Problem 71 - A Complete Resource Book in Chemistry for JEE Main

Question

The de Broglie wavelength associated with a particle of mass \(10^{-6} \mathrm{~kg}\) moving with a velocity of \(10 \mathrm{~ms}^{-1}\) is (a) \(6.63 \times 10^{-7} \mathrm{~m}\) (b) \(6.63 \times 10^{-16} \mathrm{~m}\) (c) \(6.63 \times 10^{-21} \mathrm{~m}\) (d) \(6.63 \times 10^{-29} \mathrm{~m}\)

Step-by-Step Solution

Answer

The de Broglie wavelength is \(6.63 \times 10^{-29}\) m, which corresponds to option (d).

Step 1: Understand the Formula

The de Broglie wavelength of a particle is calculated using the formula \( \lambda = \frac{h}{mv} \), where \( h \) is Planck's constant \( 6.63 \times 10^{-34} \mathrm{~Js} \), \( m \) is the mass of the particle, and \( v \) is the velocity of the particle.

Step 2: Identify Given Values

From the problem statement, we have: mass \( m = 10^{-6} \mathrm{~kg} \) and velocity \( v = 10 \mathrm{~ms}^{-1} \). Planck's constant \( h \) is \( 6.63 \times 10^{-34} \mathrm{~Js} \).

Step 3: Substitute Values into the Formula

Substitute \( h, m, \) and \( v \) into the de Broglie equation: \[ \lambda = \frac{6.63 \times 10^{-34}}{(10^{-6})(10)} \].

Step 4: Calculate the Wavelength

Perform the calculation: \[ \lambda = \frac{6.63 \times 10^{-34}}{10^{-5}} = 6.63 \times 10^{-29} \mathrm{~m} \].

Step 5: Select the Correct Answer

Compare the calculated result with the given options. The calculated de Broglie wavelength \( 6.63 \times 10^{-29} \mathrm{~m} \) matches option (d).

Key Concepts

Wave-Particle Duality

Wave-particle duality is a fundamental concept in physics that tells us particles, like electrons and photons, can exhibit both wave-like and particle-like properties. This idea helps us understand how particles behave on a very small scale, such as atoms and molecules. Think of light, for instance: it acts like a wave when it bends around corners (diffraction) and like a particle when it hits a solar panel. According to wave-particle duality, particles have a wavelength associated with their motion, which can be calculated using the de Broglie wavelength formula.

• This formula is: \( \lambda = \frac{h}{mv} \)
• Where \( \lambda \) is the wavelength, \( h \) is Planck's constant, \( m \) is mass, and \( v \) is velocity.

Wave-particle duality is an essential concept because it provides insight into the dual nature of matter and energy in the quantum realm. By seeing particles as both waves and particles, we can better understand their behavior in different scenarios. This duality is the backbone of many quantum mechanics concepts.

Quantum Mechanics

Quantum mechanics is the branch of physics that explains how the smallest particles, like electrons and photons, behave. These particles do not follow the same rules as larger objects, like balls or cars, that we encounter in everyday life. In fact, they often follow rules that seem very strange or counterintuitive. Quantum mechanics describes how particles move, interact, and exist on incredibly small scales. It incorporates wave-particle duality and other concepts to describe and predict the behavior of these particles.
The field of quantum mechanics introduced the idea that many properties of these particles, such as position or momentum, cannot be precisely measured at the same time. This is known as the Heisenberg Uncertainty Principle.

• Quantum mechanics helps us understand chemical reactions.
• It is fundamental to the development of technologies like MRI machines and lasers.

Through mathematics and experiments, quantum mechanics provides a profound understanding of the natural world, describing phenomena that classical physics cannot.

Planck's Constant

Planck's constant is a crucial number in physics that relates the energy of a photon to its frequency. It is often represented by the symbol \( h \) and is approximately equal to \( 6.63 \times 10^{-34} \) joule-seconds. This constant is named after Max Planck, one of the founding figures of quantum theory. It plays a pivotal role in equations involving wave-particle duality, including the de Broglie wavelength formula.
Planck's constant allows us to calculate the energy of particles with wave-like properties. For instance, using the energy equation \( E = hf \), where \( E \) is energy, \( h \) is Planck's constant, and \( f \) is frequency, scientists can understand how energy is quantized into discrete packets, or 'quanta.'

• Without Planck's constant, modern quantum mechanics would not exist.
• It marks the transition from classical to quantum physics.

Understanding Planck's constant thus opens the door to grasping how energy transfer operates on an atomic and subatomic scale, reshaping our comprehension of the universe.