Problem 102 - A Complete Resource Book in Chemistry for JEE Main

Question

The de Broglie wavelength associated with a ball of mass \(1 \mathrm{~kg}\) having a kinetic energy \(0.5 \mathrm{~J}\) is (a) \(6.626 \times 10^{-34} \mathrm{~m}\) (b) \(13.2 \times 10^{-34} \mathrm{~m}\) (c) \(10.38 \times 10^{-21} \mathrm{~m}\) (d) \(6.626 \AA\)

Step-by-Step Solution

Answer

The de Broglie wavelength is \(6.626 \times 10^{-34} \text{ m}\), corresponding to option (a).

Step 1: Understand the de Broglie Wavelength

The de Broglie wavelength \( \lambda \) is given by the equation \( \lambda = \frac{h}{p} \), where \( h \) is Planck's constant \( 6.626 \times 10^{-34} \text{ Js} \) and \( p \) is the momentum of the object. For an object, momentum \( p \) can also be expressed in terms of kinetic energy \( KE \) as \( p = \sqrt{2m \cdot KE} \).

Step 2: Calculate the Momentum of the Ball

First, calculate the momentum \( p \) using the ball's mass \( m = 1 \text{ kg} \) and its kinetic energy \( KE = 0.5 \text{ J} \). Use the formula \( p = \sqrt{2m \cdot KE} \):\[ p = \sqrt{2 \times 1 \times 0.5} = \sqrt{1} = 1 \text{ kg m/s} \]

Step 3: Substitute Values to Find Wavelength

Substitute the values of Planck's constant \( h = 6.626 \times 10^{-34} \text{ Js} \) and momentum \( p = 1 \text{ kg m/s} \) into the de Broglie wavelength formula:\[ \lambda = \frac{h}{p} = \frac{6.626 \times 10^{-34}}{1} = 6.626 \times 10^{-34} \text{ m} \]

Step 4: Choose the Correct Answer

Compare the calculated wavelength to the given options. The calculated de Broglie wavelength \( 6.626 \times 10^{-34} \text{ m} \) matches option (a).

Key Concepts

Planck's constant

Planck's constant is a fundamental constant in physics with the value of \( 6.626 \times 10^{-34} \text{ Js} \). Named after Max Planck, who is credited with introducing the quantum theory, it serves as a bridge between the macroscopic and quantum worlds. This constant is essential for understanding energy quantization and plays a crucial role in calculating the de Broglie wavelength.

When we deal with particles at the quantum level, Planck's constant helps us relate the wave-like and particle-like properties of matter. It appears in many critical physics equations, such as the de Broglie wavelength formula \( \lambda = \frac{h}{p} \), where \( \lambda \) is the wavelength, \( h \) is Planck's constant, and \( p \) is the momentum. Without Planck's constant, our understanding of quantum mechanics and particle dynamics would fall apart.

Momentum calculation

Momentum is a measure of the motion of an object and is calculated as the product of an object's mass and its velocity. For objects at the macro-level, it's a straightforward calculation, but when you enter the quantum realm, it dominates the behaviors we observe.

In the context of quantum mechanics, momentum is tied to the de Broglie wavelength. In exercises, we use the formula for momentum regarding kinetic energy: \( p = \sqrt{2m \cdot KE} \). Here, \( m \) is the mass of the object, and \( KE \) is its kinetic energy.

This formula is particularly useful when energy is given rather than velocity. By knowing the mass and the kinetic energy, we can find the momentum needed to determine the de Broglie wavelength. For a ball of mass \( 1 \text{ kg} \) and kinetic energy \( 0.5 \text{ J} \), the calculation yields \( p = 1 \text{ kg m/s} \), guiding us to find the wavelength.

Kinetic energy

Kinetic energy is a key concept when exploring object motion. It represents the energy that an object possesses due to its motion and can be calculated using the formula \( KE = \frac{1}{2}mv^2 \). In this exercise, however, where only kinetic energy is provided, we use it differently.

Kinetic energy plays a crucial role when calculating momentum in place of speed. Once kinetic energy is known, momentum can be expressed in a secondary formula: \( p = \sqrt{2m \cdot KE} \). This allows us to unravel scenarios where velocity information isn't outright available, as seen in this exercise with the ball.

This is vital because it links the mechanical concept of kinetic energy, which is related to how fast an object moves, with quantum mechanics, thereby enabling accurate calculation of de Broglie wavelengths.