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. 

Temperature / °C	Volume of gas / cm3
0	55
100	76

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. 

	Kinetic Energy /J	Temperature °C
A.	6 × 10−21	3000
B.	5.4 × 10−20	3600
C.	6 × 10−21	3327
D.	5.4 × 10−20	3327

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?

A fixed mass of an ideal gas undergoes an isovolumetric change. This leads to a decrease in the pressure of the gas.

Which row describes the change in internal energy of the gas and the direction of transfer of thermal energy?

	Internal Energy	Direction of thermal energy transfer
A.	greater	to the gas
B.	greater	from the gas
C.	less	to the gas
D.	less	from the gas

A fixed mass of an ideal gas is initially at an absolute temperature, T and occupies a volume, V. 

The gas is adjusted so that the final pressure of the gas is twice the magnitude of the initial pressure.

Which row of the table below, A to D, gives the correct expressions for the final volume and temperature of the gas?

	Final volume of the gas	Final temperature of the gas
A.
B.
C.
D.

The graphs, A to D, show the variation of pV with θ for one mole of two different gases, nitrogen and chlorine, where p is the pressure of the gas, V is the volume of the gas and θ is the temperature in °C.

Which graph shows the correct variation of pV with θ for 1 mole of each gas?

X and Y are two flasks containing an ideal gas that are connected by a tube that has negligible volume compared with the volume of each flask. The volume of X is twice the volume of Y.

The temperature of the gas in X is kept at 200 K and the temperature of the gas in Y is kept at 400 K.

If the mass of the gas in X is M_X and the mass of the gas in Y is M_Y, what is the ratio ?

An ideal gas of fixed mass has a pressure, p, and volume, V. The graph shows the variation of  with V at a constant temperature.

Which line represents the same gas if the thermodynamic temperature and the amount of gas are doubled?

Given the following data:

• Mass of oxygen = 16 g
• Temperature = 27 °C
• Length of side of cubic container = 10 cm
• Mass of one mole of oxygen = 32 g

What is the pressure of oxygen in a cubic container?

A sealed cylinder of radius r and length L contains an ideal gas. The gas is at pressure p and temperature T. The radius of the cylinder is halved, and the temperature is altered to maintain the same pressure.

Which of these is an expression for the new temperature of the gas?

An ideal gas of N molecules is maintained at constant volume, V. The graph shows how temperature T varies with pressure p.

What is the gradient of the graph?