An accelerated electron has a speed of \( 5 \times 10^6 \text{ m s}^{-1} \) with an uncertainty of \( 0.02\% \). The uncertainty in finding its location while in motion is \( x \times 10^{-9} \text{ m} \). The value of \( x \) is _________. (Nearest integer)
Use the following constants:
- Mass of electron (\(m_e\)) = \( 9.1 \times 10^{-31} \text{ kg} \)
- Planck's constant (\(h\)) = \( 6.63 \times 10^{-34} \text{ J s} \)
- \( \pi \) = \( 3.14 \)
Detailed Step-by-Step Solution
This problem directly applies Heisenberg's Uncertainty Principle.
Step 1: Calculate Uncertainty in Velocity (\(\Delta v\))
We are given the electron's speed \( v = 5 \times 10^6 \text{ m/s} \), and its uncertainty is \( 0.02\% \).
\( \Delta v = \frac{0.02}{100} \times (5 \times 10^6) \)
\( \Delta v = 0.0002 \times 5 \times 10^6 \)
\( \Delta v = 1000 \text{ m/s} = 10^3 \text{ m/s} \)
Step 2: Apply Heisenberg's Formula
According to the Uncertainty Principle:
Substituting \( \Delta p = m_e \cdot \Delta v \), we get:
\( \Delta x = \frac{h}{4\pi \cdot m_e \cdot \Delta v} \)
Step 3: Substitute the Values and Calculate
Let's simplify the denominator first:
\( \text{Denominator} = 12.56 \times 9.1 \times 10^{-28} \)
\( \text{Denominator} \approx 114.296 \times 10^{-28} \)
Now, divide the numerator by the denominator:
\( \Delta x = \left(\frac{6.63}{114.296}\right) \times 10^{-6} \)
\( \Delta x \approx 0.058007 \times 10^{-6} \text{ m} \)
Convert this into the standard format \( x \times 10^{-9} \text{ m} \):
Conclusion: We found that the uncertainty in location is \( 58.007 \times 10^{-9} \text{ m} \). When comparing this to \( x \times 10^{-9} \text{ m} \), the nearest integer value for \( x \) is 58.
Acing Uncertainty Principle Numericals
This is a high-yield JEE Main concept. The most common mistake students make is plugging the total velocity (\(v\)) directly into the equation instead of the uncertainty in velocity (\(\Delta v\)). Always make sure you calculate the percentage of uncertainty first before applying the Heisenberg formula.
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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