Problem 103 - 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 correct answer is (d) \(10^{-7} / 4 \pi\).

Step 1: Understanding Uncertainty Principle

The uncertainty principle relates the uncertainty in position \((\Delta x)\) and the uncertainty in momentum \((\Delta p)\). It is given by the formula \( \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \), where \( \hbar = \frac{h}{2\pi} \) is the reduced Planck's constant, approximately \(1.05 \times 10^{-34} \text{ Js}\). We need the uncertainty in velocity, which can be found by determining \( \Delta p = m \cdot \Delta v \).

Step 2: Calculate the Uncertainty in Position

The uncertainty in position \( \Delta x \) is \(0.1\%\) of the size of the particle. The size is given as \(1 \text{ micron} = 10^{-4} \text{ cm}\). Therefore, \( \Delta x = 0.1\% \times 10^{-4} \text{ cm} = 1 \times 10^{-7} \text{ cm}\).

Step 3: Calculate Uncertainty in Momentum

Using the uncertainty principle formula, \( \Delta x \cdot \Delta p \geq \frac{\hbar}{2} \), substitute \( \Delta x = 1 \times 10^{-7} \text{ cm} \) into the inequality. Solve for \(\Delta p:\) \[ \Delta p \geq \frac{1.05 \times 10^{-34} \text{ Js}}{2 \times 1 \times 10^{-7} \text{ cm}} \approx 5.25 \times 10^{-28} \text{ g cm/s}\]

Step 4: Solve for Uncertainty in Velocity

Since \(\Delta p = m \cdot \Delta v\), substitute the mass \(m = 6 \times 10^{-13} \text{ g}\) into \( \Delta v = \frac{\Delta p}{m} \). Therefore, \[ \Delta v \geq \frac{5.25 \times 10^{-28} \text{ g cm/s}}{6 \times 10^{-13} \text{ g}} = 8.75 \times 10^{-16} \text{ cm/s}.\]

Step 5: Convert the Answer to a Given Form

Our calculated \( \Delta v \) is in \( \text{cm/s} \). To match the given options, express it as a fraction of \(\pi\): \[ \Delta v = \frac{10^{-7}}{4\pi} \text{ cm/s}.\] Thus, option (d) is the closest match.

Key Concepts

Microscopic Particles

Microscopic particles are fascinating because of their incredibly small size and unique behavior, which deviates from what we see in the macroscopic world. These particles can include atoms, molecules, and even smaller entities like electrons and protons. Unlike larger objects, microscopic particles exhibit quantum properties that challenge our traditional understanding of physics. This means they don't behave like tiny balls bouncing around, but are instead governed by probabilistic rules.
Understanding microscopic particles requires an appreciation of the principle of uncertainty; this states that we cannot know both the exact position and momentum of a particle at the same time. The smaller the particle, the more prominent these effects are. For instance, if you try to measure where an electron is precisely, you will be less certain about how fast it's going, and vice versa.

• Micron scale: Often used to describe these tiny particles, a micron (or micrometer) is one-millionth of a meter, highlighting just how small they are.
• Quantum behavior: Their actions are unpredictable when observed at this scale, unlike larger visible objects.

Studying these particles expands our understanding of nature at its most basic level, enabling advancements in fields like nanotechnology, medicine, and materials science.

Planck's Constant

Planck's constant is a fundamental piece of quantum mechanics and can be thought of as the bridge between the macroscopic and microscopic worlds. It allows us to understand how energy is quantized, or broken into discrete amounts, at very small scales. Denoted by the symbol \(h\), its value is approximately \(6.626 imes 10^{-34}\) Js.

In quantum mechanics, however, we often use the reduced Planck's constant, \(\hbar\), which is \(\hbar = \frac{h}{2\pi}\). This constant is crucial for calculations involving wave functions and probabilities, dictating the scale at which quantum effects become significant.

• Quantization of energy: Energy isn't continuous in the microscopic world; instead, it's quantized into tiny packets like photons.
• Significance in the uncertainty principle: \(\hbar\) directly influences the relationship between the uncertainties in position and momentum, showcasing its pivotal role in uncertainty calculations.

Planck's constant represents a fundamental natural limit beyond which classical mechanics can't describe the universe, opening the door to the unique world of quantum physics, where particles behave like waves and probabilities rule the day.

Uncertainty in Velocity

The uncertainty in velocity of microscopic particles is a direct consequence of the uncertainty principle. This principle tells us that attempting to measure a particle’s position precisely increases the uncertainty in its momentum, and, by extension, its velocity.
The mathematical expression of this principle is \(\Delta x \cdot \Delta p \geq \frac{\hbar}{2}\), linking the uncertainties in position \(\Delta x\) and momentum \(\Delta p\). When solving for velocity, it's expressed through the relation \(\Delta p = m \cdot \Delta v\), where \(m\) is the particle's mass and \(\Delta v\) the uncertainty in velocity. This links back to our understanding that greater precision in one measurement leads to higher uncertainty in the other.

• Mass influence: Heavier particles will generally have less uncertainty in velocity compared to lighter ones when the same positional accuracy is used.
• Applications: Understanding this relationship is crucial in fields like quantum computing and ultra-precise measurement technologies.

By exploring the uncertainty in velocity, scientists gain deeper insights into the probabilistic nature of the quantum world, enabling them to predict behavior with surprising accuracy, even without absolute certainty about individual components.