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Structure of Atom question

2009 · Shift 0 · Q24
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Structure of Atom question

2009 · Shift 0 · Q24

JEE MainChemistryStructure of AtomMCQ+4 / −1
In an atom, an electron is moving with a speed of 600 m/s with an accuracy of 0.005%. Certainity with which the position of the electron can be located is (h = 6.6 ×\times× 10-34 kg m2s-1, mass of electron, em = 9.1 ×\times× 10-31 kg)
  1. A
    5.10 ×\times× 10-3 m
  2. B
    1.92 ×\times× 10-3 m
  3. C
    3.84 ×\times× 10-3 m
  4. D
    1.52 ×\times× 10-4 m
View written solutionFree

Correct answer: B

  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,

Δp=m Δv\Delta p = m\,\Delta vΔp=mΔv

So,

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

Given speed of electron:

v=600 m/sv = 600\ \text{m/s}v=600 m/s

Accuracy = 0.005%0.005\%0.005%

Thus uncertainty in velocity is

Δv=0.005% of 600\Delta v = 0.005\% \text{ of } 600Δv=0.005% of 600 Δv=0.005100×600=0.03 m/s\Delta v = \frac{0.005}{100}\times 600 = 0.03\ \text{m/s}Δv=1000.005​×600=0.03 m/s
  1. Substitute values

Given:

h=6.6×10−34 kg m2s−1h = 6.6\times 10^{-34}\ \text{kg m}^2\text{s}^{-1}h=6.6×10−34 kg m2s−1 m=9.1×10−31 kgm = 9.1\times 10^{-31}\ \text{kg}m=9.1×10−31 kg

Therefore,

Δx≥6.6×10−344π×9.1×10−31×0.03\Delta x \ge \frac{6.6\times 10^{-34}}{4\pi \times 9.1\times 10^{-31} \times 0.03}Δx≥4π×9.1×10−31×0.036.6×10−34​

First calculate the denominator:

9.1×10−31×0.03=2.73×10−329.1\times 10^{-31}\times 0.03 = 2.73\times 10^{-32}9.1×10−31×0.03=2.73×10−32 4π×2.73×10−32≈12.56×2.73×10−324\pi \times 2.73\times 10^{-32} \approx 12.56\times 2.73\times 10^{-32}4π×2.73×10−32≈12.56×2.73×10−32 ≈34.29×10−32=3.429×10−31\approx 34.29\times 10^{-32} = 3.429\times 10^{-31}≈34.29×10−32=3.429×10−31

Now,

Δx≥6.6×10−343.429×10−31\Delta x \ge \frac{6.6\times 10^{-34}}{3.429\times 10^{-31}}Δx≥3.429×10−316.6×10−34​ Δx≈1.92×10−3 m\Delta x \approx 1.92\times 10^{-3}\ \text{m}Δx≈1.92×10−3 m
  1. Match with options
Δx=1.92×10−3 m\boxed{\Delta x = 1.92\times 10^{-3}\ \text{m}}Δx=1.92×10−3 m​

So the correct option is:

B. 1.92×10−3 m1.92 \times 10^{-3}\,\text{m}1.92×10−3m

  1. Comparison with stored correct answer

Stored correct answer: B

Derived answer: B

They agree.

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