Problem 157 - A Complete Resource Book in Chemistry for JEE Main

Question

The de Broglie wavelength of a tennins ball of mass \(60 \mathrm{~g}\) moving with a velocity of 10 metres per second is approximately \([\mathbf{2 0 0 3}]\) (Planck constant \(\mathrm{h}=6.63 \times 10^{-31} \mathrm{Js}\) ) (a) \(10^{-33} \mathrm{~m}\) (b) \(10^{-31} \mathrm{~m}\) (c) \(10-16 \mathrm{~m}\) (d) \(10^{-25} \mathrm{~m}\)

Step-by-Step Solution

Answer

The de Broglie wavelength is approximately \(10^{-31} \text{ m}\), option (b).

Step 1: Identify the de Broglie Wavelength Formula

The de Broglie wavelength \( \lambda \) for an object is given by the formula \[ \lambda = \frac{h}{mv} \] where \( h \) is the Planck constant, \( m \) is the mass of the object, and \( v \) is the velocity of the object.

Step 2: Convert Units

Convert the mass of the tennis ball from grams to kilograms since SI units are required. \[ 60 \text{ g} = 0.06 \text{ kg} \]

Step 3: Substitute Values

Substitute the given values into the de Broglie wavelength formula. Here, \( h = 6.63 \times 10^{-31} \text{ Js} \), \( m = 0.06 \text{ kg} \), and \( v = 10 \text{ m/s} \). Therefore, the equation becomes: \[ \lambda = \frac{6.63 \times 10^{-31}}{0.06 \times 10} \]

Step 4: Calculate the Wavelength

Perform the division to find the wavelength: \[ \lambda = \frac{6.63 \times 10^{-31}}{0.6} = 1.105 \times 10^{-30} \text{ m} \]

Step 5: Approximate and Match Answer Choices

The calculated wavelength \( \lambda = 1.105 \times 10^{-30} \text{ m} \) can be approximated as \( \approx 10^{-31} \text{ m} \), which corresponds to option (b).

Key Concepts

Planck constant

The Planck constant, symbolized as \( h \), is a fundamental constant in physics that plays a pivotal role in quantum mechanics. It quantifies the relationship between energy and frequency in the equation \( E = hu \). This constant essentially forms the bridge between the wave and particle nature of light and matter.

The Planck constant is incredibly small, with a value of \( 6.63 \times 10^{-34} \text{ Js} \). This small magnitude is crucial in quantum mechanics, as it highlights why quantum effects are typically only observable at minuscule scales, like those involving subatomic particles. In this exercise, the Planck constant is used to calculate the de Broglie wavelength, illustrating its importance in linking motion on a macroscopic scale to quantum principles.

• Relates energy and frequency: \( E = hu \)
• Key component in quantum mechanics and de Broglie wavelength formula
• Small value indicating why quantum effects aren't seen in everyday objects

quantum mechanics

Quantum mechanics is a fundamental theory in physics that explains the nature and behavior of matter and energy on atomic and subatomic levels. Unlike classical mechanics, quantum mechanics incorporates principles of probability and uncertainty.

One of the key aspects of quantum mechanics is the wave-particle duality, which suggests that every particle or quantum entity can exhibit both wave-like and particle-like properties. The de Broglie hypothesis is a pivotal concept in quantum mechanics that proposes particles such as electrons also possess a wavelength, which is related to their momentum. This approach helps us understand the wave behavior of microscopic particles, crucial for phenomena like electron diffraction and interference seen at quantum scales.

• Deals with subatomic particles
• Wave-particle duality: matter exhibits both wave and particle characteristics
• Basis for understanding phenomena at atomic levels

wavelength calculation

Calculating the de Broglie wavelength involves a straightforward formula that links the physical properties of an object to its quantum characteristics. The formula \( \lambda = \frac{h}{mv} \) calculates the wavelength (\( \lambda \)) by dividing the Planck constant \( h \) by the momentum, which is the product of mass (\( m \)) and velocity (\( v \)).

In this formula:

• \( h \) is the Planck constant and represents the quantum nature of matter.
• \( m \) is the mass of the object, needing conversion into kilograms to match the SI system used by the constant.
• \( v \) is the velocity of the object, typically measured in meters per second (m/s).

The calculated wavelength tends to be astronomically small, especially for everyday macroscopic objects like a tennis ball, emphasizing why quantum effects are unnoticed at such scales. This exercise demonstrates the de Broglie wavelength's insightful application, although more apparent in microscopic entities like electrons rather than macroscopic ones like a tennis ball.