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 uncertainty in velocity is approximately \(10^{-7} / 4 \pi\;\mathrm{cm}\, \mathrm{s}^{-1}\) (option d).

Step 1: Understanding the Problem

We need to find the uncertainty in the velocity of a particle given the size of the particle, its mass, and the uncertainty in its position. This is a problem involving the Heisenberg Uncertainty Principle.

Step 2: Apply the Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle states that \( \Delta x \cdot \Delta v \geq \frac{h}{4 \pi m} \), where \( \Delta x \) is the uncertainty in position, \( \Delta v \) is the uncertainty in velocity, \( h \) is the Planck constant \(6.626 \times 10^{-34} \, \mathrm{Js}\), and \( m \) is the mass of the particle.

Step 3: Calculate Uncertainty in Position

The size of the particle is 1 micron \(= 1 \times 10^{-4} \mathrm{cm}\), and the uncertainty is given as 0.1% of this value. \(\Delta x = 0.1\% \times 1 \times 10^{-4} \mathrm{cm} = 1 \times 10^{-7} \mathrm{cm}\).

Step 4: Substitute Values Into the Uncertainty Formula

Using the Heisenberg Uncertainty Principle, substitute the known values: \( m = 6 \times 10^{-13} \, \mathrm{gm} = 6 \times 10^{-16} \, \mathrm{kg}\), \( h = 6.626 \times 10^{-34} \, \mathrm{Js}\), and \( \Delta x = 1 \times 10^{-7} \mathrm{cm} = 1 \times 10^{-9} \, \mathrm{m}\).

Step 5: Calculate the Uncertainty in Velocity

To find \( \Delta v \), rearrange to \( \Delta v = \frac{h}{4 \pi m \Delta x} \).Substitute the values: \\[ \Delta v = \frac{6.626 \times 10^{-34}}{4 \pi \times 6 \times 10^{-16} \times 1 \times 10^{-9}} \] \Performing the calculation gives: \( \Delta v \approx 10^{-7}/4\pi \mathrm{cm} \, \mathrm{s}^{-1} \).

Step 6: Identify the Correct Answer

Compare the calculated uncertainty in velocity to the answer choices: (a), (b), (c), and (d). The correct choice that matches our result is (d) \(10^{-7} / 4 \pi\).

Key Concepts

Uncertainty in Position

The concept of uncertainty in position is a significant part of the Heisenberg Uncertainty Principle. This principle tells us we cannot measure both the exact position and the exact momentum of a particle simultaneously.

In practical terms, measuring the position of a particle to a certain precision introduces uncertainty in its momentum. In the given exercise, the size of the particle is given as 1 micron, which is equal to 1 x 10^-4 cm. The uncertainty in the position of this particle is defined as 0.1% of its size.

By doing the math, the uncertainty in position, denoted as \( \Delta x \), is calculated:

• \( \Delta x = 0.1\% \times 1 \times 10^{-4} \text{ cm} \)
• \( \Delta x = 1 \times 10^{-7} \text{ cm} \)

This limitation on measurement precision is a fundamental concept when dealing with the quantum scale. It is what makes the study of such tiny particles both challenging and fascinating.

Uncertainty in Velocity

In the context of quantum mechanics, uncertainty in velocity is linked to the uncertainty in position through the Heisenberg Uncertainty Principle. It states that any attempt to pin down a particle's position more precisely results in increased uncertainty in its velocity.

Mathematically, uncertainty in velocity \( \Delta v \) can be expressed through the inequality:

• \( \Delta x \cdot \Delta v \geq \frac{h}{4 \pi m} \)

Here:

• \( h \) is the Planck constant
• \( m \) is the mass of the particle
• \( \Delta x \) is the uncertainty in position

Using the values from the exercise, the uncertainty in velocity is calculated by rearranging the equation to solve for \( \Delta v \):

• \( \Delta v = \frac{h}{4 \pi m \Delta x} \)

By inputting:

• \( h = 6.626 \times 10^{-34} \, \text{Js} \)
• \( m = 6 \times 10^{-16} \, \text{kg} \)
• \( \Delta x = 1 \times 10^{-9} \, \text{m}\)

We find that the resulting uncertainty in velocity is approximately \( 10^{-7}/4\pi \, \text{cm/s} \). This reflects the intertwined nature of position and velocity at a quantum level.

Planck Constant

The Planck constant \( h \) plays a pivotal role in quantum mechanics and is critical to the Heisenberg Uncertainty Principle. Named after Max Planck, it's a fundamental physical constant that describes the quantization of energy.

The Planck constant comes into play whenever quantum scale behavior is described. It allows us to relate the energy of photons to their frequency through the formula:

• \( E = h \cdot f \)

where \( E \) is energy and \( f \) is frequency.

In the context of the Heisenberg Uncertainty Principle, the Planck constant acts as a limit to the precision of simultaneous measurements of two complementary variables, like position and momentum. It ensures that at a quantum level, there's a fundamental limit to the precision with which properties of particles can be known. Specifically, the principle tells us that:

• \( \Delta x \cdot \Delta v \geq \frac{h}{4 \pi m} \)

This relationship shows that the Planck constant is the determining factor that sets the scale for the uncertainties in measurements inherent in quantum systems. Its value, \( 6.626 \times 10^{-34} \, \text{Js} \), is extraordinarily small, underpinning the differences between classical and quantum systems.