Kinetic Theory of Gases (Cambridge (CIE) A Level Physics)

A cylinder contains 5.12 mol of an ideal gas at pressure 5.60 × 10⁵ Pa and volume 3.80 × 10^–2 m³.

Calculate the thermodynamic temperature of the gas.

temperature = ....................................................... K 

The kinetic theory of gases equation can be written as

State the meaning of each of the symbols in this equation. 

Use the equation in (a) to show that the average translational kinetic energy  of a molecule of an ideal gas is given by            

The mass of a nitrogen molecule is 4.65 × 10⁻²⁶ kg.  

Use the equation in (b) to determine the root-mean-square (r.m.s.) speed u of a nitrogen molecule at 20 °C. 

Assume that nitrogen behaves as an ideal gas.

A fixed mass of nitrogen gas at initial pressure is sealed in a cylindrical container by a moveable piston at one end, as shown in Fig. 1.1.

The temperature of the gas is 20 °C.

At all times, the gas and the container remain in thermal equilibrium with the surroundings. The piston is slowly moved out of the cylinder so that the nitrogen gas is decompressed.

On Fig. 1.2, sketch the variation of pressure with the r.m.s. speed of the nitrogen molecules as the pressure decreases. 

A fixed mass of an ideal gas is at a temperature of 24 °C. The volume of the gas is 3.3 × 10⁻³ m³ and the pressure is 2.1 × 10⁵ Pa.

Calculate the number n of moles in the gas.

The mass of one molecule of the gas is 30 u. 

Determine the r.m.s speed of the gas molecules.

The temperature of the gas is increased by 72 °C. 

Calculate the value of the ratio 

State two assumptions of the simple kinetic model of a gas.

Use the kinetic model of gases and Newton’s laws of motion to explain how a gas exerts a pressure on the sides of its container.

In a sample of gas at room temperature, four of the atoms have the following speeds: 

1.42 × 10³ m s^–1

1.51 × 10³ m s^–1

1.56 × 10³ m s^–1

1.54 × 10³ m s^–1

For these four atoms, calculate the r.m.s. to three significant figures.

A single molecule is in a cube-shaped box with sides of equal length . as shown in Fig. 1.1.

The molecule has a mass  and moves with a speed  parallel to one side of the box. It collides at regular intervals with the ends of the box, exerting a force and contributing to the pressure of the gas. 

The collisions the particle has with the container walls are elastic.

Determine an expression for the change in momentum of the molecule as it hits a wall perpendicularly.

(i) Show that the time between collisions of the molecule on the container wall is given by:

[2]

(ii) Determine an expression for the change in momentum per second.

[2]

Consider N molecules identical to the single molecule in parts (a) and (b) in the same cube-shaped box.  

Each molecule has a different velocity that contributes to the pressure. 

Calculate the total pressure from the N particles in the box.

Hence, show that the kinetic theory of gases equation is:

A sealed container A has the shape of a rectangular prism and contains an ideal gas. The dimensions of the container are x, y and z as shown in Fig. 1.1.

Fig. 1.1

• The average force exerted by the gas on the bottom wall of the container is F

• There are n moles of gas in the container

• The temperature of the gas is T 

Obtain an expression for the height of the container in terms of F, n and T.

A second container B contains the same ideal gas. 

The following information is known:

• The pressure in B is a quarter of the pressure in A

• The volume of B is five times the volume of A

• Container B contains half the number of molecules as A

• The temperature of container B is 800 K

 Calculate the average translational kinetic energy of the molecules in the gas of container A.

Calculate the value of the ratio of the r.m.s. speed of the particles in containers A and B.

The kinetic theory of gases assumes that collisions between particles and between particles and the walls of their container are perfectly elastic. 

(i) Describe what is meant by a perfectly elastic collision 

[1]

(ii) Describe the nature of the forces between the molecules

[1]

(iii) Describe, with a reason, the type of speed of the molecules considered in the kinetic theory.

[2]

A gas collimator is used to measure the average velocity of particles in a gas. 

The collimator in Fig. 1.1 controls the pressure and the volume and it maintains a gas at a constant temperature.

Fig. 1.1

Explain why two narrow apertures are required in the collimator to measure the velocity of the particles inside.

A paint manufacturer is measuring the average velocity of a sample of propane particles used to make spray paint. At a temperature of 9 °C the propane molecules have a velocity of 399.00 m s⁻¹  with an uncertainty of 0.5 %. 

Propane has a molecular mass of 44.097 g mol⁻¹.  

Determine whether the propane sample behaves as an ideal gas.

A sealed box contains 3.0 moles of a monatomic ideal gas. The average translational kinetic energy of the gas is 1 × 10⁴ J.

Calculate the average temperature of the gas.

The molar mass of the gas in the box is 5.9 g mol⁻¹. Assume that there are perfectly elastic collisions between the particles and the box walls and that the particles travel perpendicular to the walls.  

Calculate the magnitude of the impulse when one gas atom collides with one wall of the box.

The pressure of the ideal gas in the box is 6.1 × 10⁵ Pa. The box is opened and additional molecules enter. The number of gas atoms is increased by y %.

The box is then resealed and the temperature of the remaining gas atoms increased by 102.5 °C. The new pressure in the box is 2% of the original pressure. The box has a volume 0.819 m³.

Find the value of y.