Problem 10 - University Physics with Modern Physics

Question

An empty cylindrical canister 1.50 \(\mathrm{m}\) long and 90.0 \(\mathrm{cm}\) in diameter is to be filled with pure oxygen at \(22.0^{\circ} \mathrm{C}\) to store in a space station. To hold as much gas as possible, the absolute pressure of the oxygen will be 21.0 atm. The molar mass of oxygen is 32.0 \(\mathrm{g} / \mathrm{mol} .\) (a) How many moles of oxygen does this canister hold? (b) For someone lifting this canister, by how many kilograms does this gas increase the mass to be lifted?

Step-by-Step Solution

Answer

The canister holds approximately 831.58 moles of oxygen, adding 26.61 kg to the mass.

Step 1: Convert Measurements to SI units

First, convert the diameter from centimeters to meters because we are using SI units in our calculations. The diameter is 90.0 cm, which is 0.90 m. Also, note that the temperature should be in Kelvin for gas calculations, which means converting 22.0°C to 295.15 K. (22.0 + 273.15).

Step 2: Calculate Volume of the Cylinder

The volume \( V \) of a cylinder is given by \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height of the cylinder. The radius is half of the diameter: \( r = 0.45 \) m. The height is \( 1.50 \) m. Therefore,\[ V = \pi \times (0.45)^2 \times 1.50 \approx 0.954 \text{ m}^3 \].

Step 3: Use the Ideal Gas Law to Find Moles

The ideal gas law is \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the universal gas constant (8.314 J/mol·K), and \( T \) is the temperature in Kelvin. Rearrange to solve for \( n \):\[ n = \frac{PV}{RT} \]Substitute \( P = 21.0 \times 101325 \) Pa (convert atm to Pa), \( V = 0.954 \) m³, and \( T = 295.15 \) K into the equation to get\[ n \approx \frac{(21.0 \times 101325) \times 0.954}{8.314 \times 295.15} \approx 831.58 \text{ moles} \].

Step 4: Calculate the Mass of the Gas

The mass \( m \) of the gas can be found by multiplying the number of moles \( n \) by the molar mass of oxygen. The molar mass of oxygen is 32.0 g/mol, which is 0.032 kg/mol in SI units. Thus, \[ m = 831.58 \times 0.032 \approx 26.61 \text{ kg} \].

Key Concepts

Gas Calculations

Gas calculations often rely on the ideal gas law, a crucial equation in chemistry. The ideal gas law can be expressed as:\[ PV = nRT \]where:

• \( P \) is the pressure of the gas
• \( V \) is the volume it occupies
• \( n \) is the number of moles
• \( R \) is the universal gas constant (8.314 J/mol·K)
• \( T \) is the temperature in Kelvin

This equation allows us to determine any one of these variables if the others are known. For example, to find the number of moles \( n \) of a gas, you can rearrange the equation as:\[ n = \frac{PV}{RT} \]Understanding how to apply this formula is essential for calculating the amount of gas in a container, especially under specific conditions of temperature and pressure. Always remember to convert all units to the International System (SI) for consistency.

Cylinder Volume Calculation

Calculating the volume of a cylinder involves understanding its geometry. The formula for the volume of a cylinder is:\[ V = \pi r^2 h \]where:

• \( \pi \approx 3.1416 \)
• \( r \) is the radius of the cylinder's base
• \( h \) is the height or length of the cylinder

The radius is half of the diameter, so if the diameter is known, remember to divide it by two to obtain the radius. For example, a diameter of 90.0 cm translates to a radius of 0.45 meters in SI units. Substituting the values into the formula allows us to compute the volume efficiently. Calculating the cylinder's volume accurately is important when determining how much gas it can contain.

Molar Mass of Oxygen

The molar mass of a substance is the mass of a given amount of that substance, typically one mole. For oxygen (\( O_2 \)), the molar mass is 32.0 g/mol or 0.032 kg/mol in SI units. The molar mass is particularly useful when converting between the mass of gas and the number of moles. For instance, if the number of moles \( n \) is known, the mass \( m \) can be calculated as:\[ m = n \cdot \text{molar mass} \]This conversion is vital when dealing with gases like oxygen, especially in applications such as filling a canister in a space station. Calculating the mass provides insights into how much the gas weighs compared to its volume or moles.