Problem 130 - A Complete Resource Book in Chemistry for JEE Main

Question

The wavelength associated with an electron (mass = \(9.11 \times 10^{-31} \mathrm{~kg}\) ) moving with a velocity of \(10^{6} \mathrm{~m} \mathrm{~s}^{-1}\) is \(\left(\mathrm{h}=6.625 \times 10^{-34} \mathrm{~J} \mathrm{~s}\right)\) (a) \(0.727 \mathrm{~nm}\) (b) \(7.27 \mathrm{~nm}\) (c) \(727 \mathrm{~nm}\) (d) \(7.27 \mathrm{~m}\)

Step-by-Step Solution

Answer

The correct answer is (a) \(0.727 \text{ nm}\).

Step 1: Understanding De Broglie Wavelength Formula

The de Broglie wavelength of a particle is given by the formula \( \lambda = \frac{h}{mv} \) where \( \lambda \) is the wavelength, \( h \) is Planck's constant, \( m \) is the mass of the particle, and \( v \) is its velocity.

Step 2: Plugging in Values into Formula

Use the given values: Planck's constant \( h = 6.625 \times 10^{-34} \) J·s, electron mass \( m = 9.11 \times 10^{-31} \) kg, and velocity \( v = 10^{6} \) m/s. Substitute these into the formula as follows: \[ \lambda = \frac{6.625 \times 10^{-34}}{9.11 \times 10^{-31} \times 10^{6}} \]

Step 3: Calculating the Wavelength

Perform the calculation: \[ \lambda = \frac{6.625 \times 10^{-34}}{9.11 \times 10^{-25}} = 7.27 \times 10^{-10} \] meters.Since 1 nanometer (nm) equals \( 10^{-9} \) meters, convert the result:\[ \lambda = 0.727 \text{ nm} \]

Step 4: Final Answer Selection

Compare the calculated wavelength \( 0.727 \text{ nm} \) with the given options, and conclude that the option (a) \( 0.727 \text{ nm} \) is correct.

Key Concepts

Planck's Constant

The link between energy and frequency in quantum mechanics is defined by Planck's constant, denoted as \( h \). This is a fundamental constant of nature that plays a crucial role in the quantization of energy. The value of Planck's constant is \( 6.625 \times 10^{-34} \) Joule seconds (J·s). This implies that the energy of a photon is directly proportional to its frequency, with Planck's constant as the proportionality factor. The equation representing this relationship is \( E = h u \), where \( E \) is the energy, \( u \) is the frequency, and \( h \) is Planck's constant.

• This constant is essential in the calculation of the de Broglie wavelength, connecting particle properties like momentum and wavelength.
• Planck's constant bridges the gap between wave and particle characteristics in the realm of quantum mechanics.

To put it simply, without Planck's constant, we would not be able to quantify the wave nature of particles that forms the basis of quantum mechanics.

Electron Mass

The mass of an electron is incredibly minute, yet it is fundamental to the structure and behavior of matter. The electron mass is approximately \( 9.11 \times 10^{-31} \) kilograms. Despite being extremely small, it significantly influences the properties of atoms and molecules.

• The mass of an electron is fundamental for calculating the de Broglie wavelength, which describes the wave-like nature of particles.
• Even though electrons are very light, their speed and behavior can deeply affect the physical and chemical properties of a substance.

Understanding electron mass helps explain phenomena like electricity flow and chemical bonding, as these particles move through materials or react with other elements.

Velocity

Velocity is a vector quantity that describes the speed of an object in a specific direction. In the context of calculating de Broglie wavelength, velocity is crucial. It defines how fast and in what direction a particle, such as an electron, is moving.The velocity of an electron in this exercise is given as \( 10^6 \) meters per second. This high velocity is typical for particles at the atomic scale.

• When calculating the de Broglie wavelength, both the magnitude and direction of velocity determine how the wave properties are expressed.
• Velocity plays a vital role in quantum mechanics, as it affects the momentum and, consequently, the wavelength of the particle.

Knowing the velocity helps us understand the particle's kinetic energy and its interactions in a field or space.