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, \(\mathrm{em}=9.1 \mathrm{x}\) \(\left.10^{-31} \mathrm{~kg}\right):\) (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 of the electron can be located with a certainty of about 1.92 4 4 4 410^{-3} m (answer b).

Step 1: Understanding the Problem

First, recognize that this problem involves Heisenberg's Uncertainty Principle, which relates the uncertainty in position (x) and the uncertainty in momentum (p). The principle is expressed as 1x 1p 4 = h/(44).

Step 2: Calculate the Uncertainty in Velocity

The electron's speed is given as 600 m/s, with an accuracy of 0.005%. Calculate the uncertainty in speed (v) using the formula v = (0.005/100) 4 4 * 600 = 0.03 m/s.

Step 3: Determine the Uncertainty in Momentum

Momentum is given by the product of mass and velocity. Thus, the uncertainty in momentum, p, is p = em 4 4 4 4v. Substituting the values, we get p = 9.1 4 4 4 410^{-31} 4 4 4 4 4 4 0.03 = 2.73 4 4 4 44 4 410^{-32} 4 4 4 4 4 4kg 4 4 4 4 4 4 4 4 4 4 4 4m/s.

Step 4: Use Heisenberg's Uncertainty Principle

Now use the uncertainty principle: 1x 44p 4 44 44 4 44 44= h/(444). Solve for x: x = h/(4444p).

Step 5: Solve for the Uncertainty in Position

Substituting the known values, x = 6.6 4 4 4 4 44 4 4 4 410^{-34}/(4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 1 4 4 4 442.73 4  4 410^{-32}) 4 4= 1.92 4 4 4 44 44 410^{-3} 4 4 4 4 4 4 4 4 4 4m.

Key Concepts

Uncertainty in Position

Heisenberg's Uncertainty Principle is central to understanding the behavior of particles at the quantum level. This principle tells us that there is a fundamental limit to how precisely we can simultaneously know a particle's position and momentum. This is expressed with the formula: \[ \Delta x \Delta p \geq \frac{h}{4\pi} \] where \( \Delta x \) is the uncertainty in position, \( \Delta p \) is the uncertainty in momentum, and \( h \) is Planck's constant.
In the exercise, calculating the uncertainty in position involves using this principle. After finding the uncertainty in momentum, we can rearrange the formula to solve for \( \Delta x \): \[ \Delta x = \frac{h}{4\pi \Delta p} \] By substituting the known values, including Planck's constant \( 6.6 \times 10^{-34} \text{ kg m}^2/\text{s} \) and the calculated uncertainty in momentum, we find that the uncertainty in determining the electron's position is \( 1.92 \times 10^{-3} \text{ m} \). This means within this range, you cannot precisely determine where the electron is at any given moment.

Uncertainty in Momentum

Uncertainty in momentum is another critical component of the Heisenberg's Uncertainty Principle. Momentum (\( p \)) is calculated by multiplying an object's mass (\( m \)) by its velocity (\( v \)). Thus, the uncertainty in momentum, \( \Delta p \), can be found by: \[ \Delta p = m \times \Delta v \] where \( \Delta v \) is the uncertainty in velocity.
Given that the electron in the problem has a mass of \( 9.1 \times 10^{-31} \text{ kg} \) and an uncertainty in speed \( \Delta v = 0.03 \text{ m/s} \), the uncertainty in momentum becomes: \[ \Delta p = 9.1 \times 10^{-31} \times 0.03 = 2.73 \times 10^{-32} \text{ kg m/s} \] This calculation helps us understand how precisely the momentum of an electron can be determined when its velocity is known within a small range.

Electron Velocity

Electron velocity is a measure of how fast an electron is moving. In this exercise, the electron has a given speed of 600 m/s but there's a specified accuracy of 0.005%. This accuracy indicates a small range of uncertainty in the speed measurement.
To find the uncertainty in velocity (\( \Delta v \)), use: \[ \Delta v = \frac{0.005}{100} \times 600 \] This results in \( \Delta v = 0.03 \text{ m/s} \). Understanding how accurately we can measure an electron's speed is essential for practical applications in quantum mechanics. A small uncertainty in velocity significantly impacts the accuracy of position and momentum measurements, as shown by the Heisenberg's Uncertainty Principle.