Question

An atom of argon 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 argon. Write your answer as a percentage of the average speed, and round it to significant digits.

Answer 1

Here is the full question:

An atom of argon has a radius of 71 .pm and the average orbital speed of the electrons in it is about 3.9 × 10⁷ m/s. Calculate the least possible uncertainty in a measurement of the speed of an electron in an atom of argon. Write your answer as a percentage of the average speed, and round it to significant digits.

Answer:

Explanation:

From the above information:

The radius of an atom of argon = 71 .pm = 71 × 10⁻¹² m

The diameter of the atom of argon Δx =142 × 10⁻¹² m

According to Heisenberg's Uncertainty Principle,

$\Delta x. \Delta p_x \geq \dfrac{h}{2}$

$\Delta x. \Delta V \geq \dfrac{h}{4 \pi m_e}$

$\Delta V \geq \dfrac{h}{4 \pi m_e \Delta_x}$

$\Delta V \geq \dfrac{6.634 \times 10^{-34} \ J.s}{4 \pi \times 9.1 \times 10^{-3} \ kg \times 142 \times 10^{-12} \ m}$

$\Delta V \geq \dfrac{6.634 \times 10^{-34} \ J.s}{1.62382641 \times 10^{-11} \ kg.m}$

$\Delta V \geq 4.08 \times 10^5 \ m/s$

where; the average speed $V_{avg}$ = 3.9  10⁷ m/s

∴

The percentage of the average speed is expressed as a fraction of:

$\% = \dfrac{4.08 \times 10^5}{3.9 \times 10^7}\times 100$

$\% =1.0462$

$V_{avg } = 1.046 \times 10^0$

$\mathbf{V_{avg }\simeq 1.0 \%}$