Problem 78 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

If \(E_{1}, E_{2}\) and \(E_{3}\) are the kinetic energies of an electron, an \(\alpha\) -particle and a proton, with same de-Broglie wavelength, then (a) \(E_{1}>E_{3}>E_{2}\) (b) \(E_{2}>E_{3}>E_{1}\) (c) \(E_{1}>E_{2}>E_{3}\) (d) \(E_{1}=E_{2}=E_{3}\)

Step-by-Step Solution

Answer

The correct order of kinetic energies with the same de-Broglie wavelength is: \(E_{1} > E_{3} > E_{2}\).

Step 1: Recall the de Broglie Wavelength Relationship

The de Broglie wavelength \(\lambda\) of a particle is inversely proportional to its momentum (p), given by \(\lambda = \frac{h}{p}\), where h is the Planck constant. Since the wavelength is the same for all particles, we can write \(p_1 = p_2 = p_3\).

Step 2: Write the Momentum as a Function of Kinetic Energy

Momentum (p) is related to the kinetic energy (E) and mass (m) of the particle by \(p = \sqrt{2mE}\). Since momentum is the same, \(\sqrt{2m_1E_1} = \sqrt{2m_2E_2} = \sqrt{2m_3E_3}\).

Step 3: Compare the Kinetic Energies

For the electron (e), alpha particle (\(\alpha\)), and proton (p), we have different masses: \(m_\alpha > m_p > m_e\). To have the same momentum, the kinetic energies must compensate for the mass difference: \(E_e > E_p > E_\alpha\). Thus, \(E_1 > E_3 > E_2\).

Key Concepts

Kinetic Energy of Particles

When we talk about the motion of particles, an essential aspect of this motion is their kinetic energy. Kinetic energy is the energy that a particle has due to its motion and can be calculated using the equation \( E = \frac{1}{2}mv^2 \) where \( E \) is the kinetic energy, \( m \) is the mass of the particle, and \( v \) is its velocity.

For particles at the atomic and subatomic levels, like electrons, protons, and alpha particles, the kinetic energy is not only a measure of how fast they move but also connects to other physical properties such as momentum and wavelength. A fascinating insight from quantum mechanics is that particles can exhibit wavelike properties, a concept central to the de Broglie hypothesis. As the exercise implies, even with the same de Broglie wavelength, particles with different masses will have different kinetic energies to maintain the same momentum, balancing the equation according to their mass.

Understanding this balance is fundamental when trying to determine the behaviors of particles at the quantum level. It allows us to predict outcomes in experiments like the famous double-slit experiment, which demonstrated the wavelike nature of electrons.

Momentum in Physics

Momentum in physics is typically understood as the product of a particle's mass and velocity, given by the formula \( p = mv \) where \( p \) denotes the momentum, \( m \) the mass, and \( v \) the velocity of the particle.

However, when considering kinetic energy, momentum can also be expressed as \( p = \sqrt{2mE} \) when \( E \) represents the kinetic energy. This relationship is pivotal when we discuss the de Broglie wavelength of different particles. If particles have the same de Broglie wavelength, as in our textbook exercise, their momenta must be equal. This condition means particles of different masses must have varying kinetic energies to achieve the same momentum.

Moreover, momentum has profound implications in both classical and quantum physics. It is conserved in isolated systems according to the conservation laws, which tell us that the total momentum before an event (like a collision) is equal to the total momentum after the event, assuming no external forces interfere. In quantum physics, momentum is tied to the wave properties of particles aligning with the de Broglie hypothesis, which is essential in understanding phenomena at the atomic scale.

Planck Constant

The Planck constant (denoted as \( h \) ) is a fundamental quantity in quantum mechanics and has the units of action (energy multiplied by time). Its value is approximately \( 6.62607015 \times 10^{-34} \) joule seconds. The Planck constant plays a critical role in the quantization of energy and is a central element in several quantum phenomena, including the de Broglie hypothesis and the Heisenberg uncertainty principle.

The de Broglie hypothesis, which relates the wavelength ( \( \lambda \) ) of a particle to its momentum ( \( p \) ), uses the Planck constant in the formula \( \lambda = \frac{h}{p} \) indicating that the product of momentum and wavelength for any particle is equal to the Planck constant. When dealing with the de Broglie wavelength of particles with the same wavelength but different masses, as presented in our exercise, the Planck constant is the equalizer, ensuring the wavelength remains constant despite variations in mass and kinetic energy.

Understanding the Planck constant helps unlock the door to the quantum world, providing a measurable connection between the behavior of macroscopic objects and the world of the very small, where particles show both wave and particle characteristics. Its discovery has opened up an era of technological advancements such as lasers, transistors, and the field of quantum computing.