Final answer:
To find the uncertainty in the position of a helium atom, we calculate the uncertainty in momentum and apply the Heisenberg Uncertainty Principle. The calculated value for the uncertainty in position does not match the given options, suggesting there might be a miscalculation or problem setup issue.
Step-by-step explanation:
To determine the uncertainty in the position of a helium atom given the uncertainty in its velocity, we can use the Heisenberg Uncertainty Principle. The principle states that the uncertainty in position (Δx) and the uncertainty in momentum (Δp) are related by ΔxΔp ≥ ℇ/2, where ℇ is the reduced Planck constant (ℇ ≈ 1.055 × 10-34 kg m2/s).
First, let's calculate the uncertainty in momentum. The uncertainty in velocity (Δ8v) is given as 1.0% of the average velocity. Therefore, Δ8v = 0.01 × 1.36 × 103 m/s = 13.6 m/s. Since momentum (p) is the product of mass (m) and velocity (v), the uncertainty in momentum (Δ8p) can be represented as Δ8p = mΔ8v. We use the mass of the helium atom in kilograms (m = 4.0026 u × 1.66 × 10-27 kg/u).
Δ8p = (4.0026 × 1.66 × 10-27 kg/u) × 13.6 m/s = 9.235 × 10-26 kg m/s. Next, we apply the Heisenberg formula to solve for Δ8x: Δ8x ≥ ℇ / (2Δ8p).
Δ8x ≥ (1.055 × 10-34 kg m2/s) / (2 × 9.235 × 10-26 kg m/s) = 5.7 × 10-10 m.
None of the options (A, B, C, D) matches exactly with our calculated value. It is possible that there might be a miscalculation or a misunderstanding in the problem setup. Please cross-verify the values and calculations.