Worksheet: Root-Mean-Square Speed of Ideal Gas Particles

In this worksheet, we will practice determining the relationship between the absolute temperature of an ideal gas and the root-mean-square speed of its particles.

Q1:

A cloud of helium gas consists of 1 . 3 5 × 1 0 helium atoms. The atoms have a root-mean-square speed of 652 m/s and the gas exerts a pressure of 4 5 2 5 0 Pa. Find the volume of the gas. Use a value of 6 . 6 4 6 5 × 1 0 kg for the mass of a helium atom. Answer to three significant figures.

Q2:

A cloud of hydrogen gas with a volume of 1.05 m3 consists of 4.22 moles of hydrogen atoms. The atoms have a root-mean-square speed of 983 m/s. Find the pressure of the gas to the nearest kilopascal. Use a value of 1.0079 g for the molar mass of hydrogen atom and a value of 8 . 3 1 / m k g s K m o l for the molar gas constant.

Q3:

A particle of mass 𝑚 moves at a speed 𝑣 between opposite faces of an empty cubic container of side length 𝑑 and volume 𝑉 , as shown in the diagram, colliding elastically with the container’s walls. The particle changes its momentum by 2 𝑚 𝑣 each time it collides with a wall and the particle collides with the wall shown being hit in the diagram once every 2 𝑑 𝑣 seconds.

Which of the following correctly represents the average force exerted on the wall of the container that the particle is shown colliding with due to the particle’s collisions?

  • A 𝑚 𝑣 𝑑
  • B 𝑚 𝑑 𝑣
  • C 𝑚 𝑣 𝑑
  • D 𝑚 𝑑 𝑣

Which of the following correctly represents the average pressure exerted on the wall of the container that the particle is shown colliding with due to the particle’s collisions?

  • A 𝑑 𝑚 𝑣
  • B 𝑚 𝑑 𝑣
  • C 𝑚 𝑣 2 𝑑
  • D 𝑚 𝑑 𝑣
  • E 𝑚 𝑣 𝑑

Q4:

A particle of mass 𝑚 moves at a speed 𝑣 between opposite faces of an empty cubic container of side length 𝑑 , as shown in the diagram, colliding elastically with the walls of the container.

Which of the following correctly represents the change in the momentum of the particle in a collision with a wall of the container?

  • A 𝑚 𝑣 2
  • B 𝑚 𝑣
  • C0
  • D 2 𝑚 𝑣

Which of the following correctly represents the time interval between the particle’s collisions with the wall of the container that the particle is shown colliding with?

  • A 2 𝑑 𝑣
  • B 𝑑 2 𝑣
  • C 2 𝑑 𝑣
  • D 𝑑 𝑣

Q5:

The product of the pressure and volume of a gas is given by the formula 𝑃 𝑉 = 2 3 𝑛 𝑚 𝑣 r m s , where 𝑛 is the number of particles in the gas, 𝑚 is the mass of a particle, and 𝑣 r m s is the root-mean-square speed of the particles. Which of the following formulas can be combined with this formula to show that the kelvin temperature of a gas is given by 𝑇 = 2 3 𝑘 𝑚 𝑣 r m s , where 𝑘 is Boltzmann’s constant?

  • A 𝑃 𝑉 𝑇 = 𝑃 𝑉 𝑇
  • B 𝑇 = 𝑇 2 7 3 K C
  • C 𝑃 𝑉 = 𝑛 𝑘 𝑇
  • D 𝑘 = 𝑅 𝑁

Q6:

A cloud of hydrogen gas with a volume of 2.56 m3 consists of 3 . 2 5 × 1 0 hydrogen atoms. The atoms have a root-mean-square speed of 776 m/s. Give your answer to three significant figures. Use a value of 1 . 6 7 3 7 × 1 0 kg for the mass of a hydrogen atom.

Q7:

A cloud of helium gas consists of 1.28 moles of helium. The atoms have a root-mean-square speed of 514 m/s and the gas exerts a pressure of 67,500 Pa. Find the volume of the gas. Use a value of 4.0026 g for the molar mass of helium and a value of 8.31 m2⋅kg/s2⋅K⋅mol for the molar gas constant. Answer to three significant figures.

  • A 6 . 6 8 × 1 0 m3
  • B 8 . 5 6 × 1 0 m3
  • C 6 . 6 8 × 1 0 m3
  • D 8 . 5 6 × 1 0 m3
  • E 5 . 2 2 × 1 0 m3

Q8:

Nitrogen atoms in a container of volume 2.35 m3 exert a pressure of 1,280 kPa. The nitrogen atoms have a root-mean-square speed of 450 m/s. Find the number of moles of nitrogen atoms in the gas. Use a value of 14.0067 g/mol for the molar mass of nitrogen and a value of 8.31 m2⋅kg/s2⋅K⋅mol for the molar gas constant. Answer to three significant figures.

Q9:

Hydrogen atoms in a gas have a root-mean-square speed of 3,460 m/s. Find the temperature of the gas. Use a value of 1 . 6 7 3 7 × 1 0 kg for the mass of a hydrogen atom and use 1 . 3 8 0 6 × 1 0 m2⋅kg/s2⋅K for the value of Boltzmann’s constant. Give your answer to three significant figures.

Q10:

Nitrogen atoms having a root-mean-square speed of 1,250 m/s exert a pressure of 115 kPa in a container of volume 0.145 m3. Find the number of atoms in the gas, giving your answer to three significant figures. Use a value of 2 . 3 2 5 8 7 × 1 0 kg for the mass of a nitrogen atom.

  • A 3 . 4 4 × 1 0 atoms
  • B 1 . 7 2 × 1 0 atoms
  • C 6 . 8 8 × 1 0 atoms
  • D 2 . 7 5 × 1 0 atoms
  • E 6 . 8 8 × 1 0 atoms

Q11:

Particles in a gas at a temperature of 355 K have a root-mean-square speed of 775 m/s. Find the mass of the particles. Use 1 . 3 8 0 6 × 1 0 m2⋅kg/s2⋅K for the value of Boltzmann’s constant. Give your answer to three significant figures.

  • A 8 . 1 6 × 1 0 kg
  • B 6 . 3 2 × 1 0 kg
  • C 1 . 9 0 × 1 0 kg
  • D 2 . 4 5 × 1 0 kg
  • E 1 . 8 4 × 1 0 kg

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