Problem 78 - A Complete Resource Book in Chemistry for JEE Main

Question

The wavelength associated with a golf ball weighing \(200 \mathrm{~g}\) and moving at a speed of \(5 \mathrm{~m} / \mathrm{h}\) is of the order (a) \(10^{-10} \mathrm{~m}\) (b) \(10^{-20} \mathrm{~m}\) (c) \(10^{-30} \mathrm{~m}\) (d) \(10^{-40} \mathrm{~m}\)

Step-by-Step Solution

Answer

The wavelength is of the order \(10^{-30} \mathrm{~m}\), so the answer is (c).

Step 1: Convert units

Convert the speed of the golf ball from meters per hour to meters per second. Since there are 3600 seconds in one hour, the speed in meters per second is: \( \frac{5}{3600} \text{ m/s} = 0.00139 \text{ m/s} \).

Step 2: Calculate the momentum of the golf ball

Calculate the momentum using the formula \( p = mv \), where \( m \) is the mass in kilograms (0.2 kg) and \( v \) is the speed in meters per second (0.00139 m/s). So \( p = 0.2 \times 0.00139 = 0.000278 \text{ kg m/s} \).

Step 3: Apply the de Broglie wavelength formula

Use the de Broglie wavelength formula \( \lambda = \frac{h}{p} \), where \( h \) is Planck's constant \(6.626 \times 10^{-34} \text{ m}^2\text{kg/s}\). Substitute \( p = 0.000278 \text{ kg m/s} \) to find \( \lambda = \frac{6.626 \times 10^{-34}}{0.000278} \approx 2.38 \times 10^{-30} \text{ m}\).

Step 4: Compare with given options

The order of magnitude of the calculated wavelength is \(10^{-30} \text{ m}\), which matches option (c) \(10^{-30} \text{ m} \).

Key Concepts

Momentum

Momentum is a fundamental concept in physics that tells us how much force and movement an object has. It is calculated using the formula \( p = mv \), where \( p \) is momentum, \( m \) is the mass in kilograms, and \( v \) is the velocity in meters per second. In our problem with the golf ball, its mass is 0.2 kg, and we've calculated its velocity to be 0.00139 m/s. After plugging in these numbers, we find the momentum to be \( 0.000278 \text{ kg}\cdot \text{m/s} \).

Here are some key points to consider:

• Momentum is directly proportional to both mass and velocity. If either increases, so does the momentum.
• Momentum has a direction as it depends on velocity, which is a vector quantity.
• Understanding momentum helps in determining how an object will move and interact with other objects.

With these points in mind, it becomes clearer why momentum is crucial in calculating the de Broglie wavelength of objects in motion.

Planck's constant

Planck's constant is a very small number but a big deal in physics! It is a fundamental constant denoted by \( h \) and is approximately \( 6.626 \times 10^{-34} \text{ m}^2\cdot\text{kg/s} \). It connects the wave and particle nature of matter and is essential when discussing quantum mechanics.

In the context of de Broglie's theory, Planck's constant helps us calculate a particle's wavelength, such as our golf ball. By using the formula \( \lambda = \frac{h}{p} \), where \( \lambda \) is the wavelength and \( p \) is the momentum, we determine how wave-like the particle is.

Some interesting facts about Planck's constant:

• It is named after Max Planck, who was pivotal in the early development of quantum theory.
• Planck's constant is crucial in the relationship between energy and frequency for photons, often written as \( E = hf \).
• The small value demonstrates why macro-objects, like our golf ball, don't exhibit noticeable wave properties in everyday life.

Understanding and using Planck's constant allows us to explore the profound and fascinating world of quantum physics.

Unit Conversion

Unit conversion is an essential skill in physics that allows us to work consistently with the units we need for calculations. In the example of our golf ball, we must convert its speed from meters per hour to meters per second to align with standard units for calculating momentum.

Some tips for successful unit conversion:

• Always check the units before starting your calculations. Ensure they are consistent.
• For time-related conversions, remember there are 3600 seconds in an hour. Use this to convert speed from m/h to m/s accurately.
• Be cautious of unit conversions in mass. Ensure the mass is in kg when calculating momentum (1 g = 0.001 kg).

Unit conversion is crucial for not only performing correct calculations but also communicating those results clearly and effectively in physics and science in general.