Ideal gases and the gas laws

In an investigation of Boyle’s law, a sealed syringe of cross-sectional area 1.0 × 10⁻⁴ m² contains 12 cm³ of gas. The gas is initially at atmospheric pressure.

A weight of 2.5 N is then suspended from the plunger of the syringe as shown in the diagram.

The temperature of the gas in the syringe and the atmospheric pressure are constant.

What is the new volume of gas in the syringe?

The graph shows the variation of pressure with volume for a constant mass of a gas.

The temperature of the gas:

The graph shows how the pressure of a fixed mass of gas varies with temperature. The gas has a constant volume.

Which expression gives a value for absolute zero in °C ?

A room contains dry air at a temperature of 20.0 °C and a pressure of 105 kPa.

Show that the amount of air in each cubic metre is about 40 mol.

The density of the dry air is 1.25 kg m⁻³.

Calculate c_rms for the air molecules. Give your answer to an appropriate number of significant figures.

Calculate, in K, the change of temperature that will double c_rms for the air molecules.

A room contains moist air at a temperature of 20 °C. A dehumidifier cools and then condenses water vapour from the moist air. The final temperature of the liquid water that collects in the dehumidifier is 10 °C. Drier air leaves the dehumidifier at a temperature of 20 °C.

Table 1 compares the air flowing into and out from the dehumidifier.

In one hour, a volume of 960 m³ of air flows through the dehumidifier. Assume that the density of the air remains constant at 1.25 kg m⁻³.

Determine how much heat energy is removed in one hour from the water vapour by the dehumidifier.

specific heat capacity of water vapour = 1860 J kg⁻¹ K⁻¹ specific latent heat of vaporisation of water = 2.3 × 10⁶ J kg⁻¹

In which process is work done by an ideal gas?

An ideal gas is enclosed in an insulated container with a small electric heater. The initial temperature of the gas is 300 K. The product of pressure and volume is 5000 J. The gas expands at constant pressure and does 1660 J of work.

What is the final temperature of the gas?

Figure 2 shows a wheel used in motorsport. A rubber tyre is fitted around a cylindrical metal rim. The tyre is filled with a gas. The dimensions shown in Figure 2 are for the volume of the gas in the tyre.

Assume that this volume remains constant throughout this question.

The mass of the wheel is measured when the gas in the tyre is at a pressure

More of the same gas is added to the tyre and the mass of the wheel is measured again.

Table 1 shows the pressure in the tyre and the mass of the wheel before and after the addition of the extra gas. The gas is kept at a constant temperature of 100 °C.

Determine, in kg mol⁻¹, the molar mass of the gas.

Motorsport regulations specify a minimum amount of gas in the tyre.

The amount of gas in the tyre is checked by measuring the pressure before the wheel is put onto the car. The regulations also specify a maximum temperature for the tyre when making this measurement.

Explain why a maximum temperature is specified.

Which can lead to a value for the absolute zero of temperature?

In the kinetic theory model, it is assumed that there are many identical particles moving at random. State two other assumptions made in deriving the equation pV = Nm (c_rms)².

Explain why molecules of a gas exert a force on the walls of a container. Refer to Newton’s laws of motion in your answer.

A sealed flask of volume 0.35 m³ contains an ideal gas at a pressure of 220 kPa. The mean kinetic energy of the gas molecules is 6.7 10⁻²¹ J.

Calculate the amount of gas in the container.

Figure 2 shows the variation of pressure with volume for a fixed mass of an ideal gas at constant absolute temperature T.

Draw, on Figure 2, the graph for the same gas at temperature 2 T.

An ideal gas, initially at 300 K, is compressed to half its original volume. It is then cooled at constant volume until the pressure is restored to its initial value.

What is the final temperature of the gas?

A fixed volume of an ideal gas is heated.

Which row gives quantities that double when the kelvin temperature of the gas doubles?

Figure 9 shows air trapped in a vertical cylinder by a valve and a piston P. The valve remains closed throughout the experiment.

A mass is placed on top of P. P moves downwards and the volume of the trapped air decreases. There are no air leaks and there is no friction between the cylinder and P.

