Liquid nitrogen has a density of 0.808 g/mL and boils at 77 K. Researchers often purchase liquid nitrogen in insulated 175 L tanks. The liquid vaporizes quickly to gaseous nitrogen (which has a density of 1.15 g/L at room temperature and atmospheric pressure) when the liquid is removed from the tank. Suppose that all 175 L of liquid nitrogen in a tank accidentally vaporized in a lab that measured 10.00 m * 10.00 m * 2.50 m. What maximum fraction of the air in the room could be displaced by the gaseous nitrogen?

Verified step by step guidance

1. Calculate the mass of liquid nitrogen using its density and volume: \( \text{mass} = \text{density} \times \text{volume} = 0.808 \, \text{g/mL} \times 175,000 \, \text{mL} \).
2. Convert the mass of liquid nitrogen to moles using the molar mass of nitrogen (N2), which is approximately 28.02 g/mol: \( \text{moles} = \frac{\text{mass}}{\text{molar mass}} \).
3. Determine the volume of gaseous nitrogen at room temperature and atmospheric pressure using the ideal gas law: \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is moles, \( R \) is the ideal gas constant, and \( T \) is temperature in Kelvin.
4. Calculate the volume of the room: \( \text{volume} = 10.00 \, \text{m} \times 10.00 \, \text{m} \times 2.50 \, \text{m} \).
5. Determine the fraction of the room's air displaced by dividing the volume of gaseous nitrogen by the volume of the room.

Key Concepts

Density and Volume Relationship

Density is defined as mass per unit volume, and it plays a crucial role in determining how much substance can occupy a given space. In this scenario, the density of liquid nitrogen (0.808 g/mL) allows us to calculate its mass when given a specific volume (175 L). Understanding this relationship is essential for converting the volume of liquid nitrogen to its gaseous form and determining how much space it will occupy when vaporized.

Gas Laws and Volume Displacement

Gas laws, particularly the ideal gas law, describe how gases behave under various conditions of temperature and pressure. When liquid nitrogen vaporizes, it expands significantly, and its volume can be calculated based on the density of gaseous nitrogen (1.15 g/L). This concept is vital for understanding how much gaseous nitrogen will be produced and how it can displace the air in the lab space.

Volume of a Rectangular Prism

The volume of a rectangular prism, such as the lab space in this scenario, is calculated by multiplying its length, width, and height. In this case, the lab measures 10.00 m x 10.00 m x 2.50 m, which provides the total volume of air that could potentially be displaced by the vaporized nitrogen. Knowing the volume of the room is essential for determining the maximum fraction of air that can be replaced by the gaseous nitrogen.