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Structure of Atom question

2021 · 25 Jul · Shift 2 · Q17
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Structure of Atom question

2021 · 25 Jul · Shift 2 · Q17

JEE MainChemistryStructure of AtomNumerical+4 / −1
An accelerated electron has a speed of 5 ×\times× 106 ms −-− 1 with an uncertainty of 0.02%. The uncertainty in finding its location while in motion is x ×\times× 10 −-− 9 m. The value of x is ‾\underline{\hspace{2cm}}​. (Nearest integer) [Use mass of electron = 9.1 ×\times× 10 −-− 31 kg, h =6.63 ×\times× 10 −-− 34 Js, π\piπ = 3.14]
Numerical answer
View written solutionFree

Correct answer: 58

  1. Use Heisenberg uncertainty principle

For position and momentum,

Δx Δp≥h4π\Delta x\, \Delta p \ge \frac{h}{4\pi}ΔxΔp≥4πh​

Since momentum p=mvp = mvp=mv, we have

Δp=m Δv\Delta p = m\,\Delta vΔp=mΔv

So,

Δx=h4πmΔv\Delta x = \frac{h}{4\pi m \Delta v}Δx=4πmΔvh​
  1. Find uncertainty in velocity

Given speed of electron,

v=5×106 m s−1v = 5\times 10^6\ \text{m s}^{-1}v=5×106 m s−1

Uncertainty is 0.02%0.02\%0.02% of the speed.

0.02%=0.02100=2×10−40.02\% = \frac{0.02}{100} = 2\times 10^{-4}0.02%=1000.02​=2×10−4

Hence,

Δv=5×106×2×10−4\Delta v = 5\times 10^6 \times 2\times 10^{-4}Δv=5×106×2×10−4 Δv=103 m s−1\Delta v = 10^3\ \text{m s}^{-1}Δv=103 m s−1
  1. Substitute in the uncertainty formula
Δx=6.63×10−344×3.14×9.1×10−31×103\Delta x = \frac{6.63\times 10^{-34}}{4\times 3.14\times 9.1\times 10^{-31}\times 10^3}Δx=4×3.14×9.1×10−31×1036.63×10−34​

First compute the denominator coefficient:

4×3.14=12.564\times 3.14 = 12.564×3.14=12.56 12.56×9.1=114.29612.56\times 9.1 = 114.29612.56×9.1=114.296

So,

Δx=6.63×10−34114.296×10−28\Delta x = \frac{6.63\times 10^{-34}}{114.296\times 10^{-28}}Δx=114.296×10−286.63×10−34​ Δx=6.63114.296×10−6\Delta x = \frac{6.63}{114.296}\times 10^{-6}Δx=114.2966.63​×10−6 Δx≈0.058×10−6\Delta x \approx 0.058\times 10^{-6}Δx≈0.058×10−6 Δx≈5.8×10−8 m\Delta x \approx 5.8\times 10^{-8}\ \text{m}Δx≈5.8×10−8 m
  1. Express in the required form

We need

Δx=x×10−9 m\Delta x = x\times 10^{-9}\ \text{m}Δx=x×10−9 m

Since

5.8×10−8=58×10−95.8\times 10^{-8} = 58\times 10^{-9}5.8×10−8=58×10−9

therefore,

x=58x = 58x=58
  1. Comparison with stored answer

Derived answer = 585858

Stored correct answer = 585858

They match.

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