The vertical distance y between the end of P and the closed end of the cylinder is measured. Additional masses are used to find out how y depends on the total mass M placed on top of P. Figure 10 shows a graph of these data.

Show that y is not inversely proportional to M. Use data points from Figure 10.

Figure 10

The masses are removed and the cylinder is inverted. P moves downwards without friction before coming to rest, as shown in Figure 11.

Explain why P does not fall out of the cylinder unless the valve is opened.

The mass of P is 0.350 kg.

Deduce y when the cylinder is in the inverted position shown in Figure 11.

Draw a line of best fit on Figure 10 to arrive at your answer.

Figure 12 shows apparatus used in schools to investigate Boyle’s law.

A fixed mass of air is trapped above some coloured oil inside a glass tube, closed at the top. A pump applies pressure to the oil and the air. The trapped air is compressed and its pressure p is read from the pressure gauge.

A scale, marked in 0.2 cm³ intervals, is used to measure the volume V of the air. A student says that the reading for V shown in Figure 12 is 35.4 cm³.

State:

• the error the student has made

• the correct reading, in cm³, of the volume.

Figure 13 shows data obtained using the apparatus in Figure 12.

Explain why the gradient of the graph in Figure 13 confirms that the air obeys Boyle’s law.

The largest pressure that can be read from the pressure gauge is 3.4 × 10⁵ Pa.

Determine, using Figure 13, the volume V corresponding to this pressure.

State one property of the air that must not change during the experiment. Go on to suggest how this can be achieved.

Figure 4 and Figure 5 show apparatus used in an experiment to confirm the distribution of atom speeds in a gas at a particular temperature.

The oven contains an ideal gas kept at a constant temperature. Atoms of the gas emerge from the oven and some pass through the narrow slit S in a rapidly rotating drum. The drum is in a vacuum.

Explain why the drum must be in a vacuum.

One atom leaves the oven, enters the drum through S and travels in a straight line across the drum. In the time taken for the atom to move from S to the detector AB, the drum rotates through 45°. The atom hits the detector at point C, as shown in Figure 5.

drum diameter = distance from S to A = 0.500 m drum rotational speed = 120 revolutions per second

Show that the atom is moving at a speed of about 500 m s⁻¹.

The speed of the atom in Question 03.2 is equal to c_rms, the root mean square speed of the atoms of the gas in the oven. The molar mass of the gas is 0.209 kg mol⁻¹.

Calculate the temperature of the gas in the oven.

The oven temperature is kept constant during the experiment but the pressure in the oven decreases as atoms leave through the exit hole.

Explain, using the kinetic theory, why the pressure decreases.

The pressure of gas in the oven is initially 5.0 × 10⁴ Pa.

Calculate, in mol, the amount of gas that has emerged from the oven.

Atoms enter the drum every time S passes the exit hole. The detector darkens at the point where an atom strikes it. After a time, the detector is removed from the drum. Figure 6 shows the appearance of the detector.

A new detector is placed in the drum and the experiment is repeated with a new sample of the same gas at a higher temperature.

Describe and explain the appearance of this detector when the experiment is repeated.

A fixed mass of gas is heated at constant volume. The graph is drawn for this process.

When an ideal gas at a temperature of 27 °C is suddenly compressed to one quarter of its volume, the pressure increases by a factor of 7

What is the new temperature of the gas?

The diagram shows two flasks X and Y connected by a thin tube of negligible volume.

The flasks contain an ideal gas.

The volume of X is twice the volume of Y. When X is at a temperature of 100 K and Y is at a temperature of 400 K there is no net transfer of particles between the flasks.

X contains gas of mass m.

What is the mass of gas in Y ?

Figure 1 shows a perfectly insulated cylinder containing 0.050 kg of liquid nitrogen at a temperature of 70 K. A heater transfers energy at a constant rate of 12 W to the nitrogen. A piston maintains the pressure at 1.0 × 10⁵ Pa during the heating process.

The nitrogen is heated from 70 K and is completely turned into a gas after 890 s.

