Worksheet: Pressure, Volume, and Temperature of an Ideal Gas

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In this worksheet, we will practice calculating the pressure, temperature, and volume of an ideal gas using pV = NkT (the ideal gas law).

Q1:

The number density of gas molecules is defined as the number of molecules present per unit volume. At a certain location in the space above Earth, the number density is 1.94×10 m−3 and the atmospheric pressure at that location is 3.45×10 N/m2. What is the temperature at this location?

Q2:

Airtight bags containing food that are taken to high elevations become puffed up because the air inside the bags has expanded. A bag of pretzels was packed at a pressure of 1.00 atm and a temperature of 25.0∘C. The bag is taken to a summer picnic in Santa Fe, New Mexico, where the air temperature is 35.0∘C. The volume of the air in the bag is 1.24 times its volume when packed. What is the pressure of the air at the picnic?

Q3:

A vacuum chamber does not produce a true vacuum, but can produce a very low pressure of 1.00×10 N/m2 at a temperature of 20.0∘C. How many molecules per cubic centimeter remain in the vacuum chamber?

  • A 2 . 4 7 × 1 0
  • B 2 . 3 2 × 1 0
  • C 1 . 6 8 × 1 0
  • D 2 . 1 9 × 1 0
  • E 1 . 9 0 × 1 0

Q4:

A company advertises that it delivers helium at a gauge pressure of 1.72×10 Pa in a cylinder of volume 43.8 L. How many balloons can be inflated to a volume of 4.00 L with that amount of helium? Assume the pressure inside the balloons is 1.01×10 Pa and the temperature in the cylinder and the balloons is 25.0∘C.

  • A 1 . 8 7 × 1 0
  • B 1 . 5 5 × 1 0
  • C 2 . 2 3 × 1 0
  • D 2 . 5 0 × 1 0
  • E 2 . 0 2 × 1 0

Q5:

A bicycle tire contains 2.00 L of gas at an absolute pressure of 7.00×10 N/m2 and a temperature of 18.0∘C. An amount of gas that has a volume of 1.00×10 cm3 at atmospheric pressure is released from the tire, having negligible effect on the tire’s temperature and volume. What is the pressure of the gas in the tire after the gas is removed?

  • A 6 . 9 5 × 1 0 Pa
  • B 6 . 3 6 × 1 0 Pa
  • C 6 . 7 3 × 1 0 Pa
  • D 7 . 6 6 × 1 0 Pa
  • E 7 . 3 2 × 1 0 Pa

Q6:

A gas-filled incandescent light bulb is manufactured so that the gas inside the bulb is at atmospheric pressure when the bulb has a temperature of 20.0∘C. Find the gauge pressure inside such a bulb when its average temperature is 60.0∘C, neglecting any change in volume due to thermal expansion or gas leaks.

Q7:

A high-pressure gas cylinder contains 50.0 L of toxic gas at a pressure of 1.40×10 N/m2 and a temperature of 25.0∘C. The cylinder is cooled to dry ice temperature (−78.5)∘C to reduce the leak rate and pressure so that it can be safely repaired.

What is the final pressure in the tank, assuming a negligible amount of gas leaks while being cooled and that there is no phase change?

  • A 1 0 . 0 × 1 0 Pa
  • B 9 . 0 4 × 1 0 Pa
  • C 1 0 . 4 × 1 0 Pa
  • D 9 . 4 0 × 1 0 Pa
  • E 1 0 . 8 × 1 0 Pa

What is the final pressure if one tenth of the gas escapes?

  • A 8 . 5 0 × 1 0 Pa
  • B 8 . 3 8 × 1 0 Pa
  • C 8 . 4 2 × 1 0 Pa
  • D 8 . 2 8 × 1 0 Pa
  • E 8 . 2 2 × 1 0 Pa

To what temperature must the tank be cooled to reduce the pressure to 1.00 atm (assuming the gas does not change phase and that there is no leakage during cooling)?

Q8:

One process for decaffeinating coffee uses carbon dioxide (𝑀=44.0/)gmol at a molar density of 14,600 mol/m3 and a temperature of 60.0∘C. The van der Waals constants for carbon dioxide are 𝑎=0.3658⋅/Pammol and 𝑏=4.286×10/mmol. Using the van der Waals equation, find the pressure of CO2 at that temperature and density.

  • A 3 . 1 0 × 1 0 Pa
  • B 3 . 0 0 × 1 0 Pa
  • C 3 . 1 7 × 1 0 Pa
  • D 3 . 2 3 × 1 0 Pa
  • E 3 . 0 7 × 1 0 Pa

Q9:

A tank with a volume of 6.75 m3 contains 5.23×10 mol of compressed oxygen. Determine the gauge pressure inside the tank if the root-mean-square speed of the oxygen molecules is 524 m/s. Use a value of 32.0 g/mol for the molar mass of molecular oxygen.

  • A 2 . 0 8 × 1 0 Pa
  • B 3 . 3 3 × 1 0 Pa
  • C 2 . 6 6 × 1 0 Pa
  • D 2 . 1 9 × 1 0 Pa
  • E 2 . 2 6 × 1 0 Pa

Q10:

A mole of dry air consists of 78% nitrogen, 21% oxygen, and 1% argon, with trace amounts of other gases. A tank of compressed dry air at a temperature of 293 K has a volume of 1.76 cubic feet. The compressed air has a gauge pressure of 2,200 pounds per square inch. How many moles of oxygen does the compressed air contain?

