Problem 130 - A Complete Resource Book in Chemistry for JEE Main

Question

According to de Broglie concept, all material particles posses wave character as well as particle character. The wave associated with a moving particle is called matter wave. The wavelength of the matter wave is given by the equation \(\lambda=\frac{\mathrm{h}}{\mathrm{p}}=\frac{\mathrm{h}}{\mathrm{mv}}\) where \(\mathrm{p}\) is the momentum of the particle, " \(\mathrm{m}\) ' is the mass of the particle and ' \(\mathrm{v}^{\prime}\) is the velocity of the particle. ' \(\mathrm{h}\) ' is called Planck's constant. The wavelength associated with an electron (mass = \(9.11 \times 10^{-11} \mathrm{~kg}\) ) moving with a velocity of \(10^{5} \mathrm{~m} \mathrm{~s}^{-1}\) is \(\left(\mathrm{h}=6.625 \times 10^{-34} \mathrm{~J} \mathrm{~s}\right)\) (a) \(0.727 \mathrm{~nm}\) (b) \(7.27 \mathrm{~nm}\) (c) \(727 \mathrm{~nm}\) (d) \(7.27 \mathrm{~m}\)

Step-by-Step Solution

Answer

The wavelength is 7.27 nm, so the correct answer is (b) 7.27 nm.

Step 1: Recap de Broglie Wavelength Formula

The de Broglie wavelength \( \lambda \) of a particle is given by the formula \( \lambda = \frac{h}{p} \), where \( p \) is the momentum of the particle. The momentum \( p \) can also be expressed as the product of mass \( m \) and velocity \( v \), so the formula becomes \( \lambda = \frac{h}{mv} \).

Step 2: Identify Given Values

The mass \( m \) of the electron is \( 9.11 \times 10^{-31} \text{ kg} \). The velocity \( v \) is \( 10^5 \text{ m/s} \). Planck's constant is given as \( h = 6.625 \times 10^{-34} \text{ J·s} \).

Step 3: Substitute Values into Formula

Substitute the known values into the de Broglie wavelength formula: \[ \lambda = \frac{6.625 \times 10^{-34}}{9.11 \times 10^{-31} \times 10^5} \].

Step 4: Calculate the Wavelength

First compute the denominator: \( 9.11 \times 10^{-31} \times 10^5 = 9.11 \times 10^{-26} \). Now divide the numerator by the denominator: \[ \lambda = \frac{6.625 \times 10^{-34}}{9.11 \times 10^{-26}} = 7.27 \times 10^{-9} \text{ m} \].

Step 5: Convert Wavelength to Nanometers

Convert the wavelength from meters to nanometers knowing that \( 1 \text{ m} = 10^9 \text{ nm} \): \[ \lambda = 7.27 \times 10^{-9} \text{ m} = 7.27 \text{ nm} \].

Step 6: Select the Correct Answer

The calculated wavelength is \( 7.27 \text{ nm} \), corresponding to option (b) in the provided choices.

Key Concepts

Matter Waves

Matter waves are a fascinating concept introduced by physicist Louis de Broglie. They illustrate the dual nature of matter, enabling particles such as electrons to exhibit both wave-like and particle-like properties. This idea emerged from the realm of quantum mechanics and radically shifted our understanding of how particles behave on a microscopic scale. According to this concept, every particle in motion is associated with a wave, known as a de Broglie wave or matter wave.

• They are central to quantum mechanics and explain phenomena such as electron diffraction and interference, which are typical wave-like behaviors.
• These waves have wavelengths that depend on the momentum of the particle.
• The lower the mass and higher the velocity, the smaller the wavelength of the matter wave.

Understanding matter waves is crucial because it reveals why particles like electrons exhibit behavior that cannot be described by classical physics alone.

Planck's Constant

Planck's constant, represented by the symbol \( h \), is a fundamental constant in physics that plays a critical role in quantum mechanics. It is a proportionality factor that relates the energy of a photon to the frequency of its associated electromagnetic wave.

Significance in Quantum Mechanics

Planck's constant is essential in quantifying the behavior of particles at atomic and subatomic levels. When Planck discovered \( h \), it was initially used to explain blackbody radiation, but it soon became foundational to the entire field of quantum theory.

• The value of Planck's constant is approximately \( 6.626 imes 10^{-34} ext{ J·s} \).
• In the context of the de Broglie wavelength, Planck's constant helps calculate the wavelength associated with a particle's wave nature.
• Because it is so small, quantum effects governed by \( h \) typically only become noticeable at the atomic scale.

Planck's constant denotes the scale at which classical physics gives way to quantum mechanics, marking a complete shift in our understanding of reality at small scales.

Electron Wave Properties

Electrons, like all matter, exhibit wave properties that are described by their de Broglie wavelength. The wave nature of electrons is crucial for many areas of scientific research and technology. It helps explain phenomena that would otherwise be impossible to understand using classical physics.

Wave Behavior of Electrons

• The wave property of electrons allows them to form the basis of technologies like electron microscopes, which rely on the interference and diffraction of electron waves to image objects at tiny scales.
• Electrons have small mass and can exhibit prominent wave properties even at relatively low velocities, as shown in the provided exercise example.
• The calculated wavelength of an electron can be exceedingly small, which allows us to observe very detailed structures at the atomic level.

Understanding electron wave properties not only aids in comprehending quantum mechanics but is also critical for advances in fields like nanotechnology and quantum computing.