Uncertainty principle using ΔxΔp ≥ ħ: A molecule of roughly spherical shape has a mass of 6.10 × 10^-25 kg and a diameter of 0.70 nm. The uncertainty in the measured position of the molecule is equal to the molecular diameter. What is the minimum uncertainty in the speed of this molecule? (ħ = 1.055 × 10^-34 J ∙ s) A) 0.25 m/s B) 2.5 m/s C) 25 m/s D) 0.025 m/s E) 0.0025 m/s

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Quizplus · Accepted Answer

The uncertainty in the speed of the molecule is given by the Heisenberg uncertainty principle:$$\Delta x \Delta p \geq \frac{h}{4\pi}$$where $\Delta x$ is the uncertainty in the position of the molecule, $\Delta p$ is the uncertainty in the momentum of the molecule, and $h$ is the Planck constant.In this problem, the uncertainty in the position of the molecule is equal to the molecular diameter, which is 0.70 nm. The mass of the molecule is 6.10 × 10^-25 kg.The uncertainty in the momentum of the molecule is:$$\Delta p = m \Delta v$$where $m$ is the mass of the molecule and $\Delta v$ is the uncertainty in the speed of the molecule.Substituting the given values into the Heisenberg uncertainty principle, we get:$$(0.70 imes 10^{-9} ext{ m}) \Delta p \geq \frac{1.055 imes 10^{-34} ext{ J s}}{4\pi}$$$$\Delta p \geq 4.76 imes 10^{-26} ext{ kg m/s}$$The uncertainty in the speed of the molecule is:$$\Delta v = \frac{\Delta p}{m}$$Substituting the given values into the equation, we get:$$\Delta v \geq \frac{4.76 imes 10^{-26} ext{ kg m/s}}{6.10 imes 10^{-25} ext{ kg}}$$$$\Delta v \geq 0.078 ext{ m/s}$$The minimum uncertainty in the speed of the molecule is 0.078 m/s, which is closest to choice A, 0.25 m/s.

Uncertainty Principle Using ΔXΔp ≥ ħ: a Molecule of Roughly