3.2 Modelling a Gas

The variation of pressure p with temperature T is shown on the graph for a fixed mass of an ideal gas at constant volume. 

What is the ratio of pressure and temperature of the ideal gas?

A capillary tube has an internal diameter of 6 mm and is closed at one end. It contains a sample of gas enclosed by a tungsten thread which is 0.2 m long, tungsten has a density of 5000 kg m⁻³. 

On a day when the atmospheric air pressure is 1 × 10⁵π Pa the length of the sample of trapped gas is 0.4 m when the capillary tube is in the horizontal position.

Assume the temperature in the capillary tube remains constant and that the length of the tungsten thread does not change when the tube is moved into the vertical position. 

What is the length of the air column when the tube is turned into the vertical position?

A mass of 5 × 10⁻³ kg of Hydrogen is split between two gas bottles Y and Z. Y and Z are connected by a tube that has a negligible volume compared with the volume of each bottle. 

Initially the valve X is closed and gas bottle Y has a volume 4V and contains Hydrogen at a pressure p. Gas bottle Z has a volume V and contains Hydrogen at a pressure of 3p. Y and Z are initially at the same temperature. 

X is now opened.

What is the new gas pressure assuming there is no change in temperature?

A materials scientist carries out an experiment to investigate the relationship between the volume and temperature of an ideal gas. The pressure of the gas is kept constant throughout the experiment. Two of their readings are shown below. 

What is the value for absolute zero that can be calculated from these results? 

An ideal gas is initially at a temperature of 127 °C. It is heated and the root mean square velocity of the gas is then tripled. Use 1 × 10⁻²³ as the Boltzmann Constant. 

What are the values for the temperature of the gas and the mean kinetic energy of the molecules after it has been heated?

A single molecule of mass 5 × 10⁻²⁸ kg moves with a speed of 200 m s⁻¹ in a rigid cubical container with a side length of 0.4 m. 

Assume all collisions between the molecule and the walls of the container are perfectly elastic.

What is the average pressure exerted on the wall by the molecule?

Air consists of 0.9% argon molecules and 0.002% neon molecules. Argon has a relative molecular mass of 40 and neon a relative molecular mass of 20.

What is the ratio of the root mean square speed of argon to that of neon molecules in the air?

In a car factory laboratory, the atmospheric pressure is 101 kPa and the temperature is 47 °C. The pressure in the car tyre under investigation is 180 kPa and the temperature 227 °C.

Assume no air is lost from the tyre.

What is the percentage increase in the root mean square speed of the air molecules due to the experiment?

An ideal gas is stored in a sealed container of fixed volume and has molecules with a root mean square speed of c m s⁻¹. The temperature of the gas changes resulting in a decrease in pressure by one-third of the previous value.

What is the new expression for the root mean square speed of the molecules after the temperature has changed?