Chemistry · Dual nature of matter, de Broglie's relationship, Heisenberg uncertainty principle
The uncertainty in position of an electron \) moving with a velocity accurate up
The uncertainty in position of an electron \( \left(\boldsymbol{m}=\mathbf{9 . 1} \times \mathbf{1 0}^{-\mathbf{2 8}} \mathbf{g m}\right) \) moving with a velocity \( 3 \times 10^{4} \mathrm{cm} / \mathrm{s} \) accurate upto \( 0.001 \% \) will be:
- A. \quad 3.84 \mathrm{cm} \)
- B. 1.92 \( \mathrm{cm} \)
- C. \( 7.68 \mathrm{cm} \)
- D. 5.76 cm
Step-by-step solution
Given: m = 9.1 × 10^{-28} g, v = 3 × 10^4 cm/s, Δv = 0.001% of v = 10^{-5} × 3×10^4 = 0.3 cm/s. Using Heisenberg's uncertainty principle: Δx · Δp ≥ h/(4π) with Δp = m Δv = (9.1×10^{-28}) × 0.3 = 2.73×10^{-28} g cm/s. Planck's constant h = 6.626×10^{-27} erg·s. Thus Δx ≥ h/(4π Δp) = (6.626×10^{-27}) / (4π × 2.73×10^{-28}) ≈ 1.92 cm.
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