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

Question

According to de Broglie concept, all material particles posses wave character as well as particle character. The wave associated with a moving particle is called matter wave. The wavelength of the matter wave is given by the equation \(\lambda=\frac{\mathrm{h}}{\mathrm{p}}=\frac{\mathrm{h}}{\mathrm{mv}}\) where \(\mathrm{p}\) is the momentum of the particle, " \(\mathrm{m}\) ' is the mass of the particle and ' \(\mathrm{v}^{\prime}\) is the velocity of the particle. ' \(\mathrm{h}\) ' is called Planck's constant. Particle A moving with a certain velocity has de Broglie wavelength of \(1 \mathrm{~A}\). If the particle B has mass \(20 \%\) and velocity \(80 \%\) of that of \(A\), the de Broglie wavelength of 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 \(6.25 \text{ Å} (d)\).

Step 1: Identify Given Values

We know, the de Broglie wavelength \( \lambda_A = 1 \text{ Å} \). The mass of B, \( m_B = 0.2m_A \), and the velocity of B, \( v_B = 0.8v_A \). We need to find \( \lambda_B \).

Step 2: Understand Relationship

The de Broglie wavelength equation states \( \lambda = \frac{h}{mv} \). Hence, for particle A, \( \lambda_A = \frac{h}{m_Av_A} \). For particle B, the equation is \( \lambda_B = \frac{h}{m_Bv_B} \).

Step 3: Use Proportionality

Since \( \lambda_A = \frac{h}{m_Av_A} \), we can write \( \lambda_B = \frac{h}{(0.2m_A)(0.8v_A)} \).

Step 4: Simplify Expression for \(\lambda_B\)

Substitute \( m_B = 0.2m_A \) and \( v_B = 0.8v_A \) into \( \lambda_B \): \( \lambda_B = \frac{h}{0.16m_Av_A} \). This simplifies to \( \lambda_B = \frac{1}{0.16} \lambda_A \).

Step 5: Calculate \(\lambda_B\)

Since \( \lambda_A = 1 \text{ Å} \), we have \( \lambda_B = \frac{1}{0.16} \times 1 \). Calculating gives \( \lambda_B = 6.25 \text{ Å} \).

Step 6: Select Correct Option

Given the options, \( \lambda_B = 6.25 \text{ Å} \) matches option (d).

Key Concepts

Matter Waves

In quantum mechanics, the term "matter waves" describes the concept that all particles display wave-like characteristics. This dual nature arises from the de Broglie hypothesis, which suggests that every moving particle or object has an associated wave. The wavelike behavior is especially significant for small particles like electrons.

• Particle-Wave Duality: Traditionally, light and matter were thought to be distinct – light exhibiting wave characteristics and matter having a solid form. However, the discovery of matter waves showed that particles have a wavelength, linking wave properties to all matter.
• Wavelength Calculation: The wavelength associated with a particle, known as the de Broglie wavelength, is determined by the particle’s momentum: \[ \lambda = \frac{h}{p} \] where \(\lambda\) is the wavelength, \(h\) is Planck's constant, and \(p\) is the momentum.
• Applications: Matter waves explain phenomena like the diffraction patterns observed when electrons pass through thin slits, which would be impossible if electrons solely exhibited particle-like behavior.

Matter waves redefine our understanding of the microscopic world by integrating wave properties into the realm of particles. It’s a cornerstone in the quantum mechanics framework.

Planck's Constant

Planck's constant is a fundamental quantity in quantum mechanics, named after physicist Max Planck. It is a crucial element in the de Broglie equation, linking a particle's kinetic properties to its wave characteristics.

• Planck's constant \(h\) is approximately \(6.626 \times 10^{-34} \text{ Js}\). It embodies the concept that energy is quantized, only changing in discrete amounts rather than continuously.
• In the de Broglie equation, Planck's constant allows us to determine the wavelength of matter waves: \[ \lambda = \frac{h}{mv} \] where \(\lambda\) is the wavelength, \(m\) is mass, and \(v\) is velocity.
• Planck's constant redefined how physicists understood energy at atomic levels. It revealed that energy and matter behaviors are interlinked in ways classical physics couldn’t account for.

Understanding Planck's constant is essential in studying quantum phenomena and interpreting experiments in modern physics.

Momentum

Momentum in physics refers to the quantity of motion an object possesses and is generally defined as the product of mass and velocity.

• Within the scope of the de Broglie equation, momentum \(p\) is pivotal since the wavelength of matter waves is inversely proportional to momentum: \[ \lambda = \frac{h}{p} \] whereby a higher momentum results in a shorter wavelength.
• Momentum is a vector quantity, meaning it has both magnitude and direction. This makes it crucial for understanding how different forces affect an object's motion.
• In quantum mechanics, momentum helps describe particles’ motion through space, playing a significant role in behaviors like tunneling and the Heisenberg Uncertainty Principle.

Momentum bridges classical mechanics and quantum concepts, providing a pathway to analyze and predict the behavior of particles with wave-like properties.