Question

If the position of a chlorine ion in a membrane is measured to an accuracy of 1.00 μm, what is its minimum uncertainty in velocity, given its mass is 5.86×10^(-26) kg?

a) 1.07 x 10^(-3) m/s
b) 2.45 x 10^(-3) m/s
c) 3.78 x 10^(-3) m/s
d) 5.12 x 10^(-3) m/s

Answer 1

Using Heisenberg's Uncertainty Principle, the minimum uncertainty in velocity for a chlorine ion with a given uncertainty in position and mass is calculated to be 1.07 x 10^-3 m/s, which aligns with option (a).

Step-by-step explanation:

The question involves calculating the minimum uncertainty in velocity for a chlorine ion using Heisenberg's Uncertainty Principle. This principle states that there is a fundamental limit to the precision with which certain pairs of physical properties, such as position and momentum (or velocity, in this case), can be known simultaneously. The uncertainty in position (Δx) is given as 1.00 μm, and the mass (m) of the chlorine ion is provided as 5.86 x 10-26 kg.

To find the minimum uncertainty in velocity (Δv), we use the uncertainty principle formula:

Δx * m * Δv ≥ h / (4 * π)

where h is Planck's constant (h = 6.626 x 10-34 m2 kg / s). Substituting the known values:

(1.00 x 10-6 m) * (5.86 x 10-26 kg) * Δv ≥ 6.626 x 10-34 m2 kg / s / (4 * π)

Solving for Δv, we get:

Δv ≥ (6.626 x 10-34 m2 kg / s) / (4 * π * 1.00 x 10-6 m * 5.86 x 10-26 kg)

Δv ≥ 1.07 x 10-3 m/s

Therefore, the minimum uncertainty in velocity for the chlorine ion is 1.07 x 10-3 m/s, which corresponds to option (a).