Question

A cryogenic storage container holds liquid helium, which boils at \(4.22 \mathrm{~K}\). The container's outer shell has an effective area of \(0.500 \mathrm{~m}^{2}\) and is at \(3.00 \cdot 10^{2} \mathrm{~K}\). Suppose a student painted the outer shell of the container black, turning it into a pseudo- blackbody. a) Determine the rate of heat loss due to radiation. b) 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 \(20.9 \mathrm{~kJ} / \mathrm{kg} .\) The density of liquid helium is \(0.125 \mathrm{~kg} / \mathrm{L}\).

Short answer

Answer: The rate of heat loss due to radiation is approximately \(2025.11 \mathrm{~W}\). 2. What is the rate at which the volume of liquid helium in the container decreases due to boiling off? Answer: The rate at which the volume of the liquid helium in the container decreases as a result of boiling off is approximately \(7.750 \times 10^{-4}~L/s\).

Step-by-step solution

Determine the heat loss due to radiation

Given the effective area of the outer shell - \(A\), and the temperature - \(T\), we can use the Stefan-Boltzmann Law. The formula for heat loss due to radiation is - \(Q = σ A T^4\), where \(σ\) is the Stefan-Boltzmann constant (\(5.67 * 10^{-8} W/m^2K^4\)). So, when the temperature is given as \(3.00 * 10^2 K\), and the area of the outer shell is \(0.500 m^2\), we can find the heat loss due to radiation. \(Q = (5.67 * 10^{-8}) * (0.500) * (3.00 * 10^2)^4\)

Calculate the rate of heat loss due to radiation

Now, we calculate the rate of heat loss due to radiation by plugging the values into the formula. \(Q = (5.67 * 10^{-8}) * (0.500) * (3.00 * 10^2)^4\) \(Q \approx 2025.11~W\) The rate of heat loss due to radiation is approximately \(2025.11 \mathrm{~W}\).

Find the mass loss rate of liquid helium due to boiling off

Given the latent heat of vaporization (\(L_h\)) of liquid helium is \(20.9 \mathrm{~kJ} / \mathrm{kg}\), we can find the mass loss rate by dividing the heat loss rate (\(Q\)) by the latent heat of vaporization. Mass loss rate = \(\frac{Q}{L_h}\) Convert the latent heat of vaporization to \(W\), \(L_{h(converted)} = 20.9 \times 10^3~W/kg\) Mass loss rate = \(\frac{2025.11}{20.9 \times 10^3}\) Mass loss rate \(\approx 9.688 \times 10^{-5}~kg/s\)

Determine the rate of decrease in volume of liquid helium

We are given the density (\(ρ\)) of liquid helium as \(0.125 \mathrm{~kg} / \mathrm{L}\). We can find the volume loss rate by dividing the mass loss rate by the density. Volume loss rate = \(\frac{mass~loss~rate}{ρ}\) Volume loss rate = \(\frac{9.688 \times 10^{-5}}{0.125}\) Volume loss rate \(\approx 7.750 \times 10^{-4}~L/s\) The rate at which the volume of the liquid helium in the container decreases as a result of boiling off is approximately \(7.750 \times 10^{-4}~L/s\).

Key concepts

Stefan-Boltzmann Law

The Stefan-Boltzmann Law plays a critical role in understanding how objects emit radiation, a fundamental aspect of thermal physics. It states that the power emitted per unit area of a black body is proportional to the fourth power of the temperature. Mathematically, it's given by the equation \[\begin{equation}P = \.sigma A T^4\end{equation}\] where \( P \) is the power radiated per unit area, \( \.sigma \) is the Stefan-Boltzmann constant (approximately \(5.67 \times 10^{-8} W/m^2 K^4\)), \( A \) is the area of the emitting surface, and \( T \) is the absolute temperature in kelvins.In practical terms, this law indicates that objects at a higher temperature emit more radiation compared to those at a lower temperature. This principle is essential for a variety of applications, from estimating the Earth's heat balance to designing energy-efficient furnaces.

Latent Heat of Vaporization

Latent heat of vaporization is the heat required to convert a unit mass of a liquid into a vapor without a temperature change. This is energetically demanding because it breaks intermolecular forces without raising the kinetic energy of the molecules - they don't get 'hotter', but they do gain energy to move into a gaseous state. For water, this is what's happening when boiling: at 100 degrees Celsius, additional heat cooks off water as steam instead of making the liquid water hotter.The latent heat of vaporization is different for every substance, reflecting the unique intermolecular forces of each. In the context of cryogenic storage and liquids like helium, this concept becomes critical. As the liquid inside a cryogenic container absorbs heat, it reaches a point where it starts vaporizing, a process that carries away a significant amount of heat, affecting the efficiency of the storage.

Cryogenic Storage

Cryogenic storage refers to the preservation of materials at extremely low temperatures, typically below \-150\ufffdC (\(123.15 K\)). The most common examples involve liquefied gases, such as nitrogen, oxygen, and helium, which are stored as liquids because compressing them into a liquid state reduces the storage volume massively compared to their gaseous form.Successful cryogenic storage demands efficient insulation to minimize heat transfer because even a small amount of heat leakage can lead to the vaporization of the stored liquid, known as 'boil-off'. Advanced materials and vacuum technology are often used to achieve this. Also, considering the concept of latent heat of vaporization, containers are designed to manage the gas that boils off, as it can be used to cool the remaining liquid, thereby increasing efficiency.

Thermal Physics

Thermal physics involves the study of temperature, heat, and their effects on matter. It encompasses concepts from kinetic theory, thermodynamics, and statistical mechanics. Getting to grips with thermal physics involves understanding how energy in the form of heat is transferred and transformed.Key processes of this energy exchange include conduction, convection, radiation, and phase transitions (like vaporization). Understanding these processes is essential not only in academic contexts but also in real-world applications from refrigerator design to environmental science. For instance, the exercise involving a cryogenic storage container requires knowing how heat loss occurs through radiation (according to the Stefan-Boltzmann Law) and how the latent heat of vaporization impacts the storage efficiency of materials, like liquid helium.