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 \( 6.626 \times 10^{-34} \mathrm{~m} \), so the correct answer is (a).

Step 1: Understand the de Broglie Wavelength Formula

The de Broglie wavelength \( \lambda \) is calculated using the formula: \( \lambda = \frac{h}{p} \) where \( h \) is the Planck’s constant \( 6.626 \times 10^{-34} \mathrm{~J~s} \) and \( p \) is the momentum of the object.

Step 2: Calculate the Momentum of the Ball

The momentum \( p \) is the product of mass \( m \) and velocity \( v \). Since momentum can also be expressed in terms of kinetic energy \( KE \) and mass, it is given by \( p = \sqrt{2mKE} \). Given the ball's mass \( m = 1 \mathrm{~kg} \) and the kinetic energy \( KE = 0.5 \mathrm{~J} \), the momentum is \( p = \sqrt{2 \times 1 \times 0.5} = 1 \mathrm{~kg~m/s} \).

Step 3: Calculate the de Broglie Wavelength

Using the de Broglie wavelength formula \( \lambda = \frac{h}{p} \), substitute \( h = 6.626 \times 10^{-34} \mathrm{~J~s} \) and \( p = 1 \mathrm{~kg~m/s} \), resulting in \( \lambda = \frac{6.626 \times 10^{-34}}{1} = 6.626 \times 10^{-34} \mathrm{~m} \).

Step 4: Identify the Correct Answer

Referencing the given options, find that option (a) matches the calculated wavelength \( 6.626 \times 10^{-34} \mathrm{~m} \). Therefore, the correct choice is (a).

Key Concepts

Planck's Constant

Planck's constant is a fundamental figure in quantum mechanics. Represented by the letter \( h \), it has the value of \( 6.626 \times 10^{-34} \mathrm{~J \, s} \). This constant is essential in the formula for calculating the de Broglie wavelength, which describes the wave properties of particles. It's a small number, which reflects the tiny scales at which quantum mechanical effects become noticeable.

Planck's constant is central to several significant theories and equations like Planck's radiation law, E=hf, where \( E \) is energy and \( f \) is frequency, and the Heisenberg uncertainty principle. These underscore the quantized nature of energy and the dual wave-particle behavior of light and matter.

Understanding Planck's constant helps grasp the scale and nature of quantum phenomena. It's an intrinsic part of our universe, underlying the physics at atomic and subatomic levels.

Momentum Calculation

Momentum, a key concept in physics, is a measure of the quantity of motion an object has. It is calculated by multiplying an object's mass by its velocity. The formula is \( p = mv \). For particles with kinetic energy, momentum can also be derived using \( p = \sqrt{2mKE} \), where \( m \) is mass and \( KE \) is kinetic energy.

This calculation is crucial when determining the de Broglie wavelength of an object, as the wavelength is inversely proportional to the momentum. To understand momentum in the broader sense, consider the law of conservation of momentum, which states that the total momentum of a closed system remains constant. This principle is pivotal in collision and explosion analyses, helping us analyze how objects move and interact.

Knowing how to calculate momentum is not only essential for physics problems but also provides insight into various real-world scenarios, from sports to car crashes.

Kinetic Energy

Kinetic energy is a measure of the energy of motion. Any object that is moving has kinetic energy, computed using the formula \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the velocity of the object. In the context of the de Broglie wavelength problem, kinetic energy allows us to express momentum without directly knowing velocity.

Kinetic energy is crucial because it changes with the square of the velocity; a small increase in speed results in a significant increase in kinetic energy. Understanding this concept is important in many fields, such as engineering, where calculating the kinetic energy of moving parts is necessary for safety and design purposes.

• In everyday life, kinetic energy is what powers vehicles down the road, what children burn off on playground adventures, and what keeps the merry-go-round spinning.
• It's the energy an object possesses due to its motion, and it transfers between objects during interactions, like balls striking or cars colliding.

In essence, kinetic energy helps us quantify the dynamic activities occurring around us constantly.