To determine the rate at which helium is leaking from a thermally insulated vessel into a vacuum, we can apply the principles of fluid dynamics, specifically using the concept of effusion. When a gas escapes through a small hole into a vacuum, the speed of the gas molecules can be described by the root mean square speed, which is derived from the kinetic theory of gases.
Understanding the Kinetic Theory of Gases
The kinetic theory provides a framework for understanding the behavior of gases at the molecular level. According to this theory, the average kinetic energy of gas molecules is directly proportional to the temperature of the gas. The formula for the root mean square speed (v_rms) of gas molecules is given by:
v_rms = √(3kT/m)
Where:
- v_rms = root mean square speed of the gas molecules
- k = Boltzmann's constant (1.38 x 10^-23 J/K)
- T = absolute temperature in Kelvin
- m = mass of a single molecule of the gas in kilograms
Calculating the Mass of a Helium Molecule
Helium (He) has an atomic mass of approximately 4 g/mol. To convert this to kilograms, we use:
m = 4 g/mol × (1 kg/1000 g) × (1 mol/6.022 x 10^23 molecules) ≈ 6.64 x 10^-27 kg
Applying the Formula
Now, substituting the values into the root mean square speed formula:
T = 1000 K
m ≈ 6.64 x 10^-27 kg
We can now calculate:
v_rms = √(3 × (1.38 x 10^-23 J/K) × (1000 K) / (6.64 x 10^-27 kg))
Calculating the numerator:
3 × (1.38 x 10^-23) × (1000) = 4.14 x 10^-20 J
Now, dividing by the mass:
4.14 x 10^-20 J / (6.64 x 10^-27 kg) ≈ 6.23 x 10^6 m²/s²
Taking the square root gives:
v_rms ≈ √(6.23 x 10^6) ≈ 2495 m/s
Final Considerations
The calculated speed of approximately 2495 m/s represents the average speed of helium molecules escaping through the hole. This speed is crucial for determining the rate of effusion, which can be further analyzed using Graham's law if comparing with other gases or under different conditions.
In summary, the rate at which helium is leaking from the vessel can be approximated by the root mean square speed of the gas molecules, which in this case is about 2495 m/s, not the 2236 m/s you mentioned. This discrepancy could arise from variations in assumptions or calculations, so it's always good to double-check each step for accuracy.
