Liquid helium, with a boiling point of 4.2 K, is used in ultralow-temperature experiments and also for cooling the superconducting magnets used in MRI imaging in medicine. Storing liquid helium so far below room temperature is a challenge because even a small 'heat leak' will boil the helium away. A standard helium dewar, shown in FIGURE P19.67, has an inner stainless-steel cylinder filled with liquid helium surrounded by an outer cylindrical shell filled with liquid nitrogen at –196°C. The space between is a vacuum. The small structural supports have very low thermal conductivity, so you can assume that radiation is the only heat transfer between the helium and its surroundings. Suppose the helium cylinder is 16 cm in diameter and 30 cm tall and that all walls have an emissivity of 0.25. The density of liquid helium is 125 kg/m³ and its heat of vaporization is 2.1×10⁴ J/kg. What is the mass of helium in the filled cylinder?

Verified step by step guidance

1. Step 1: Begin by calculating the volume of the helium cylinder. The cylinder has a diameter of 16 cm and a height of 30 cm. The formula for the volume of a cylinder is V = πr²h, where r is the radius and h is the height. Convert the diameter to radius (r = diameter/2) and ensure all units are in meters before substituting into the formula.
2. Step 2: Once the volume is calculated, use the density of liquid helium to find the mass. The relationship between mass, density, and volume is given by m = ρV, where m is the mass, ρ is the density, and V is the volume. Substitute the density of liquid helium (125 kg/m³) and the calculated volume into this formula.
3. Step 3: Ensure unit consistency throughout the calculation. The radius and height should be converted to meters (1 cm = 0.01 m) before calculating the volume. This ensures the volume is in cubic meters, which matches the units of density (kg/m³).
4. Step 4: Perform the multiplication to find the mass of helium in the cylinder. This step involves multiplying the calculated volume by the given density.
5. Step 5: The result from the previous step gives the mass of helium in kilograms. This is the final value for the mass of helium in the filled cylinder.

Key Concepts

Density and Volume

Density is defined as mass per unit volume, typically expressed in kg/m³. To find the mass of a substance, you can use the formula: mass = density × volume. In this context, the volume of the helium cylinder can be calculated using the formula for the volume of a cylinder, V = πr²h, where r is the radius and h is the height. Understanding how to manipulate these relationships is crucial for determining the mass of liquid helium in the cylinder.

Heat Transfer and Emissivity

Heat transfer can occur through conduction, convection, and radiation. In this scenario, radiation is the primary mode of heat transfer due to the vacuum between the helium and its surroundings. Emissivity is a measure of how effectively a surface emits thermal radiation, ranging from 0 to 1. An emissivity of 0.25 indicates that the walls of the cylinder emit only 25% of the thermal radiation of a perfect black body, which is important for understanding heat loss in the system.

Heat of Vaporization

The heat of vaporization is the amount of energy required to convert a unit mass of a substance from liquid to gas at its boiling point. For liquid helium, this value is 2.1×10^4 J/kg. This concept is essential when considering the energy required to maintain the liquid state of helium in the presence of heat leaks, as any heat absorbed will lead to vaporization, affecting the mass of helium available for experiments.