Electron moving with a speed of 600 m/s with an accuracy of 0.005% | Chemca Solution

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Question AIEEE

In an atom, an electron is moving with a speed of 600 m/s with an accuracy of 0.005%. Certainty with which the position of the electron can be located is:

Given: \( h = 6.6 \times 10^{-34} \text{ kg m}^2 \text{s}^{-1} \), mass of electron \( (m_e) = 9.1 \times 10^{-31} \text{ kg} \)

Detailed Step-by-Step Solution

This problem directly applies Heisenberg's Uncertainty Principle, which states that it is impossible to determine simultaneously, the exact position and exact momentum (or velocity) of an electron.

Step 1: Calculate the uncertainty in velocity (\(\Delta v\))

We are given the velocity \( v = 600 \text{ m/s} \), and the accuracy is \( 0.005\% \). The uncertainty in velocity is:

\( \Delta v = \frac{0.005}{100} \times 600 \)

\( \Delta v = 0.00005 \times 600 \)

\( \Delta v = 0.03 \text{ m/s} \)

Step 2: Apply Heisenberg's Uncertainty Equation

The mathematical form of Heisenberg's Uncertainty Principle is:

\( \Delta x \cdot \Delta p \ge \frac{h}{4\pi} \)

Since momentum \( p = m \cdot v \), the uncertainty in momentum \( \Delta p = m \cdot \Delta v \). Substituting this into the equation:

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

Rearranging the formula to solve for the uncertainty in position (\(\Delta x\)):

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

Step 3: Substitute values and calculate

Now, plug the given values into our rearranged equation:

\( \Delta x = \frac{6.6 \times 10^{-34}}{4 \times 3.14 \times (9.1 \times 10^{-31}) \times 0.03} \)

First, calculate the denominator:

\( 4 \times 3.14 \times 9.1 \times 0.03 = 3.42864 \)

Now, substitute this back into the equation:

\( \Delta x = \frac{6.6 \times 10^{-34}}{3.42864 \times 10^{-31}} \)

\( \Delta x = \frac{6.6}{3.42864} \times 10^{-3} \)

\( \Delta x \approx 1.92 \times 10^{-3} \text{ m} \)

Conclusion: The certainty with which the position of the electron can be located is \( 1.92 \times 10^{-3} \text{ m} \). Therefore, the correct option is (c).

Conquer Competitive Chemistry

Heisenberg's Uncertainty Principle is a cornerstone of quantum mechanics and is frequently tested in engineering (JEE Main/AIEEE) and medical (NEET) entrance exams. The key to solving these numericals quickly is mastering the manipulation of scientific notation exponents.

If you want to review the theoretical aspects of orbitals, quantum mechanics, and probability density, we highly recommend reading our detailed guide on the Structure of Atom Class 11 Chemistry.

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