Q11:

An airplane passenger’s stomach contains 170 cm3 of air just before the plane takes off from a sea-level airport. What will the volume of air in her stomach be during the flight if the airplane’s cabin pressure is 8.30×10 N/m2?

  • A107 cm3
  • B208 cm3
  • C207 cm3
  • D480 cm3
  • E430 cm3

Q12:

The deep space between galaxies is not a complete vacuum. The intergalactic medium can be modeled as an ideal gas with a density of various molecular and sub-molecular scale particles of 204 particles/m3, at an average temperature of 3.4 K.

What pressure is produced by the intergalactic medium?

  • A 4 7 × 1 0 Pa
  • B 9 . 6 × 1 0 Pa
  • C 1 . 3 × 1 0 Pa
  • D 2 . 1 × 1 0 Pa
  • E 6 . 9 × 1 0 Pa

What volume is occupied by 1.00 mol of intergalactic medium particles?

  • A 1 . 9 8 × 1 0 m3
  • B 3 . 0 0 × 1 0 m3
  • C 1 . 5 8 × 1 0 m3
  • D 4 . 1 2 × 1 0 m3
  • E 2 . 9 5 × 1 0 m3

What is the side length of a cube that has the volume required to contain 1.00 mol of intergalactic medium particles?

Q13:

The air inside a hot-air balloon has a temperature of 387 K and a pressure of 103.3 kPa, the same as that of the air outside. Using the molar composition of air as 78.0%N2, 21.0%O2, and 1.00%Ar, find the density of the air inside the balloon. Use a value of 28.0 g/mol for the molar mass of N2, 32.0 g/mol for the molar mass of O2, and 39.3 g/mol for the molar mass of Ar.

Q14:

A cubic container of volume 2.25 L holds 0.750 mol of nitrogen gas at a temperature of 27.3∘C. What is the magnitude of the force exerted on one wall of the container by the gas?

Q15:

2.25 kg of steam is contained in a region with a volume of 0.448 m3 and at a pressure of 1.67×10 Pa. The steam can be modeled as having a molar mass of 18.0 g/mol.

If the steam is modeled as an ideal gas, what is the steam’s temperature?

The van der Waals constants for water are 𝑎=0.5537 Pa⋅m6/mol2 and 𝑏=3.049×10 m3/mol. If the van der Waals equation of state is used to calculate the temperature of the steam, what is the calculated temperature?

Q16:

Treating air as if it was an ideal gas and a single substance, calculate the number of moles of air that would be contained in the 2.00 L volume of air in the lungs of the average person. Assume the air to be at atmospheric pressure and a temperature of 37.0∘C.

  • A 7 . 6 0 × 1 0
  • B 7 . 7 5 × 1 0
  • C 8 . 1 7 × 1 0
  • D 7 . 8 4 × 1 0
  • E 8 . 0 2 × 1 0

Q17:

A tire with a volume of 35.0 L contains 4.86 mol of a gas at a temperature of 23.0∘C. An additional 1.25 L of gas is added to the tire, increasing its pressure while the temperature and volume of the gas remain constant.

What is the gauge pressure in the tire before the additional gas is added?

  • A 2 . 0 1 × 1 0 Pa
  • B 1 . 0 1 × 1 0 Pa
  • C 2 . 1 2 × 1 0 Pa
  • D 2 . 4 1 × 1 0 Pa
  • E 2 . 4 1 × 1 0 Pa

What is the gauge pressure in the tire after the additional gas is added?

  • A 3 . 1 1 × 1 0 Pa
  • B 2 . 1 5 × 1 0 Pa
  • C 2 . 4 4 × 1 0 Pa
  • D 2 . 4 9 × 1 0 Pa
  • E 2 . 4 1 × 1 0 Pa

Q18:

When an air bubble rises from the bottom to the top of a freshwater lake, its volume increases by 80%. If the temperatures at the bottom and the top of the lake are 4.0 and 10∘C, respectively, how deep is the lake?

Q19:

A car’s tires have a gauge pressure of 3.20×10 N/m2 when at a temperature of 30.0∘C. What is the gauge pressure of the tires at a temperature of −25.0∘C? Assume the tires have not gained or lost any air.

  • A 2 . 0 1 × 1 0 Pa
  • B 2 . 9 1 × 1 0 Pa
  • C 2 . 8 8 × 1 0 Pa
  • D 2 . 6 2 × 1 0 Pa
  • E 2 . 4 4 × 1 0 Pa

Q20:

In a common demonstration, a hard-boiled egg is pushed through the neck of a glass bottle that is noticeably narrower than the egg’s width. A bottle used in such a demonstration has a volume of 0.750 L and the temperature inside the bottle is maintained at 85.0∘C. The pressure inside the bottle is maintained at a constant 1.00 atm. An egg is then placed on the bottle’s neck and the bottle is cooled until its internal temperature is 21.0∘C. The pressure inside the bottle falls as the air inside the bottle reduces in temperature and the greater external air pressure pushes the egg through the bottle’s neck.

How many moles of air does the bottle contain?

What is the air pressure inside the bottle after it has been cooled, just before the egg starts to move down the bottle’s neck?

Q21:

A gas in a container that has a volume of 2.45 L is at a temperature of 42∘C and a pressure of 8.24×10 N/m2. How many moles of gas are in the container?

Q22:

A rigid vessel is maintained at a pressure of 10 MPa and a temperature of 300∘C. The vessel contains steam that has a mass of 0.80 kg. What is the volume of the vessel?

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