6.5 Ideal Gases

A cylinder fitted with an airtight piston, contains an ideal gas at a temperature of 20°C.

When the pressure, p, in the cylinder is 3 × 10⁴ Pa the volume, V, is 2.0 × 10^–3 m³.This is shown in Figure 1.

Figure 1

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

Draw a graph by plotting two or three additional points on the axes given in Figure 1, to show the relationship between pressure and volume as the piston is slowly pushed in.

Calculate the number of gas molecules in the cylinder.

The cylinder (Cylinder A) is connected to a second cylinder (Cylinder B), which is initially fully compressed. Cylinder B has a diameter that is twice the diameter of the cylinder A, as shown in Figure 2. The total number of molecules in the system remains the same.

Figure 2

Cylinder A is pushed down by a distance, , causing cylinder B moves upwards a distance, . The pressure and temperature within the cylinders remain constant.

Determine the ratio : .

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

Assume that the volume of the connecting tube is negligible.

 A student is investigating how the volume of an ideal gas is dependent on its temperature. A gas syringe is connected through a delivery tube to a conical flask. The gas syringe is frictionless, so that the gas pressure within the system remains equal to atmospheric pressure, p = 101 kPa. The flask is immersed in an ice bath, as shown in Figure 1. 

Figure 1

The total volume of the conical flask and delivery tube is 283 cm³, and after settling in the ice bath whilst the ice is melting, the gas syringe measures a volume of 12 cm³. 

Determine the total number of moles of the ideal gas within the flask, tube and syringe.

The ice bath is heated at a constant rate. 4 minutes are needed for the ice to melt, and then 8 minutes later, the water in the bath begins boiling. After 4 minutes of boiling, the heater is turned off. 

Determine the volume of the gas throughout the period of heating and plot a graph using the axes provided in Figure 2 below to show how the volume of the gas changes with time. 

Figure 2

The relationship between the volume of a gas and the temperature of a gas is important for hot air balloons. A hot air balloon is open at the base, so air can move in and out of the balloon. A burner at the base of the balloon is used to heat the air in the balloon, as shown in Figure 1. 

Figure 1

As the balloon rises the air pressure and air temperature around the balloon decreases. The mass of the balloon can be reduced by releasing sand from the balloon. 

Discuss the forces upon the hot air balloon due to changes in the temperature within the balloon and the changes in air pressure outside the balloon, and with reference to the ideal gas equation. In your answer you should: 

• Explain how the burner is used so the balloon can rise.
• Explain how the forces upon the balloon change with altitude and as the mass of the balloon decreases. 

You may wish to draw a diagram to clarify your answer.

The quality of your written communication will be assessed in your answer.

Figure 1 shows the cross-section of a bicycle pump with a cylindrical barrel. The piston has been pulled to the position marked X and the outlet of the pump sealed.

Figure 1

The lengthL of the column of trapped air is 12 cm and the volume of the gas is 1.4 × 10⁻⁴ m³ when the piston is at position X. Under these conditions the trapped air is at a pressure p of 1.5 × 10⁵ Pa and its temperature is 23°C.

Assume the trapped air consists of identical molecules and behaves like an ideal gas in this question.

   Calculate the internal diameter of the barrel.

The piston is pushed slowly inwards until the length L of the column of trapped air is 2.4 cm. Figure 2 shows how the pressure p of the trapped air varies as L is changed during this process.

Figure 2

Use data from Figure 2 to show that p is inversely proportional to L.

The piston on the bike pump is moved to a position Y, such that the volume of the trapped air is V. A balloon containing particles of the same ideal gas is connected as shown in Figure 2 with the seal between the pump and the balloon closed. The pressure and the temperature in both the balloon and the pump are p and T respectively. The initial volume of the balloon is 2V. 

Figure 2

When the tap is opened, the temperature of the gas particles does not change, while the pressure in both the balloon and the glass jar becomes 2p.

Show that the new volume of the balloon is equal to .

A sealed containerAin the shape of a rectangular prism contains an ideal gas. The dimensions of the container are l, w and h, as shown in Figure 1 

Figure 1

The average force exerted by the gas molecules on the bottom wall of the container is F. There are n moles of gas in the container and the temperature of the gas is T. 

