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 approximately \(6.63 \times 10^{-29} \, \mathrm{m}\), so the correct answer is (d).

Step 1: Understanding the de Broglie Wavelength

The de Broglie wavelength (bby) of a particle is given by the formula \(\lambda = \frac{h}{mv}\), where \(h\) is Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\)), \(m\) is the mass of the particle, and \(v\) is its velocity. We will use this formula to calculate the wavelength for the given mass and velocity.

Step 2: Substitute Known Values

Substitute the given values into the de Broglie equation: \(\lambda = \frac{6.626 \times 10^{-34} \, \text{Js}}{10^{-6} \, \text{kg} \times 10 \, \text{ms}^{-1}}\). This simplifies to \(\lambda = \frac{6.626 \times 10^{-34}}{10^{-5}}\).

Step 3: Perform the Calculation

Calculate the value: \(\lambda = \frac{6.626 \times 10^{-34}}{10^{-5}} = 6.626 \times 10^{-29} \, \text{m}\).

Step 4: Select the Correct Answer

Compare the calculated value to the options given in the problem. The result \(6.626 \times 10^{-29} \, \text{m}\) matches option (d).

Key Concepts

Planck's Constant

Planck's constant () is one of the most important constants in quantum mechanics. Its value is approximately 6.626 x 10^{-34} Joule-seconds (Js) and is symbolized by the letter \(h\). This constant is fundamental in determining the energy of photons (light particles) using the equation \(E = h u\), where \(u\) is the frequency of the radiation. It represents the idea that energy is quantized, meaning it exists in discrete units, rather than a continuous range.

In the context of de Broglie's wavelength, Planck's constant plays a key role in bridging the concept of wave-particle duality. It allows us to calculate the wavelength of a particle with known mass and velocity using the formula \(\lambda = \frac{h}{mv}\). Here, \(m\) is the mass, \(v\) is velocity, and \(\lambda\) represents the particle's wavelength.

Planck's constant is small because it deals with very short distances and tiny energy quanta common in the quantum world, thereby making it less noticeable at everyday scales. Nevertheless, it is indispensable in understanding and calculating microscopic phenomena.

mass and velocity relation

The relationship between mass and velocity is crucial in determining the behavior of particles and their associated de Broglie wavelength. In the formula \(\lambda = \frac{h}{mv}\), both mass \(m\) and velocity \(v\) are inversely proportional to the wavelength \(\lambda\). This implies several interesting points:

• **Increased Mass or Velocity:** As either the mass or velocity of a particle increases, the de Broglie wavelength decreases. This means larger or faster particles have shorter wavelengths.
• **Decreased Mass or Velocity:** Conversely, a smaller mass or lower velocity results in a longer wavelength. This is particularly evident in microscopic particles, where quantum effects like wave-particle duality are observable.

This inverse relationship is key to understanding why we don't observe wave-like properties in larger objects with greater masses and velocities. For everyday objects, the de Broglie wavelength is so minuscule it seems nonexistent, reinforcing classical physics over quantum mechanical behaviors seen at the atomic and subatomic levels.

wave-particle duality

Wave-particle duality is a cornerstone concept in quantum mechanics, suggesting that particles can exhibit both wave-like and particle-like properties. This duality was famously proposed by Louis de Broglie, who suggested that every moving particle or object has an associated wave.

Key Points of Wave-Particle Duality:

• **Photons and Matter Waves:** Light displays wave-particle duality, behaving as a wave in phenomena like interference and diffraction, and as particles (photons) in the photoelectric effect. De Broglie extended this notion to matter, suggesting matter waves are analogous to light waves.
• **Demonstration in Particles:** Electrons are quintessential to demonstrating wave-particle duality. When electrons pass through a double slit, they form an interference pattern, a hallmark of wave behavior, even though electrons are particles.

Wave-particle duality expands our understanding of nature beyond Newtonian mechanics, as it reveals the complex, dual nature of matter and energy at small scales. This fundamental concept allows scientists to predict and explain physical phenomena that don't fit into classical physics, highlighting the need for quantum physics.