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

Step 1: Calculate Momentum

To find the de Broglie wavelength, we first need to calculate the momentum of the ball. The kinetic energy \( KE \) is given by \( KE = \frac{1}{2} mv^2 \). Given \( KE = 0.5 \) J and mass \( m = 1 \) kg, we can find the velocity \( v \) as follows: \( v = \sqrt{2 \cdot KE / m} = \sqrt{2 \times 0.5 / 1} = 1 \text{ m/s} \). The momentum \( p \) is then calculated as \( p = mv = 1 \cdot 1 = 1 \text{ kg m/s} \).

Step 2: Apply de Broglie Wavelength Formula

The de Broglie wavelength \( \lambda \) is given by the formula \( \lambda = \frac{h}{p} \), where \( h = 6.626 \times 10^{-34} \text{ Js} \) is Planck's constant and \( p \) is the momentum we calculated. Substituting the values, we have \( \lambda = \frac{6.626 \times 10^{-34}}{1} = 6.626 \times 10^{-34} \text{ m} \).

Step 3: Compare with Options

Now that we have calculated the de Broglie wavelength as \( 6.626 \times 10^{-34} \text{ m} \), we compare this with the given options. The matching option is (a) \( 6.626 \times 10^{-34} \mathrm{~m} \).

Key Concepts

Kinetic Energy

Kinetic energy is a concept that describes the energy an object possesses due to its motion. When you see an object moving, like a rolling ball, it stores energy that can do work. This energy is called kinetic energy, and it is given by the formula:

• \( KE = \frac{1}{2} mv^2 \)

Here, \( m \) is the mass of the object and \( v \) is its velocity. The formula shows that kinetic energy is proportional to both the mass of the object and the square of its velocity. Thus, as the speed of the object increases, its kinetic energy increases exponentially.
For example, in our original exercise, the ball has a kinetic energy of \( 0.5 \) J with a mass of \( 1 \) kg. Using the equation, by finding the velocity, you directly see how much energy a moving object possesses due to its motion.

Momentum

Momentum is a fundamental concept in physics that describes the quantity of motion an object has. It is the product of an object's mass and velocity. The formula for momentum \( p \) is:

• \( p = mv \)

This relationship shows that both the mass and velocity of an object affect its momentum. An increase in either mass or velocity will result in a higher momentum, implying more motion. Momentum is a crucial component when considering the de Broglie wavelength of an object.
In the problem, after calculating the velocity of the ball as \( 1 \text{ m/s} \), the momentum is found to be \( 1 \text{ kg m/s} \) since both mass and velocity were known. This value is essential because the de Broglie wavelength is directly calculated using this momentum value.

Planck's Constant

Planck's constant is a vital figure in quantum mechanics and is denoted by \( h \). This constant is extremely small, with a value of \( 6.626 \times 10^{-34} \text{ Js} \). It serves as the bridge between the wave and particle nature of matter, helping describe phenomena at atomic and subatomic levels.
Planck's constant is key when discussing the de Broglie wavelength, which relates the wavelength associated with a moving particle to that particle's momentum. The de Broglie wavelength \( \lambda \) formula is:

• \( \lambda = \frac{h}{p} \)

Here, \( h \) is Planck's constant, and \( p \) is the momentum of the object. This formula shows how quantum mechanics suggests that every object exhibits wave-like properties, albeit very slight for macroscopic objects due to the tiny magnitude of Planck's constant. In our exercise, using both Planck's constant and the calculated momentum enables the determination of the de Broglie wavelength.