Show that the height of the container, h =

Another container, B, has the same ideal gas within it. The pressure in B is a third of the pressure in container A. The volume of B is five times the volume of A, and there are four times fewer molecules in B than in A. 

The temperature of container B is 500 K. 

   Calculate the temperature, in °C, of container A.

The temperature of different container C, is 50°C.  The container is cubic in shape. At this temperature, the pressure exerted by the ideal gas is 1.5 × 10⁵ Pa. 

If the height of the container is 3 cm. 

   Calculate the number of molecules of gas in the container.

The pressure of the gas is measured at different temperatures while volume and number of moles of the gas remains constant.

Draw a graph on the grid in Figure 2 to show how the pressure varies with the temperature.

Figure 2

State two equations or laws which ideal gases obey.

A car tyre of volume 3.1 × 10^–2 m³ contains air at a pressure of 400 kPa and a temperature of 240 K. The mass of one mole of air is 2.9 × 10^–2 kg. 

Assuming that the air behaves as an ideal gas, calculate: 

            (i)         The amount of moles of air. 

            (ii)        The mass of the air. 

            (iii)       The density of the air.

A bicycle tyre with 0.53 moles of air has a volume of 2.30 × 10^–3 m³ when the temperature is 246 K. 

Calculate the pressure inside the bicycle tyre.

After the bicycle has been ridden, the temperature of the air in the tyre is 295 K. 

Calculate the new pressure in the tyre assuming the volume is unchanged.

Give your answer to an appropriate number of significant figures.

The ‘laws of football’ require the ball to have a circumference between 680 mm and 700 mm. The pressure of the air in the ball is required to be between 0.60 × 10⁵ Pa and 1.10 × 10⁵ Pa above atmospheric pressure. 

A ball is inflated when the atmospheric pressure is 3.00 × 10⁵ Pa and the temperature is 15 °C. When inflated the mass of air inside the ball is 17.4 g and the circumference of the ball is 685 mm. 

Assume that air behaves as an ideal gas and that the thickness of the material used for the ball is negligible. 

Deduce whether or not the inflated ball satisfies the pressure requirements according to the ‘laws of football’. 

   Molar mass of air = 29 g mol^–1

   Atmospheric pressure = 1.00 × 10⁵ Pa

‘The volume of an ideal gas is directly proportional to its temperature’, is an incomplete statement of Charles’s law. 

State two conditions necessary to complete the statement.

A volume of 6.6 × 10^–4 m³ of air at a pressure of 6.4 × 10⁵ Pa and a temperature of 390 K is trapped in a cylinder. Under these conditions the volume of air occupied by 1.0 mol is 5.0 × 10^–3 m³. The air in the cylinder is cooled and at the same time expanded slowly by a piston. 

The initial condition and final condition of the trapped air are shown in Figure 1. 

Figure 1

In the following calculations treat air as an ideal gas having a molar mass of 0.029 kg mol^–1. 

Calculate the final volume of the air trapped in the cylinder.

Calculate the initial density of air trapped in the cylinder.

Figure 1 shows a p–V graph for a fixed mass of gas. The volume increases from  to   while the pressure falls from to   . 

Figure 1

Which one of the paths A, B, C or D will result in the greatest amount of work being done by the gas? Explain your answer.

An ideal gas at a temperature of 45 °C is trapped in a metal cylinder of volume 0.70 m³ at a pressure of 2.6 × 10⁶ Pa. 

Calculate the number of molecules of gas contained in the container.

The density of the gas is 21 kg m^–3. 

Calculate the molar mass of the gas in the cylinder. 

State an appropriate unit for your answer.

The cylinder is taken to high altitude where the temperature is −60 °C and the pressure is 4.1 × 10⁴ Pa. A valve on the cylinder is opened to allow gas to escape. 

Calculate the mass of gas remaining in the cylinder when it reaches equilibrium with its surroundings. 

Give your answer to an appropriate number of significant figures.

State the Pressure law of ideal gases.

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

Calculate the temperature of gas in the container in ºC

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

Figure 1

Draw a graph on the grid in Figure 1 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.