Problem 132 - A Complete Resource Book in Chemistry for JEE Main

Question

Particle A moving with a certain velocity has de Broglie wavelength of \(1 \AA\). If the particle B has mass \(20 \%\) and velocity \(80 \%\) of that of \(\mathrm{A}\), the de Broglie wavelength of \(\mathrm{B}\) will be (a) \(1.6 \AA\) (b) \(16 \AA\) (c) \(4.0 \AA\) (d) \(6.25 \AA\)

Step-by-Step Solution

Answer

The de Broglie wavelength of particle B is \(6.25 \AA\) (option d).

Step 1: Understanding de Broglie wavelength formula

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

Step 2: Information for particle A

Particle A has a de Broglie wavelength of \(1 \AA\). Thus, according to the formula, \( \lambda_A = \frac{h}{m_Av_A} = 1 \AA\).

Step 3: Establishing relationships for particle B

Particle B has a mass of \(0.2m_A\) (20% of A) and a velocity of \(0.8v_A\) (80% of A). Therefore, the de Broglie wavelength of B can be expressed as: \( \lambda_B = \frac{h}{m_Bv_B} = \frac{h}{0.2m_A \times 0.8v_A}\).

Step 4: Substitute and simplify for \( \lambda_B \)

Substitute the expressions for \(m_B\) and \(v_B\) into the de Broglie equation for \(\lambda_B\): \[ \lambda_B = \frac{h}{0.16 \times m_Av_A} = \frac{1}{0.16} \times \frac{h}{m_Av_A} = \frac{1}{0.16} \times \lambda_A \] Since \( \lambda_A = 1 \AA \), we have \[ \lambda_B = \frac{1}{0.16} \times 1 \AA = 6.25 \AA \].

Key Concepts

Planck's constant

Planck's constant is a fundamental constant in physics, denoted by the symbol \( h \). It is essential in the study of quantum mechanics because it relates the energy of a photon to its frequency. The constant has a value of approximately \( 6.626 \times 10^{-34} \) Js. In the context of the de Broglie wavelength, Planck's constant helps us connect the wavelength of a particle, such as an electron or a proton, to its momentum. In the de Broglie equation, \( \lambda = \frac{h}{mv} \), where \( \lambda \) is the wavelength, \( m \) is the mass, and \( v \) is the velocity of the particle, Planck's constant serves as the bridge between the wave and particle nature of matter. Understanding Planck's constant allows us to recognize that particles at the atomic and subatomic level can exhibit wave-like properties, such as interference and diffraction. This forms the basis for much of modern physics and is crucial in fields like nanotechnology and quantum computing.

Particle Velocity

The velocity of a particle is a crucial component in determining its de Broglie wavelength. Velocity, represented by \( v \), refers to the speed and direction of the particle's movement. In simple terms, it's how fast and in which direction the particle is traveling. Velocity is a vector quantity, meaning it has both a magnitude (speed) and a direction.In the calculation of de Broglie wavelength using \( \lambda = \frac{h}{mv} \), velocity influences the wavelength inversely. This means:

• If the velocity is high, the resulting wavelength is shorter.
• If the velocity is low, the wavelength is longer.

For example, in the comparison of particles A and B from the original problem, particle B has a velocity of \( 0.8v_A \). Since velocity plays a critical role in determining the wavelength, any change in velocity alters the de Broglie wavelength, highlighting the importance of accurately measuring or considering the velocity in quantum behaviors.

Mass and Wavelength Relationship

The mass of a particle plays a significant role in determining its de Broglie wavelength. Mass is represented by \( m \) in the de Broglie equation \( \lambda = \frac{h}{mv} \). Similar to velocity, mass affects the wavelength inversely. This means:

• The greater the particle's mass, the shorter its wavelength becomes.
• The smaller the mass, the longer the wavelength.

In our exercise, particle B has a mass that is \( 20\% \) of particle A's mass, i.e., \( m_B = 0.2m_A \). Decreasing the mass of B compared to A significantly affects its wavelength, as we see in the solution.Understanding this mass-wavelength relationship is fundamental in fields such as particle physics and quantum mechanics, where particles can have very small masses, resulting in noticeable wave-like properties. This concept underpins the behavior of electrons in structures like atoms, affecting phenomena such as electron diffraction and the operation of electron microscopes.