3.2 Modelling a Gas

A cylinder is fixed with an airtight piston containing an ideal gas of temperature 20 °C.

When the pressure, P in the cylinder is 3 × 10⁴ Pa the volume, V is 2.0 × 10⁻³ m³. 

Calculate the number of gas molecules present in the cylinder.

The piston is slowly pushed in and the temperature of the gas remains constant. 

Draw a graph by plotting three additional points on the axis to show the relationship between pressure and volume as the piston is slowly pushed in. 

The cylinder, cylinder X is connected now to a second cylinder, cylinder Y which is initially fully compressed. Cylinder Y has a diameter two times that of the diameter of cylinder X. The total number of molecules in the system remains the same. 

Cylinder X is pushed down by a distance Δh_x causing Y to move up a distance Δh_y. The pressure and temperature within the system both remain constant. 

Determine the ratio Δh_x : Δh_y.

Initially, the gas molecules are divided between both cylinders. The diameter, d, of cylinder X, is 16 cm. The piston in cylinder X is compressed at a constant rate until all of the gas is moved into cylinder Y over a period of 5 seconds. 

Assume that the volume of the connecting tube is negligible. 

A gas syringe is connected through a delivery tube to a conical flask, which is immersed in an ice bath. The syringe is frictionless so the gas pressure within the system remains equal to the atmospheric pressure 101 kPa.  

The total volume of the conical flask and delivery tube is 275 cm³, and after settling in the ice bath whilst the ice is melting the gas syringe has a volume of 15 cm³. 

Calculate the total number of moles contained within the system.

When the ice bath is heated at a constant rate it takes the following time to melt the ice and heat the water:

• Time for ice to melt is 3 minutes
• Time from ice melting to water boiling is 10 minutes
• Time for water to boil is 3 minutes

(ii)

Sketch a graph to show this process.

A burner in the base of a hot air balloon is used to heat the air inside the balloon. 

The mass of the balloon can be reduced by releasing sand from the basket of the balloon. 

A sealed container C has the shape of a rectangular prism and contains an ideal gas. The dimensions of the container are l, w and h. 

• 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 in terms of F, n and T for the height of the container.

A second container D contains the same ideal gas. The pressure in D is a fifth of the pressure in C and the volume of D is four times the volume of C. In D there are three times fewer molecules than in C. 

The temperature of cylinder D is 600 K. 

Calculate the temperature of cylinder C in °C.

The temperature of a different container E is 60 °C. At this temperature, the pressure exerted by the ideal gas is 1.75 × 10⁵ Pa. The container is a cube and has a height of 4 cm. 

Calculate the number of molecules of gas in this container.

In a different container F the pressure of the gas is measured at different temperatures whilst the volume and number of moles are kept the same. 

Plot a graph to show how the pressure varies with temperature for this gas.

This question is about an ideal gas in a container.

An ideal gas is held in a glass gas syringe.

Calculate the temperature of 0.726 mol of an ideal gas kept in a cylinder of volume 2.6 × 10^–3 m³ at a pressure of 2.32 × 10⁵ Pa.

The average kinetic energy of the gas is directly proportional to one particular property of the gas. 

[1]

Calculate the average kinetic energy, , per molecule of the gas.

 [1]

Energy is supplied to the gas at a rate of 0.5 J s^–1 for 4 minutes. The specific heat capacity of the gas is 519 J kg^–1 K^–1 and the atomic mass number is 4 u.

Calculate the change in kinetic energy per molecule of the gas.

The gas is heated until its temperature doubles.

Determine the factor the average speed of the molecules increases by.

This question is about the specific heat capacity of an ideal gas.

Outline two assumptions made in the kinetic model of an ideal 

Xenon–131 behaves as an ideal gas over a large range of temperatures and pressures.

One mole of Xenon–131 is stored at 20 °C in a cylinder of fixed volume. The Xenon gas is heated at a constant rate and the internal energy increased by 450 J. The new temperature of the Xenon gas is 41.7 °C.

[1]

[2]

[2]

The volume of the sealed container is 0.054 m³.

Calculate the change in pressure of the gas due to the energy supplied in part (b).

One end of the container is replaced with a moveable piston. The piston is compressed until the pressure of the container is 67000 Pa.

Determine the new volume of the container.

This question is about an experiment to investigate the variation in the pressure p of an ideal gas with changing volume V.

The gas is trapped in a cylindrical tube of radius 0.5 cm above a column of oil.

The pump forces the oil to move up the tube and so reduces the volume of the gas. The scientist measures the pressure p of the gas and the height H of the column of gas.

Calculate the volume of the gas when the height is 1 cm.

When the system is at a constant temperature of 20 °C, the pressure is 9600 Pa.

Calculate: 

[2]

The scientist plots their results of p against  on a graph.

Explain the shape of the graph and why this is to be expected.

When conducting the experiment, the scientist waits for a period of time between taking each reading. 

[1]

[2]

State the Pressure law of ideal gases.

The pressure exerted by an ideal gas containing 9.7 × 10²⁰ molecules in a container of volume 1.5 × 10^–5 m³ is 2.8 × 10⁵ Pa.

Calculate the temperature of the gas in the container in °C.

The pressure of the gas is measured at different temperatures whilst the volume of the container and the mass of the gas remain constant.

On the grid, sketch a graph to show how the pressure varies with the temperature.

The container described in part (a) has a release valve that allows gas to escape when the pressure exceeds 3.5 × 10⁵ Pa.

Calculate the number of gas molecules that escape when the temperature of the gas is raised to 380°C.