Problem 155 - A Complete Resource Book in Chemistry for JEE Main

Question

Uncertainty in position of a particle of \(25 \mathrm{~g}\) in space is \(10^{-5} \mathrm{~m} .\) Hence uncertainty in velocity \(\left(\mathrm{ms}^{-1}\right)\) is (Planck constant \(\mathrm{h}=6.6 \times 10^{-34} \mathrm{Js}\) ) \(\quad\) [2002] (a) \(2.1 \times 10^{-28}\) (b) \(2.1 \times 10^{-34}\) (c) \(0.5 \times 10^{-34}\) (d) \(5.0 \times 10^{-24}\)

Step-by-Step Solution

Answer

The uncertainty in velocity is approximately \(8.4 \times 10^{-28} \text{ m/s}\), which doesn't match any of the provided answer options.

Step 1: Identify the Known Values

In this exercise, we know that the mass of the particle \(m\) is \(25 \text{ g} = 0.025 \text{ kg}\) and the uncertainty in position \(\Delta x\) is \(10^{-5} \text{ m}\). The Planck constant \(h\) is given as \(6.6 \times 10^{-34} \text{ Js}\).

Step 2: Apply Heisenberg's Uncertainty Principle

Heisenberg's uncertainty principle is given by the equation: \(\Delta x \cdot \Delta p \geq \frac{h}{4\pi}\), where \(\Delta p\) is the uncertainty in momentum. Since \(\Delta p = m \cdot \Delta v\), we can rewrite the equation as: \(\Delta x \cdot m \cdot \Delta v \geq \frac{h}{4\pi}\).

Step 3: Rearrange the Equation for Velocity Uncertainty

We need to find \(\Delta v\), the uncertainty in velocity. Rearranging the equation, we get: \(\Delta v \geq \frac{h}{4\pi m \cdot \Delta x}\). Substitute the known values into the equation.

Step 4: Substitute Values and Calculate

Substituting the values we have: \(\Delta v \geq \frac{6.6 \times 10^{-34}}{4\pi \times 0.025 \times 10^{-5}}\). Calculate the value of \(\Delta v\).

Step 5: Simplify and Solve

Calculate \(4\pi \times 0.025 \times 10^{-5} = 3.14159 \times 0.025 \times 10^{-5} \approx 7.85 \times 10^{-7}\). Then, calculate \(\Delta v \geq \frac{6.6 \times 10^{-34}}{7.85 \times 10^{-7}}\). This gives \(\Delta v \approx 8.4 \times 10^{-28} \text{ m/s}\).

Step 6: Compare with Given Options

The calculated uncertainty in velocity \(8.4 \times 10^{-28} \text{ m/s}\) matches with one of the answer choices. Compare this value with the options given in the exercise.

Key Concepts

Quantum Mechanics

Quantum mechanics is a fascinating branch of physics that seeks to explain the behavior of particles at the smallest scales, like atoms and subatomic particles. It departs from classical mechanics, which deals with visible and macroscopic phenomena, and instead handles phenomena that are inherently probabilistic.
One of the pillars of quantum mechanics is the concept that particles don't have precise positions or velocities until we measure them. This idea is rooted in Heisenberg's Uncertainty Principle, which tells us we cannot know both the exact position and momentum of a particle at the same time. This principle leads to many interesting and non-intuitive phenomena that are key to understanding the quantum world.
In summary, quantum mechanics challenges our classical intuitions by introducing concepts like wave-particle duality and the probabilistic nature of the microscopic world, which have deep implications for fields like quantum computing and particle physics.

Momentum Uncertainty

Momentum uncertainty is a critical aspect of Heisenberg's Uncertainty Principle. In essence, it states that there is a limit to how precisely we can know a particle's momentum and position simultaneously. When we try to measure a particle's position more accurately, we increase the uncertainty in its momentum, and vice versa.
Momentum (\(p\)) is the product of a particle's mass (\(m\)) and its velocity (\(v\)). Therefore, the uncertainty in momentum (\(\Delta p\)) can be expressed in relation to the uncertainties in velocity (\(\Delta v\)) and mass. Since mass is typically a fixed quantity, the uncertainty in momentum primarily arises from the velocity's uncertainty.
In the context of our problem, this uncertainty translates into a limitation, set by Planck's constant, that prevents the accurate measurement of both a particle's position and momentum beyond a certain threshold, making the Heisenberg's principle a fundamental concept in quantum mechanics.

Velocity Uncertainty

Velocity uncertainty refers to the inability to precisely know the speed of a particle due to Heisenberg's Uncertainty Principle. In our exercise, we utilized this principle to calculate the uncertainty in velocity (\(\Delta v\)) of a particle given its uncertainty in position (\(\Delta x\)).
By rearranging the equation derived from Heisenberg's principle \(\Delta x \cdot m \cdot \Delta v \geq \frac{h}{4\pi}\), we found a method to estimate this uncertainty. The smaller the uncertainty in position, the larger the uncertainty in velocity, demonstrating the inherent trade-off between measuring these quantities.
Understanding this velocity uncertainty is crucial in fields that rely on precise measurements at quantum scales, like electron microscopy and other areas where predicting a particle's behavior is essential. It embodies the idea that at a microscopic level, the universe is less certain and predictable than once thought, influencing how scientists think about observation and measurement in the quantum realm.