Uncertainty principle using ΔpΔx ≥ ħ/2: 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) 0.12 m/s B) 1.2 m/s C) 12 m/s D) 0.012 m/s E) 0.0012 m/s

Question

Quizplus · Accepted Answer

We have the formula ΔpΔx ≥ ħ/2, where Δp is the uncertainty in momentum, Δx is the uncertainty in position, and ħ is the reduced Planck constant.

Since the molecule is roughly spherical, we can approximate its diameter as the uncertainty in position: Δx = 0.70 nm.

We can rearrange the formula to solve for the minimum uncertainty in momentum, which is equal to the minimum uncertainty in speed since momentum is mass times velocity:

Δp = ħ/2Δx

Δp = (1.055 × 10^-34 J ∙ s)/(2 × 0.70 × 10^-9 m)

Δp ≈ 7.53 × 10^-26 kg ∙ m/s

The minimum uncertainty in speed can be calculated using the formula:

Δv = Δp/m

where m is the mass of the molecule:

Δv = (7.53 × 10^-26 kg ∙ m/s) / (6.10 × 10^-25 kg)

Δv ≈ 0.123 m/s

Therefore, the minimum uncertainty in speed is approximately 0.12 m/s, or choice letter A.

Uncertainty Principle Using ΔPΔx ≥ ħ/2: a Molecule of Roughly