Problem 173 - A Complete Resource Book in Chemistry for JEE Main

Question

Calculate the wavelength (in monometer) associated with a proton moving at \(1.0 \times 10^{3} \mathrm{~ms}^{-1}\) (Mass of proton \(=1.67 \times 10^{-27} \mathrm{~kg}\) and \(\left.\mathrm{h}=6.63 \times 10^{-34} \mathrm{Js}\right): \quad[\mathbf{2 0 0 9}]\) (a) \(0.40 \mathrm{~nm}\) (b) \(2.5 \mathrm{~nm}\)

Step-by-Step Solution

Answer

The wavelength is approximately 0.40 nm, which is option (a).

Step 1: Identify the Formula to Use

To find the wavelength associated with a moving particle, we use the de Broglie wavelength formula: \[\lambda = \frac{h}{mv}\]where \(\lambda\) is the wavelength, \(h\) is Planck's constant, \(m\) is the mass of the particle, and \(v\) is the velocity.

Step 2: Substitute Known Values

We have \(h = 6.63 \times 10^{-34}\, \text{Js}\), \(m = 1.67 \times 10^{-27}\, \text{kg}\), and \(v = 1.0 \times 10^{3}\, \text{ms}^{-1}\). Substitute these values into the formula:\[\lambda = \frac{6.63 \times 10^{-34}}{1.67 \times 10^{-27} \times 1.0 \times 10^{3}}\]

Step 3: Calculate the Wavelength

Perform the calculation:\[\lambda = \frac{6.63 \times 10^{-34}}{1.67 \times 10^{-24}} \approx 3.97 \times 10^{-10}\, \text{m}\]This converts to 0.397 nm since \( \text{1 nm} = 10^{-9} \text{m} \).

Step 4: Choose the Correct Answer

The calculated wavelength is approximately 0.397 nm, which matches closest to option (a) 0.40 nm.

Key Concepts

Understanding Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes the nature and behavior of matter and energy on the atomic and subatomic level. It provides a mathematical framework for understanding various physical phenomena, which classical physics cannot explain. Quantum mechanics introduces the concept that particles can exhibit both wave-like and particle-like properties. This duality is essential in explaining behaviors that occur on a microscopic scale.

Key principles of quantum mechanics include:

• Wave-Particle Duality: Particles such as electrons and protons display characteristics of both waves and particles. This duality is central to quantum mechanics.
• Uncertainty Principle: Developed by Werner Heisenberg, it states that one cannot simultaneously know the exact position and momentum of a particle.
• Quantization: Energy levels of electrons in atoms are quantized, meaning they exist in discrete levels rather than a continuous range.

By applying these principles, quantum mechanics helps us explore and predict the behavior of particles in various conditions, such as determining the de Broglie wavelength of particles like protons.

Calculating Proton Wavelength

The wavelength of a proton can be determined using the de Broglie wavelength formula. The concept of wavelength is significant in quantum mechanics since it relates to the wave-like nature of particles. The de Broglie hypothesis proposed that any particle with momentum can be described by a wave function. This is captured through the formula:

• \[\lambda = \frac{h}{mv}\]
• Where:
  • \(\lambda\) is the wavelength,
  • \(h\) is Planck's constant, \(6.63 \times 10^{-34}\; \text{Js}\),
  • \(m\) is the mass of the particle,
  • \(v\) is the velocity.

When calculating the de Broglie wavelength of a proton moving at \(1.0 \times 10^{3}\; \text{ms}^{-1}\), we substitute the known values into the formula. This allows us to observe the wave properties of protons, much like photons of light. The de Broglie wavelength highlights the quantum mechanical aspect of all matter, demonstrating that wave properties are inherent in particles although more noticeable in microscopic objects like protons.

Importance of Planck's Constant

Planck's constant is a fundamental quantity in physics, symbolized as \(h\), equal to \(6.63 \times 10^{-34}\; \text{Js}\). It plays a critical role in quantum mechanics and is pivotal in the de Broglie wavelength formula. This constant represents the proportionality between the energy of a photon and its frequency.Planck's constant is not just a numerical factor in equations. It essentially determines the scale at which quantum mechanical effects become significant. For example, it is tiny enough to show that macroscopic objects don't display noticeable quantum effects, while microscopic particles like electrons or protons do exhibit such characteristics.In the context of the de Broglie wavelength, Planck's constant bridges the gap between classical and quantum physics. It allows us to calculate how microscopic particles behave like waves, thereby providing a clearer picture of the dual nature of matter. Understanding Planck's constant is crucial for students delving into quantum mechanics to comprehend the wave-particle duality fully.