Problem 131 - A Complete Resource Book in Chemistry for JEE Main

Question

The de Broglie wave length of a moving particle of mass \(1 \mathrm{~g}\) is \(6.625 \times 10^{-23} \mathrm{~m}\). The velocity of the particle is (a) \(100 \mathrm{~cm} \mathrm{~s}^{-1}\) (b) \(100 \mathrm{~m} \mathrm{~s}^{-1}\) (c) \(10 \mathrm{~m} \mathrm{~s}^{-1}\) (d) \(1000 \mathrm{~m} \mathrm{~s}^{-1}\)

Step-by-Step Solution

Answer

The velocity is much lower than the given options, suggesting a reevaluation or reconfirmation of approximations.

Step 1: Understand the de Broglie Wavelength Formula

The de Broglie wavelength \( \lambda \) of a particle is given by the formula: \( \lambda = \frac{h}{mv} \), where \( h \) is the Planck's constant \( (6.626 \times 10^{-34} \text{ J s}) \), \( m \) is the mass of the particle, and \( v \) is the velocity of the particle.

Step 2: Convert the Mass Unit

The mass of the particle is given as \( 1 \text{ g} \). Convert this mass into kilograms since the SI unit for mass in the de Broglie formula is kilograms. So, \( 1 \text{ g} = 1 \times 10^{-3} \text{ kg} \).

Step 3: Substitute Known Values into the Formula

We know the wavelength \( \lambda = 6.625 \times 10^{-23} \) m and the mass \( m = 1 \times 10^{-3} \) kg. The Planck's constant \( h = 6.626 \times 10^{-34} \text{ J s} \). Substitute these into the formula: \[ 6.625 \times 10^{-23} = \frac{6.626 \times 10^{-34}}{(1 \times 10^{-3}) v} \]

Step 4: Solve for Velocity \( v \)

Rearrange the equation to solve for \( v \): \[ v = \frac{6.626 \times 10^{-34}}{6.625 \times 10^{-23} \times 1 \times 10^{-3}} \] Calculate \( v \): \[ v = \frac{6.626 \times 10^{-34}}{6.625 \times 10^{-26}} = 1 \times 10^{-8} \text{ m/s} \].

Step 5: Match the Calculated Velocity with Given Options

The calculated velocity \( 1 \times 10^{-8} \text{ m/s} \) does not match any given options exactly. Perform this to check any possible miscalculation by re-evaluating concepts: Given the orders of magnitude, recheck Apple's constants and approximations, aimed to better translation.

Key Concepts

Planck's Constant

To comprehend the de Broglie wavelength, it is vital to understand Planck's constant, denoted by \( h \). Max Planck, a German physicist, introduced this constant when studying black-body radiation; its value is \( 6.626 \times 10^{-34} \text{ J s} \). Planck's constant plays a fundamental role in quantum mechanics because it relates the energy of a photon to the frequency of its electromagnetic wave. Moreover, it is a crucial part of the de Broglie equation which links wave behavior to particles. This constant is incredibly tiny, underscoring how quantum effects are pronounced only at very small scales. Therefore, when calculating the de Broglie wavelength, Planck's constant ensures these intricate quantum behaviors are accurately described. The precise value of Planck's constant allows physicists and students to perform exact calculations about particle wavelengths, bridging the gap between traditional physics and quantum physics.

Mass Conversion

In physics, appropriately converting units is essential, especially when applying equations. The de Broglie wavelength formula, \( \lambda = \frac{h}{mv} \), necessitates the mass of a particle in kilograms, consistent with the SI unit system. However, common measurements might be presented in grams, as in this case where the particle's mass is \( 1 \text{ g} \). To align with the SI system, convert grams to kilograms by dividing by 1000 (since \( 1 \text{ kg} = 1000 \text{ g} \)), giving \( 1 \times 10^{-3} \text{ kg} \).
This conversion is crucial in ensuring the de Broglie formula applies correctly. Any oversight in unit conversion can lead to discrepancies in calculations or interpretations of results. By standardizing units to the SI system, accurate and reliable calculations for phenomena like quantum particle behavior become feasible. This practice highlights the importance of precision and attention to detail in scientific calculations.

Velocity Calculation

Solving for velocity in this problem requires rearranging the de Broglie wavelength formula. Given \( \lambda = \frac{h}{mv} \), solve for velocity \( v \) by rearranging it as \( v = \frac{h}{m\lambda} \). With known values for \( \lambda \), \( m \), and \( h \), substitute these into the equation:

• \( \lambda = 6.625 \times 10^{-23} \text{ m} \)
• \( m = 1 \times 10^{-3} \text{ kg} \)
• \( h = 6.626 \times 10^{-34} \text{ J s} \)

Plugging in these figures, the equation becomes:
\[ v = \frac{6.626 \times 10^{-34}}{6.625 \times 10^{-23} \times 1 \times 10^{-3}} \]
Executing the calculation gives \( v = 1 \times 10^{-8} \text{ m/s} \). Although this answer doesn't match any of the provided options in the problem statement, it emphasizes the need for precision and understanding of the underlying principles in physics problems. Revisiting calculations and comprehending the possible factors, like approximations, can help ensure solutions align with expected results.