A cryogenic storage container holds liquid helium, which boils at 4.20 K. Suppose a student painted the outer shell of the container black, turning it into a pseudo-blackbody, and that the shell has an effective area of 0.791 m2 and is at 2.95·102 K. What is the rate at which the volume of the liquid helium in the container decreases as a result of boiling off? The latent heat of vaporization of liquid helium is 20.9 kJ/kg. The density of liquid helium is 0.125 kg/L.
To find the rate at which the volume of the liquid helium decreases as a result of boiling off, we can use the Stefan-Boltzmann law to calculate the rate at which energy is radiated from the pseudo-blackbody shell, and then divide that by the latent heat of vaporization to find the rate of decrease in the volume of the liquid helium.
The Stefan-Boltzmann law relates the rate of energy radiated by a blackbody to its temperature and surface area. However, since the container is a pseudo-blackbody, we need to multiply the energy radiated by the Stefan-Boltzmann constant (σ) and the emissivity (ε) of the shell.
The rate at which energy is radiated by the shell is given by:
P = ε * σ * A * T^4
where P is the power of radiation, ε is the emissivity, σ is the Stefan-Boltzmann constant (approximately 5.67 × 10^-8 W/m^2·K^4), A is the effective area of the shell (0.791 m^2), and T is the temperature of the shell (2.95 × 10^2 K).
Substituting the given values into the equation:
P = ε * (5.67 × 10^-8 W/m^2·K^4) * (0.791 m^2) * (2.95 × 10^2 K)^4
Next, we need to calculate the total energy radiated by the shell over a certain period of time. To do this, we need to multiply the power of radiation (P) by the time period (Δt).
E = P * Δt
The change in volume of the liquid helium (ΔV) is related to the energy radiated by the shell (E) by the equation:
E = ΔV * Latent Heat of Vaporization
Rearranging the equation, we get:
ΔV = E / Latent Heat of Vaporization
Substituting the values of E and the Latent Heat of Vaporization (20.9 kJ/kg) into the equation, you can calculate the rate at which the volume of the liquid helium decreases.
Note: Be sure to convert the units appropriately to maintain consistency throughout the calculations.