Question

What should be the increase in kinetic energy, of electron in order to decrease its de-Broglie wavelength from \(100 \mathrm{~nm}\) to \(50 \mathrm{~nm} ?\) (a) \(0.451 \mathrm{keV}\) (b) \(4.51 \times 10^{-4} \mathrm{eV}\) (c) \(4.51 \times 10^{-3} \mathrm{eV}\) (d) \(0.0451 \mathrm{eV}\)

Short answer

The increase in kinetic energy of the electron should be \(0.451 \mathrm{keV}\).

Step-by-step solution

Understand the de-Broglie Wavelength Formula

The de-Broglie wavelength \(\lambda\) of a particle is given by the equation \(\lambda = \frac{h}{p}\), where \(h\) is the Planck constant and \(p\) is the momentum of the particle. The momentum \(p\) can be expressed in terms of the mass \(m\) and velocity \(v\) of the electron by \(p = mv\). Since the kinetic energy \(KE\) can be given by \(\frac{1}{2}mv^2\), these expressions will be used to relate the de-Broglie wavelength and kinetic energy.

Determine the Initial and Final Momenta

From the de-Broglie wavelength formula, initial and final momenta can be calculated by rearranging the formula to \(p = \frac{h}{\lambda}\). Calculate the initial momentum \(p_{i}\) using the initial wavelength \(\lambda_{i} = 100 \mathrm{~nm}\) and the final momentum \(p_{f}\) using the final wavelength \(\lambda_{f} = 50 \mathrm{~nm}\).

Relate Changes in Momentum to Changes in Kinetic Energy

Knowing the initial and final momenta, calculate their associated kinetic energies using the formula \(KE = \frac{p^2}{2m}\), where \(m\) is the mass of the electron. The increase in kinetic energy is the difference between the final and initial kinetic energies: \(\Delta KE = KE_{f} - KE_{i}\).

Perform the Calculation

Substitute the values for the initial and final momenta into the kinetic energy formula and subtract the initial kinetic energy from the final kinetic energy to find the increase.

Key concepts

Physical Chemistry

Physical Chemistry is a branch of chemistry dedicated to the study of how matter behaves on a molecular and atomic level and how chemical reactions occur. It combines principles of physics and chemistry to understand the physical properties of molecules, the forces that act upon them, and the energy changes involved in reaction processes.

One of the fascinating aspects of physical chemistry is the exploration of particle behavior, such as that of electrons, at the quantum level. The de-Broglie wavelength, which is the wavelength associated with a particle, is a concept that merges the wave and particle theories of matter. This wavelength is important because it helps in understanding the wave-like behavior of particles when they are in motion and is crucial in applications such as electron microscopy and the development of various types of spectroscopy.

Kinetic Energy Calculation

Kinetic energy is the energy that a particle possesses due to its motion. In the context of de-Broglie wavelength problems, we calculate the kinetic energy (KE) of an electron, which is one of the fundamental particles in atoms. The kinetic energy is expressed mathematically as \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass of the electron and \( v \) is the velocity.

The kinetic energy calculation becomes very intriguing when dealing with microscopic particles like electrons because their behavior is governed by quantum mechanics rather than classical mechanics. Even tiny changes in kinetic energy can have a significant impact on properties like the de-Broglie wavelength, thus influencing an electron’s behavior in an observable manner, such as in diffraction patterns. Understanding how to calculate and manipulate the kinetic energy of electrons is crucial for explaining and predicting results in experimental physical chemistry.

Momentum of an Electron

Momentum, in classical physics terms, is the product of the mass and velocity of an object; for an electron, it is represented as \( p = mv \), where \( p \) is the momentum, \( m \) is the mass, and \( v \) is the velocity. However, when we dive into the world of quantum mechanics, momentum takes on a new layer of complexity because particles like electrons exhibit both particle and wave-like properties.

Through the de-Broglie hypothesis, the momentum of an electron is linked to its de-Broglie wavelength by the formula \( \lambda = \frac{h}{p} \), where \( h \) is Planck's constant and \( \lambda \) is the de-Broglie wavelength. This relationship is fundamental in quantum mechanics and is key to understanding phenomena such as electron diffraction and the Heisenberg uncertainty principle. Calculating the momentum of an electron is, therefore, a central skill in physical chemistry and is vital for interpreting a wide range of experimental data.