Question

Alveoli are the tiny sacs of air in the lungs whose average diameter is 5x10-5m. Consider an oxygen molecule (5.3x10-26kg) trapped within a sac. Calculate the uncertainty in the velocity of oxygen molecules.

Answer 1

Uncertainty principle:

1. The uncertainty principle states that it is impossible to exactly determine the position and velocity of a quantum particle at the same time.

$$\begin{gathered} \Delta x\mathit{}\mathit{\times }\mathit{}\Delta p\ge \frac{h}{4\pi } \\ \Delta x\mathit{}\mathit{\times }\mathit{}\Delta mv\mathit{}\ge \frac{h}{\mathit{4}\pi }\because \Delta p\mathit{=}\Delta mv \\ \Delta x\mathit{}\mathit{\times }\mathit{}\Delta m\mathit{\times }\mathit{}\Delta v\mathit{}\ge \frac{h}{\mathit{4}\pi } \end{gathered}$$

Where $\Delta x$ signifies change in position, $\Delta p$signifies change in momentum, $\Delta v$ is change in velocity and $h$is the plank's constant.

Step 1: Given data

Uncertainty in the position $\Delta x=5\times {10}^{-5}m$

Step 2: Applying the formula

$\Delta x\times \Delta v\ge \frac{h}{4\pi m}$

Step 3: Solving the equation

$\Delta v\ge 6.626\times {10}^{-34}\times 4\times 3.14\times 5.3\times {10}^{-26}\times 5\times {10}^{-5}$

Final answer: The uncertainty in the velocity of oxygen molecules is $\Delta v\ge 1.99m/sec.$