Final answer:
To find the temperature, pressure, and volume of helium gas with a rms speed of 1857 m/s and a mean free path equal to the diameter of a sphere, we convert the volume to m³ and use the kinetic theory of gases, involving the mean free path, rms speed, and ideal gas law. The actual calculations would require additional information like the number of moles of helium or specific mean free path conditions.
Step-by-step explanation:
To determine at what temperature, pressure, and volume a sample of helium gas must be when the root mean square (rms) speed is 1857 m/s and its mean free path is comparable to the diameter of a 151.2 cm³ sphere, we can use the kinetic theory of gases and formulas relating these variables.
First, we convert the volume of the sphere from cm³ to m³, which is 1.512 × 10⁻´ m³. The mean free path (λ) is the average distance a molecule travels between collisions, and it can be formulated in terms of diameter (d), so λ = d. The mean free path depends on the number of particles, temperature, and pressure, according to the equation:
λ = (kT)/(sqrt(2)πd²P)
where k is the Boltzmann constant, T is the temperature in Kelvin, P is the pressure in Pascals, and d is the diameter of the molecules.
Considering the sphere's diameter as the mean free path, we can set up equations involving the root mean square speed (Urms), which for helium gas is related to temperature by:
Urms = sqrt((3kT)/M)
where M is the molar mass of helium.
We can then use these formulas, along with the ideal gas law, to find the temperature, pressure, and volume conditions for the helium gas sample. However, the actual calculations are complex and would need more information such as the actual number of moles of helium or the conditions to meet a specific mean free path.