Equation of State (Cambridge (CIE) A Level Physics)

The ideal gas equation in terms of the number of molecules, N is written as 

where k is the Boltzmann constant.

Show that the ideal gas equation can also be written as

A cylinder for an ideal gas has a radius of 120 mm and a height of 1350 mm. The gas is at a temperature of 14 °C and a pressure of 1.7 × 10⁷ Pa.

Show that the amount of gas in the cylinder is 430 mol.

The gas leaks slowly from the cylinder at an average rate of 8.3 × 10¹⁵ atoms s^–1 so that, after a time of 51 days, the pressure reduces. The temperature remains constant.  

Calculate the percentage of the number of moles lost from the cylinder.

Calculate the number of molecules in a gas containing 120 g of . 

A different ideal gas has an initial volume of 6.00 × 10⁴ m³ at a pressure of 1.42 × 10⁵ Pa and a temperature of 8 °C 

It is heated at a constant volume so that, in its final state, the pressure is 1.70 × 10⁵ m³ at a temperature of 63 °C. 

Show that these two states prove it behaves as an ideal gas.

Atoms of a real gas each have a diameter of 0.12 nm. 

Estimate an accurate value for the volume occupied by 0.54 moles of this gas.

volume = ........................... m³

Explain the differences in the volumes of the atoms in part (b) and part (c).

(i) State what is meant by an ideal gas.

[2]

(ii) Sketch a graph that represents the definition from part (i).

[2]

The air in a bicycle tire has a constant volume of 3.5 × 10⁶ mm³. The pressure of 0.29 mol of the air is 2.0 × 10⁵ Pa. The air may be considered to be an ideal gas. 

Calculate the temperature of the air in the tire.

 temperature = .......................... °C 

The pressure of the tire is increased using a pump. On each stroke of the pump, 0.0045 mol of air is forced into the tire. 

Calculate the number of moles of air in the tire when the pressure increases to 2.7 × 10⁵ Pa and the temperature increases by 10 °C.

Calculate the number of strokes of the pump required to increase the pressure to 2.7 × 10⁵ Pa and the temperature by 10 °C.

The pressure exerted by an ideal gas of 8.4 × 10²⁰ molecules in a container of volume 1.6 × 10^–5 m³ is 3.2 × 10⁵ Pa. 

Calculate the temperature of the 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. 

Fig 1.1

Draw a graph on the grid in Fig 1.1 to show how the pressure varies with the temperature.

The container described in part (b) has a release valve that allows gas to escape when the pressure exceeds 4.0 × 10⁵ Pa. 

Calculate the number of gas molecules that escape when the temperature of the gas is raised to 520 °C.

State two equations or laws which ideal gases obey.

A car tyre of volume 4.2 × 10^–2 m³ contains air at a pressure of 300 kPa and a temperature of 290 K. The mass of one mole of air is 3.1 × 10^–2 kg.

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

(i) The amount of moles of air.

[3]

(ii) The mass of the air.

[1]

(iii) The density of the air.

[1]

A bicycle tyre with 0.47 moles of air has a volume of 1.90 × 10^–3 m³ when the temperature is 252 K. 

Calculate the pressure inside the bicycle tyre.

After the bicycle has been ridden, the temperature of the air in the tyre is 301 K. 

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

Give your answer to an appropriate number of significant figures.

An airship floating high up in the Earth's atmosphere is kept in position due to the balance of weight and buoyancy forces.

At one point in the flight, the hydrogen gas has a temperature of 8 °C at a pressure of 4.2 × 10⁵ Pa.

The mass of the hydrogen in the ship is 1224 kg.

The atomic mass of hydrogen is 1.00794 g mol⁻¹.  

Calculate the density of the hydrogen gas in the airship.

Calculate the surface area of the inside surface of the airship at this same point in the flight.

An error with the airship causes it to fall vertically from stationary to the ground. The hydrogen inside has a specific heat capacity of 14.51 J kg⁻¹ K⁻¹. Due to the nature of the error the Physicists observe that the pressure, volume and specific heat capacity of the hydrogen within the airship remain constant but the temperature changes as it falls.  

The mass of the material in the airship is 7320 kg and it hits the ground with a velocity of 200 m s⁻¹. 

Calculate the temperature of the hydrogen in the airship at the point just before it hits the ground. Give your answer to as many significant figures as required.

(i) Explain what happened to the airship in part (c)

[5]

(ii) Explain why there has to be a problem with the data obtained from the airship's fall to Earth.

[1]

A gas syringe is connected through a delivery tube to a conical flask which is immersed in a beaker of boiling water. The water is being heated by a constant bunsen burner flame as shown in Fig. 1.1. The syringe is frictionless so the gas pressure within the system remains equal to the atmospheric pressure 1.02 × 10³ Pa. 

 Fig 1.1

The total volume of the conical flask and delivery tube is 325 cm³, and after settling in the boiling water the gas syringe has a volume of 30 cm³.

Calculate the total number of moles contained within the system.

The bunsen burner is turned off and the whole system is placed in a freezer. The conical flask is left to cool in the beaker of water which turns into ice.  

It takes 5 minutes for all the water to stop boiling and then 3 hours to cool until the point before the water turns to ice. It then takes a further 1.5 hours for all the water to turn into ice around the conical flask.  

Sketch, on Fig. 1.2, a graph to show this process.

Fig 1.2

The mass of the ice and the water in the beaker is 240 times the mass of the gas in the system. 

Calculate the root mean square speed (r.m.s.) of the particles in the system when the conical flask is surrounded by ice in the beaker in the freezer.  

• Specific latent heat of vapourisation = 2160 kJ kg⁻¹

• Specific latent heat of fusion = 324 kJ kg⁻¹

• Specific heat capacity of water = 4084 J kg⁻¹ °C⁻¹