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 correct answer is (d) \(6.63 \times 10^{-29} \mathrm{~m}\).

Step 1: Identify the Formula

The de Broglie wavelength \( \lambda \) for a particle is given by 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 its velocity.

Step 2: Insert Known Values

Substitute the given values into the formula: - Mass \( m = 10^{-6} \mathrm{~kg} \)- Velocity \( v = 10 \mathrm{~ms}^{-1} \)- Planck's constant \( h = 6.63 \times 10^{-34} \mathrm{~Js} \) So, \( \lambda = \frac{6.63 \times 10^{-34}}{10^{-6} \times 10} \).

Step 3: Calculate the Wavelength

Perform the calculation: \[ \lambda = \frac{6.63 \times 10^{-34}}{10^{-5}} = 6.63 \times 10^{-29} \mathrm{~m} \] The de Broglie wavelength is \( 6.63 \times 10^{-29} \mathrm{~m} \).

Key Concepts

Planck's Constant

Planck's constant, denoted by the symbol \( h \), is a fundamental constant in quantum mechanics. It is crucial in explaining the behavior of particles at the smallest scales. This constant has the value \( 6.63 \times 10^{-34} \mathrm{~Js} \). In simple terms, Planck's constant relates the energy of a photon to the frequency of its electromagnetic wave. This relationship is given by the equation \( E = h \cdot f \), where \( E \) is energy and \( f \) is frequency.
When it comes to the de Broglie wavelength, Planck's constant plays a central role. The de Broglie wavelength expression is \( \lambda = \frac{h}{mv} \), showing that the wavelength depends inversely on both the mass and velocity of the particle but directly on the constant \( h \). What this means is that for very small masses or high velocities, the wavelength becomes significant, making Planck's constant an essential part of understanding wave-particle duality in quantum mechanics.
Planck's constant is a pivotal element in several areas of physics, from understanding blackbody radiation to the development of quantum mechanics. It is a constant reminder of the peculiar nature of quantum objects, where energy isn't continuous but composed of discrete packets called quanta.

Mass of Particle

Mass refers to the amount of matter contained in a particle. In the context of the de Broglie wavelength, mass is a critical factor. The formula \( \lambda = \frac{h}{mv} \) tells us that the wavelength \( \lambda \) is inversely proportional to the mass \( m \). This means that the heavier the particle, the shorter its wavelength. Conversely, lighter particles have longer wavelengths, making quantum effects more noticeable.
The mass of a particle is measured in kilograms (kg) in the International System of Units (SI). For particles that are extremely small, like atoms or electrons, their mass might be given in more suitable units like atomic mass units (amu) or even electronvolts (eV) when dealing with subatomic particles. In our example, the particle in question has a mass of \( 10^{-6} \mathrm{~kg} \). This small mass value results in a wavelength that is big enough to demonstrate wave properties.
Mass is an intrinsic property of any particle and affects not only its de Broglie wavelength but also its inertia and gravitational interactions. Understanding mass in relation to quantum mechanics shows the unique balance and duality between wave and particle nature.

Velocity of Particle

Velocity is a vector quantity that denotes the speed of a particle and its direction of motion. Within the scope of the de Broglie wavelength, velocity has an equally vital role. The wavelength \( \lambda \) decreases as the velocity \( v \) of the particle increases, as seen in the equation \( \lambda = \frac{h}{mv} \). This inverse relationship implies that a faster-moving particle has a shorter wavelength.
Velocity is measured in meters per second (m/s) in physics. It determines not just how quickly a particle moves from one point to another but also influences its kinetic energy. The example provided includes a velocity of \( 10 \mathrm{~ms}^{-1} \). This moderate speed, especially for microscopic particles, highlights the wavelike characteristics predicted by de Broglie's hypothesis.
Understanding velocity's impact on the de Broglie wavelength is crucial as it reveals how motion at microscopic scales often defies classical intuition. Quantum behaviors become prominent with particular combinations of mass and velocity, underlying the essence of quantum physics where particles exhibit both wave and particle characteristics.