Question

An atom of lead has a radius of and the average orbital speed of the electrons in it is about . Calculate the least possible uncertainty in a measurement of the speed of an electron in an atom of lead. Write your answer as a percentage of the average speed, and round it to significant digits.

Answer 1

The question is incomplete. Here is the complete question.

An atom of lead has a radius of 154 pm and the average orbitalspeed of the electron in it is about 1.8x$10^{8}$ m/s. Calculate the least possible uncertainty in a measurement of the speed of an electron in an atom of lead. Write your answer as a percentage of the average speed, and round it to significant 2 digits.

Answer: v% = 0.21 m/s

Explanation: To calculate the uncertainty, use Heisenberg's Uncertainty Principle, which states that:  ΔpΔx≥$\frac{h}{4\pi }$

where h is Planck's constant and it is equal to 6.626.$10^{-34}$m²kg/s.

Since p (momentum) is p = m.v:

mΔv.Δx ≥ $\frac{h}{4\pi }$

Δv = $\frac{h}{4\pi.x.m }$

Given that: r = x = 1.54.$10^{-10}$m and mass of an electron is m=9.1.$10^{-31}$kg

Δv = $\frac{6.626.10^{-34} }{4.3.14.1.54.10^{-10}.9.1.10^{-31}}$

Δv = 0.0376.$10^{7}$

As percentage of average speed:

Δv.$\frac{1}{v}$.100% = $\frac{0.0376.10^{7} }{1.8.10^{8} }$.10² = 0.021.10 = 0.21%

The least possible uncertainty in a speed of an electron is 0.21%.