Twenty-five milliliters of liquid nitrogen (density = 0.807 g/mL) is poured into a cylindrical container with a radius of 10.0 cm and a length of 20.0 cm. The container initially contains only air at a pressure of 760.0 mmHg (atmospheric pressure) and a temperature of 298 K. If the liquid nitrogen completely vaporizes, what is the total force (in lb) on the interior of the container at 298 K?

Verified step by step guidance

Calculate the mass of liquid nitrogen using its volume and density: \( \text{mass} = \text{volume} \times \text{density} \).

Convert the mass of nitrogen to moles using its molar mass (28.02 g/mol for \( N_2 \)).

Use the ideal gas law \( PV = nRT \) to find the pressure exerted by the vaporized nitrogen. Here, \( n \) is the number of moles, \( R \) is the ideal gas constant (0.0821 L·atm/mol·K), and \( T \) is the temperature in Kelvin.

Calculate the total pressure inside the container by adding the initial atmospheric pressure to the pressure exerted by the nitrogen gas.

Determine the total force on the interior of the container using the formula \( F = P \times A \), where \( A \) is the surface area of the interior of the cylindrical container. Convert the force from Newtons to pounds (1 N = 0.224809 lb).

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. This law is essential for understanding how gases behave under varying conditions. In this scenario, it will help calculate the pressure exerted by the vaporized nitrogen in the container after it transitions from liquid to gas.

Density and Mass Calculation

Density is defined as mass per unit volume (density = mass/volume). Knowing the density of liquid nitrogen allows us to calculate its mass when given a specific volume. This mass is crucial for determining the number of moles of nitrogen gas produced upon vaporization, which will be used in the Ideal Gas Law.

Force and Pressure Relationship

Force exerted by a gas on the walls of a container is related to pressure and area through the equation F = P × A, where F is force, P is pressure, and A is the area of the container's interior. Understanding this relationship is vital for calculating the total force on the container's interior after the nitrogen has vaporized and exerted pressure.

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