Problem 172 - A Complete Resource Book in Chemistry for JEE Main

Question

In an atom, an electron is moving with a speed of \(600 \mathrm{~m} / \mathrm{s}\) with an accuracy of \(0.005 \%\). Certainity with which the position of the electron can be located is \(\left(\mathrm{h}=6.6 \times 10^{-34} \mathrm{~kg} \mathrm{~m}^{2} \mathrm{~s}^{-1}\right.\), mass of electron, \(\mathrm{em}=9.1 \mathrm{x}\) \(\left.10^{-31} \mathrm{~kg}\right)\) [2009] (a) \(5.10 \times 10^{-3} \mathrm{~m}\) (b) \(1.92 \times 10^{-3} \mathrm{~m}\) (c) \(3.84 \times 10^{-3} \mathrm{~m}\) (d) \(1.52 \times 10^{-3} \mathrm{~m}\)

Step-by-Step Solution

Answer

The position uncertainty is \(1.92 \times 10^{-3} \text{ m}\), option (b).

Step 1: Understanding the Variables

We have the speed of the electron given as \(v = 600 \text{ m/s}\), with an uncertainty \( \Delta v = 0.005\% \) of its speed, which converts to a fractional uncertainty. Therefore, \(\Delta v = 0.00005 \times 600 = 0.03 \text{ m/s}\).

Step 2: Apply the Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle is expressed as \(\Delta x \cdot \Delta p \geq \frac{h}{4\pi}\). We need to find \(\Delta x\), which represents the uncertainty in the position. Here, \(\Delta p\) is the uncertainty in momentum, \(\Delta p = \Delta v \cdot m = 9.1 \times 10^{-31} \text{ kg} \cdot 0.03 \text{ m/s}\).

Step 3: Calculate Momentum Uncertainty

Substitute the mass of the electron and the speed uncertainty, \(\Delta p = 9.1 \times 10^{-31} \cdot 0.03 = 2.73 \times 10^{-32} \text{ kg m/s}\).

Step 4: Solve for Position Uncertainty

Insert the given values into the Heisenberg equation \(\Delta x \cdot 2.73 \times 10^{-32} \geq \frac{6.6 \times 10^{-34}}{4\pi}\). Simplifying gives \(\Delta x \geq \frac{6.6 \times 10^{-34}}{4\pi \times 2.73 \times 10^{-32}}\).

Step 5: Perform the Calculations

Calculate \(\Delta x\):\[\Delta x \geq \frac{6.6 \times 10^{-34}}{4\times 3.14159 \times 2.73 \times 10^{-32}} \approx 1.92 \times 10^{-3} \text{ m}\]

Step 6: Verification

This matches option (b), confirming that our calculations align with the possible choices provided.

Key Concepts

Electron Speed

An electron speed in an atom determines how fast the electron moves around the nucleus. In this exercise, the electron moves at a speed of 600 meters per second. Understanding this speed is important because:

• The electron's speed affects its momentum, as momentum is the product of mass and velocity.
• Any uncertainty in the speed will impact the electron's momentum uncertainty.
• Therefore, it's essential to accurately measure or estimate the speed for precise calculations.

Speed uncertainty usually arises from measurement inaccuracies or fluctuations within the system. In this problem, the speed uncertainty is given as 0.005% of the electron's speed. Calculating this gives us an uncertainty in speed (9vloweremoji) equal to 0.03 m/s.
The accurate knowledge of the electron's speed is crucial for further calculations, particularly when applying the Heisenberg Uncertainty Principle.

Position Uncertainty

Position uncertainty quantifies how precisely we can determine an object's location at a given time. According to the Heisenberg Uncertainty Principle, there is a fundamental limit to the precision with which certain pairs of physical properties, like position and momentum, can be known simultaneously.

• In simple terms, the more accurately we know an object's position, the less accurately we know its momentum, and vice-versa.
• Position uncertainty (9x) plays a critical role in quantum mechanics and affects calculations of observable measurements.

For our electron, position uncertainty is derived from the uncertainty in momentum. Using the relationship 9x 9p 7 5 44,9p is calculated first, which helps in calculating 9x. In this exercise, after performing the calculations with the Heisenberg equation, we find that the uncertainly in position must be at least 1.92 x 10^-3 m.

Momentum Uncertainty

Momentum uncertainty stems from the inherent limitations in measuring the position or speed of a particle with exact precision. Momentum (9p*) is a product of the particle's mass and velocity. Therefore, any small uncertainty in speed will result in uncertainty in momentum.

• 9p = m 9v, where 9v is the uncertainty in speed and m is the mass of the electron.
• In our problem, the electron's mass is 9.1 x 10^-31 kg, and its speed uncertainty is 0.03 m/s.

Substituting these values, we find the momentum uncertainty to be about 2.73 x 10^-32 kg m/s. This value feeds directly into the Heisenberg Uncertainty Principle equation, 9x 9p 7 5 44, allowing us to explore the limits of measurement precision in the microscopic world.