p} \), which can be rewritten as \( v = \dfrac{h}{\lambda \cdot m} \).
• This equation shows that velocity is inversely proportional to wavelength—larger wavelengths correspond to slower velocities.

By converting the wavelength to meters and plugging into the equation, we solve for the electron's velocity. Understanding how to manipulate and use these equations is key to solving quantum mechanical problems like this.

Planck's Constant

Planck's constant, \( h \), is a fundamental constant in quantum mechanics, playing a crucial role in linking the particle and wave nature of matter. It represents the scale at which quantum effects become significant.

• The value of Planck's constant is approximately \( 6.626 \times 10^{-34} \text{Js} \).
• In equations like the de Broglie wavelength, it helps quantify the mass-energy interaction in terms of wave-particle duality.
• Without Planck's constant, expressions relating momentum and velocity to a particle's wave nature would be impossible.

Understanding Planck's constant gives insight into why particles on a quantum scale exhibit wave-like characteristics, directly affecting calculations involving electron velocities and wavelengths.

Mass of an Electron

The mass of an electron is another critical component when calculating its velocity using the de Broglie equation. Electrons, being one of the fundamental particles, have a relatively small mass, which is crucial in quantum mechanics.

• The mass of an electron is approximately \( 9.109 \times 10^{-31} \text{kg} \).
• It's small compared to other particles, making quantum mechanical effects more pronounced.
• Using this mass in calculations allows us to observe the implications of wave-particle duality on such a small scale.

This value is used in the formula \( v = \dfrac{h}{\lambda \cdot m} \) to determine how massive the particle in question is, which in turn affects the velocity for given wavelengths.

Momentum

Momentum in physics describes the motion of a particle, defined as the product of its mass and velocity. It is a core concept that connects classical mechanics to quantum mechanics via the de Broglie equation.

• Momentum \( p \) is given by the formula \( p = m \cdot v \).
• It is directly related to a particle's wavelength through de Broglie's equation: \( \lambda = \dfrac{h}{p} \).
• In this context, a smaller de Broglie wavelength implies higher momentum, as their relationship is inversely proportional.

Grasping momentum's role allows us to understand the dynamic behavior of electrons, especially how their motion translates to observable wave-like properties in quantum physics.