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Chapter 18: Problem 10

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?

Short Answer

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

Step by step solution

01

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掳C to 295.15 K. (22.0 + 273.15).
02

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,\[ V = \pi \times (0.45)^2 \times 1.50 \approx 0.954 \text{ m}^3 \].
03

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路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), \( V = 0.954 \) m鲁, and \( T = 295.15 \) K into the equation to get\[ n \approx \frac{(21.0 \times 101325) \times 0.954}{8.314 \times 295.15} \approx 831.58 \text{ moles} \].
04

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, \[ m = 831.58 \times 0.032 \approx 26.61 \text{ kg} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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路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.

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Most popular questions from this chapter

A metal tank with volume 3.10 \(\mathrm{L}\) will burst if the absolute pressure of the gas it contains exceeds 100 atm. (a) If 11.0 mol of an ideal gas is put into the tank at a temperature of \(23.0^{\circ} \mathrm{C},\) to what temperature can the gas be warmed before the tank ruptures? You can ignore the thermal expansion of the tank. (b) Based on your answer to part (a), is it reasonable to ignore the thermal expansion of the tank? Explain.

(a) For what mass of molecule or particle is \(v_{\mathrm{rms}}\) equal to 1.00 \(\mathrm{mm} / \mathrm{s}\) at 300 \(\mathrm{K} ?\) (b) If the particle is an ice crystal, how many molecules does it contain? The molar mass of water is 18.0 \(\mathrm{g} / \mathrm{mol}\) . (c) Calculate the diameter of the particle if it is a spherical piece of ice. Would it be visible to the naked eye?

If a certain amount of ideal gas occupies a volume \(V\) at STP on earth, what would be its volume (in terms of \(V )\) on Venus, where the temperature is \(1003^{\circ} \mathrm{C}\) and the pressure is 92 atm?

Gaseous Diffusion of Uranium. (a) A process called gaseous diffusion is often used to separate isotopes of uranium that is, atoms of the elements that have different masses, such as 235 \(\mathrm{U}\) and 238 \(\mathrm{U} .\) The only gaseous compound of uranium at ordinary temperatures is uranium hexafluoride, UF \(_{6}\) . Speculate on how 235 \(\mathrm{UF}_{6}\) and \(^{238} \mathrm{UF}_{6}\) molecules might be separated by diffusion. (b) The molar masses for \(^{235} \mathrm{UF}_{6}\) and 238 \(\mathrm{UF}_{6}\) molecules are 0.349 \(\mathrm{kg} / \mathrm{mol}\) and \(0.352 \mathrm{kg} / \mathrm{mol},\) respectively. If uranium hexafluoride acts as an ideal gas, what is the ratio of the root-meansquare speed of \(^{235} \mathrm{UF}_{6}\) molecules to that of \(^{238} \mathrm{UF}_{6}\) molecules if the temperature is uniform?

Helium gas with a volume of \(2.60 \mathrm{L},\) under a pressure of 0.180 atm and at a temperature of \(41.0^{\circ} \mathrm{C},\) is warmed until both pressure and volume are doubled. (a) What is the final temperature? (b) How many grams of helium are there? The molar mass of helium is 4.00 \(\mathrm{g} / \mathrm{mol} .\)
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