Problem 157 - A Complete Resource Book in Chemistry for JEE Main

Question

The de Broglie wavelength of a tennins ball of mass \(60 \mathrm{~g}\) moving with a velocity of 10 metres per second is approximately \(\mathbf\) (Planck constant \(\mathrm{h}=6.63 \times 10^{-31} \mathrm{Js}\) ) (a) \(10^{-33} \mathrm{~m}\) (b) \(10^{-31} \mathrm{~m}\) (c) \(10-16 \mathrm{~m}\) (d) \(10^{-25} \mathrm{~m}\)

Step-by-Step Solution

Answer

The de Broglie wavelength is approximately \(10^{-31} \text{ m}\) (option b).

Step 1: Understanding the De Broglie Wavelength Formula

The de Broglie wavelength \(\lambda\) of a particle is calculated using the formula \(\lambda = \frac{h}{mv}\), where \(h\) is Planck's constant, \(m\) is the mass of the particle, and \(v\) is its velocity. To find the de Broglie wavelength for the tennis ball, we need the values of these parameters.

Step 2: Convert the Mass into Kilograms

The mass of the tennis ball is given as \(60 \text{ g}\). Since we need the mass in kilograms for our calculation, convert grams to kilograms:\(60 \text{ g} = 0.060 \text{ kg}\).

Step 3: Use the Given Values to Calculate the Wavelength

Substitute the given values into the de Broglie equation. We have:- Planck's constant \(h = 6.63 \times 10^{-31} \text{ Js}\)- Mass \(m = 0.060 \text{ kg}\)- Velocity \(v = 10 \text{ m/s}\)Calculate \(\lambda\):\[ \lambda = \frac{6.63 \times 10^{-31}}{0.060 \times 10} = \frac{6.63 \times 10^{-31}}{0.6} \].

Step 4: Simplify the Wavelength Expression

Perform the division to simplify the expression:\[ \lambda = \frac{6.63 \times 10^{-31}}{0.6} = 1.105 \times 10^{-31} \text{ m} \].

Step 5: Identify the Closest Option

Compare the calculated wavelength \(1.105 \times 10^{-31} \text{ m}\) to the provided options:(a) \(10^{-33} \text{ m}\)(b) \(10^{-31} \text{ m}\)(c) \(10^{-16} \text{ m}\)(d) \(10^{-25} \text{ m}\)The closest match is option (b) \(10^{-31} \text{ m}\).

Key Concepts

Planck's constant

Planck's constant is a fundamental quantity in physics representing the smallest action or quantum of action possible in the universe. It is typically denoted by the letter \(h\) and has a value of \(6.63 \times 10^{-34} \text{ Js}\). However, the original problem provided it as \(6.63 \times 10^{-31} \text{ Js}\), likely a typo or simplified for certain contexts.

• It was first introduced by Max Planck in the early 20th century during his study of black-body radiation.
• Planck's constant serves as a bridge between the classical and quantum physics worlds, supporting the discrete quantum nature of energy.

The significance of Planck's constant cannot be overstated as it features prominently in equations measuring the dual wave-like and particle-like behavior of particles. Specifically, it is used in the de Broglie wavelength formula, showcasing that every moving particle exhibits a wavelength dependent on its velocity and mass.
Understanding Planck's constant thus provides essential insight into the quantum mechanics realm.

Particle physics

Particle physics is the branch of physics that studies the nature of particles, the building blocks of matter. It dives deep into understanding subatomic particles like electrons, protons, neutrons, and the forces that govern their interactions.

• These particles are categorized into two primary groups: fermions and bosons.
• Fermions include particles that constitute matter, such as quarks and leptons.
• Bosons are particles that mediate forces, such as photons for the electromagnetic force.

Particle physics seeks to explain the fundamental workings of the universe. The exploration includes examining how particles acquire mass, understanding antimatter, and measuring quantum phenomena. Understanding how fundamental forces interact through particles is critical to advancements in technologies and theories about the universe, making particle physics a cornerstone of modern science.

Wave-particle duality

Wave-particle duality is a fundamental concept in quantum mechanics proposing that every particle or quantic entity may exhibit both wave and particle properties. This dual nature was first revealed when Faraday and Bohr studied the behavior of electrons.

• Louis de Broglie suggested that particles, like electrons, have inherent wave characteristics.
• Phenomena like diffraction and interference demonstrate wave properties while events such as collisions exhibit particle characteristics.

The duality concept challenges traditional Newtonian physics that assumed entities behaved strictly as either particles or waves. This led to the development of quantum mechanics, reshaping how scientists perceive and model the microscopic world.
It illustrates the principle that the probability wave predicts the chance of particle detection at any location, framing foundational ideas about the mechanics of particles at an atomic level.

Quantum mechanics

Quantum mechanics is a branch of physics that describes the behavior of matter and energy at very small scales, typical of atoms and subatomic particles. At its core, quantum mechanics dethrones the certainty present in classical physics, introducing probability and chance.

• Rather than observing absolute states, quantum mechanics uses probabilities to determine states.
• It outlines phenomena like superposition, entanglement, and the uncertainty principle.
• The famous Schrödinger's equation serves as a quantum system description tool.

Quantum mechanics also emerged to explain experimental results that could not be rationalized by classical theories. Its principles allow for predicting and quantifying behaviors and properties of nature at small scales.
Teaching quantum mechanics acknowledges that particles can occupy multiple states at once, having vast implications for fields such as chemistry, computer science, and material sciences.