Problem 104 - A Complete Resource Book in Chemistry for JEE Main

Question

The size of a microscopic particle is one micron and its mass is \(6 \times 10^{-13} \mathrm{gm}\). If its position may be measured to within \(0.1 \%\) of its size, the uncertainty in velocity, in \(\mathrm{cm} \mathrm{s}^{-1}\), is approximately (a) \(10^{-6} / 3 \pi\) (b) \(10^{-7} / 2 \pi\) (c) \(10^{-5} / 4 \pi\) (d) \(10^{-7} / 4 \pi\)

Step-by-Step Solution

Answer

The uncertainty in velocity is approximately (d) \(\frac{10^{-7}}{4\pi}\) \(\text{cm s}^{-1}\).

Step 1: Identify given values and known formula

We are provided with the particle's size: 1 micron = \(1 imes 10^{-4}\) cm and its mass: \(6 \times 10^{-13}\) gm. The uncertainty in position is \(0.1\%\) of its size, which translates to \(\Delta x = 0.001 \times 1 \times 10^{-4}\) cm. We need to find \(\Delta v\), the uncertainty in velocity, using the Heisenberg uncertainty principle, which is given by \(\Delta x \cdot \Delta p \geq \frac{h}{4\pi}\), where \(\Delta p = m \cdot \Delta v\).

Step 2: Calculate uncertainty in position \(\Delta x\)

The uncertainty in position \(\Delta x\) is \(0.1\%\) of 1 micron or \(1 \times 10^{-4}\) cm, which is \(\Delta x = 0.001 \times 1 \times 10^{-4} = 10^{-7}\) cm.

Step 3: Apply the Heisenberg uncertainty principle

Substitute \(\Delta x = 10^{-7}\) cm and factor in the mass \(m = 6 \times 10^{-13}\) gm (converted to kg). The equation becomes \(\Delta x \cdot m \cdot \Delta v \geq \frac{h}{4\pi}\), where \(h = 6.626 \times 10^{-34}\) Js. Convert the mass to kg (since \(1 \text{ gm} = 10^{-3} \text{ kg}\)), thus the mass in kg is \(6 \times 10^{-16}\) kg.

Step 4: Solve for \(\Delta v\), the uncertainty in velocity

Substitute \(\Delta x = 10^{-7}\) cm (\(10^{-9}\) meters) and \(m = 6 \times 10^{-16}\) kg into the uncertainty formula: \(\Delta v \geq \frac{h}{4\pi \cdot \Delta x \cdot m}\). Solve for \(\Delta v\): \[\Delta v \geq \frac{6.626 \times 10^{-34}}{4\pi \times 10^{-9} \times 6 \times 10^{-16}} \approx \frac{10^{-7}}{4\pi}\] Verify the units remain as \(\text{cm s}^{-1}\) since we derived \(\Delta x\) with respect to centimeters.

Key Concepts

Microscopic Particles

Microscopic particles are incredibly small elements of matter, often on the scale of micrometers or less. They include entities such as atoms, molecules, and other particles like protons or neutrons. These particles are fundamental building blocks of matter and are pivotal in understanding the physical and chemical properties of substances.

A particle with a size of one micron, like the one mentioned in the exercise, is equal to one-millionth of a meter, showcasing its minuscule nature. To provide a clearer picture:

• One micron is equivalent to one-thousandth of a millimeter.
• Due to their small size, these particles require special techniques, like electron microscopes, for observation.
• They exhibit unique behaviors that aren’t observed in macroscopic or everyday objects, which we can often explain by quantum mechanics.

Understanding these tiny particles helps scientists explain complex phenomena and develop technologies across various fields.

Uncertainty in Velocity

The concept of uncertainty in velocity is rooted in the Heisenberg Uncertainty Principle from quantum mechanics. This principle states that we cannot precisely measure both the position and the momentum (which involves velocity) of a particle simultaneously. This comes from the wave-like behavior of microscopic particles.

In the exercise, the particle's position was measured within a fraction of its size, creating an uncertainty in its position. This uncertainty directly affects our ability to precisely know the velocity:

• If we measure position more accurately, the velocity becomes more uncertain.
• The calculation showed us that the uncertainty in velocity is significant for a microscopic particle measured with high precision in position.
• This illustrates why at the quantum level, predictions involve probabilities rather than certainties.

The principle doesn't reflect limitations in measurement technologies but rather a fundamental property of the universe.

Quantum Mechanics

Quantum mechanics is a branch of physics that deals with the behavior of microscopic particles, including their interactions and inherent uncertainties. It challenges the classical views of physics that apply to macroscopic objects, offering a new paradigm.

• Quantum mechanics introduces concepts like wave-particle duality, where particles such as electrons exhibit both wave and particle characteristics.
• Understanding the behavior and properties of particles at this scale requires accepting the probabilistic nature of their location and speed.
• The Heisenberg Uncertainty Principle, used in the exercise, is foundational in describing the limitations in measuring quantum systems.

This field underpins technologies like semiconductors and lasers and is key in areas like quantum computing, pushing the boundaries of modern science and technology. Quantum mechanics not only alters the way we see submicroscopic events but also influences how we perceive reality at a fundamental level.