To find the smallest possible length of the box inside which a helium atom could be located with certainty, we can use the uncertainty principle from quantum mechanics. The principle states:

$\Delta x \Delta p \geq \frac{\hbar}{2}$

where:

• $\Delta x$ is the uncertainty in position,
• $\Delta p$ is the uncertainty in momentum,
• $\hbar$ is the reduced Planck's constant ($\approx 1.055 \times 10^{-34} \, \text{Js}$).

1. Calculate the momentum ($p$) of a helium atom: $p = m \cdot v$ where $m$ is the mass of the helium atom and $v$ is its speed.

   The molar mass of helium is approximately $4 \, \text{g/mol}$. The mass of one helium atom is: $m = \frac{4 \times 10^{-3} \, \text{kg}}{6.022 \times 10^{23} \, \text{atoms/mol}} \approx 6.646 \times 10^{-27} \, \text{kg}$

   The speed ($v$) is given as $787 \, \text{m/s}$.

   Thus, the momentum $p$ is: $p = (6.646 \times 10^{-27} \, \text{kg}) \cdot (787 \, \text{m/s}) \approx 5.24 \times 10^{-24} \, \text{kg m/s}$

2. Calculate the uncertainty in momentum ($\Delta p$): The speed is measured to within $0.10\%$: $\Delta v = 0.001 \times 787 \, \text{m/s} \approx 0.787 \, \text{m/s}$ Therefore, the uncertainty in momentum is: $\Delta p = m \cdot \Delta v = (6.646 \times 10^{-27} \, \text{kg}) \cdot (0.787 \, \text{m/s}) \approx 5.24 \times 10^{-27} \, \text{kg m/s}$

3. Apply the uncertainty principle: Rearranging the uncertainty principle gives: $\Delta x \geq \frac{\hbar}{2 \Delta p}$ Plugging in values: $\Delta x \geq \frac{1.055 \times 10^{-34} \, \text{Js}}{2 \cdot 5.24 \times 10^{-27} \, \text{kg m/s}} \approx 1.01 \times 10^{-8} \, \text{m} \approx 10.1 \, \text{nm}$

4. Convert $\Delta x$ to a multiple of $r_{He}$: The radius of a helium atom is given as $31 \, \text{pm} = 3.1 \times 10^{-11} \, \text{m}$. Therefore, we calculate: $\frac{\Delta x}{r_{He}} = \frac{10.1 \times 10^{-9} \, \text{m}}{3.1 \times 10^{-11} \, \text{m}} \approx 325.8$

   Rounding to 2 significant figures gives: $\Delta x \approx 330 \, r_{He}$

Thus, the smallest possible length of the box is:

$\boxed{330 \, r_{He}}$

Would you like any further details or have questions about the calculations? Here are some related questions you might consider:

1. How does the uncertainty principle apply to other particles?
2. What are the implications of this uncertainty for quantum mechanics?
3. How would the calculation change for a different gas?
4. What other methods are used to measure atomic sizes?
5. Can the uncertainty principle be observed directly in experiments?

Tip: Always ensure to keep track of units in physics problems to avoid errors in calculations!