Calculate the specific heat capacity of liquid nitrogen. Give an appropriate unit for your answer.

specific latent heat of vaporisation of nitrogen = 2.0 × 10⁵ J kg⁻¹

boiling point of nitrogen = 77 K

The work done by the nitrogen in the cylinder when expanding due to a change of state is X. The energy required to change the state of the nitrogen from a liquid to a gas is Y.

Deduce which is greater, X or Y.

density of liquid nitrogen at its boiling temperature = 810 kg m⁻³

density of nitrogen gas at its boiling temperature = 3.8 kg m⁻³

State what is meant by the internal energy of a gas.

Absolute zero of temperature can be interpreted in terms of the ideal gas laws or the kinetic energy of particles in an ideal gas.

Describe these two interpretations of absolute zero of temperature.

A mixture of argon atoms and helium atoms is in a cylinder enclosed with a piston. The mixture is at a temperature of 310 K.

Calculate the root mean square speed (c_rms) of the argon atoms in the mixture.

molar mass of argon = 4.0 × 10⁻² kg mol⁻¹

Compare the mean kinetic energy of the argon atoms and the helium atoms in the mixture.

Explain, in terms of the kinetic theory model, why a pressure is exerted by the gas on the piston.

The mixture of gases in the cylinder stays the same.

Explain, using the kinetic theory model, two changes that can be made independently to reduce the pressure exerted by the gas.

ideal gas at constant volume.

What is the gradient of the graph?

Two flasks X and Y are filled with an ideal gas and are connected by a tube of negligible volume compared to that of the flasks. The volume of X is twice the volume of Y. X is held at a temperature of 150 K and Y is held at a temperature of 300 K

This question is about an experiment to estimate absolute zero.

Figures 9a to 9d show the stages in the procedure carried out by a student.

An empty flask fitted with a tube and an open valve is placed in water bath H containing hot water. The air inside the flask is allowed to come into thermal equilibrium with the water. The valve is then closed, trapping a certain volume of air, as shown in Figure 9a.

The flask is inverted and placed in water bath C in which the water is at room temperature. The air inside the flask is again allowed to come into thermal equilibrium with the water, as shown in Figure 9b.

The valve is opened and some water enters the flask, as shown in Figure 9c.

The depth of the inverted flask is adjusted until the level of water inside the flask is the same as the level in the water bath. The valve is then closed, trapping the air and the water inside the flask, as shown in Figure 9d.

Explain why the volume of the air in the flask in Figure 9c is less than the volume of the air in the flask in Figure 9d.

Explain why Charles’s Law can be applied to compare the air in the flask in Figure 9a with the air in the flask in Figure 9d.

The flask is removed from water bath C and the valve and stopper are removed. The volume of the water in the flask is V₁ The flask is then completely refilled with water and the valve and stopper replaced. The volume of the water now in the flask is V₂ The volumes V₁ and V₂ are shown by the shaded parts in Figure 10.

Explain how V₁ and V₂ can be determined.

In your answer you should

• identify a suitable measuring instrument

• explain a suitable procedure to eliminate possible systematic error.

Plot on Figure 11 points to show the volume V and the temperature θ of the air in the flask when

• the flask is as shown in Figure 9a

• the flask is as shown in Figure 9d.

The temperature of the hot water bath is 86 C

Add a best fit line to your graph in Figure 11 to show how V should vary with θ according to Charles’s Law.

Determine the value of absolute zero in C using your graph in Figure 11.

value of absolute zero =

A number of assumptions are made when explaining the behaviour of a gas using the molecular kinetic theory model.

State one assumption about the size of molecules.

Figure 1 shows how the pressure changes with volume for a fixed mass of an ideal gas.

At A the temperature of the gas is 27^oC. The gas then undergoes two changes, one from A to B and then one from B to C.

Calculate the number of gas molecules trapped in the cylinder using information from the initial situation at A.

Calculate, in K, the change in temperature of the gas during the compression that occurs between A and B.

Deduce whether the temperature of the gas changes during the compression from B to C.

Compare the work done on the gas during the change from A to B with that from B to C on Figure 1.

What is the number of molecules of CO₂ in 2.2 kg of the gas?