The diameter of a dust particle of mass g, is 2 Å. The speed of this dust particle is measured with — Atomic Structure Chemistry Question
Question
The diameter of a dust particle of mass $10^{-3}$ g, is 2 Å. The speed of this dust particle is measured with the uncertainty of $\frac{3.313}{\pi} \times 10^{-3}$ m/s. The minimum uncertainty in measuring the position of the duct particle (in order of $10^{-26}$ m) is
💡 Solution & Explanation
Mass $m = 10^{-3} \text{ g} = 10^{-6}$ kg. Heisenberg's uncertainty principle: $\Delta x \cdot \Delta p \ge \frac{h}{4\pi}$. Thus, $\Delta x = \frac{h}{4\pi m \Delta v}$. Substituting the given values: $\Delta x = \frac{6.626 \times 10^{-34}}{4\pi \times 10^{-6} \times \left(\frac{3.313}{\pi} \times 10^{-3}\right)} = \frac{6.626 \times 10^{-34}}{4 \times 3.313 \times 10^{-9}} = \frac{6.626 \times 10^{-34}}{13.252 \times 10^{-9}} = 0.5 \times 10^{-25} = 5 \times 10^{-26}$ m. The value is 5. Therefore, correct answer is 5.