Kinetic theory of gases

One of these assumptions is that particle motion is random.

State what is meant by random particle motion.

State two other assumptions required for this derivation.

When the temperature of a fixed mass of an ideal gas increases at constant pressure, the gas must expand.

Explain this expansion using the kinetic theory model.

Helium behaves as an ideal gas. The temperature of a sample of helium gas is increased from 27 °C to 77 °C.

Calculate the change in the average kinetic energy of a helium atom in the sample.

When an ideal gas is heated at constant volume, all of the energy supplied by heating is used to increase the average kinetic energy of the particles.

When water vapour is heated at constant volume, its specific heat capacity

Deduce whether water vapour behaves as an ideal gas when heated at constant volume.

mass of a water molecule = 3.0 × 10⁻²⁶ kg

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

Three molecules have speeds 2.00 v, 4.00 v and 5.00 v.

What is the c_rms speed of these molecules?

State what is meant by the internal energy of an ideal gas.

Figure 1 shows a single gas particle P of an ideal gas inside a hollow cube.

The cube has side length l and volume V. P has mass m and is travelling at a velocity c perpendicular to side W.

Explain why P has a change in momentum of −2 mc during one collision with W.

P collides repeatedly with W. Show that the frequency f of collisions is .

Deduce an expression, in terms of m, c and V, for the contribution of P to the pressure exerted on W. Refer to appropriate Newton’s laws of motion.

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.

Carbon is used as the moderator in some thermal nuclear reactors.

Identify one other material commonly used as a moderator.

State two benefits of slowing down the neutrons released during fission.

The collision of a neutron with the nucleus of a moderator atom is modelled using two gliders on a horizontal frictionless air track.

In Figures 13 and 14 the glider N of mass m_N represents the neutron and the glider M of mass m_M represents the moderator nucleus.

Figure 13 shows glider N travelling with initial speed u towards the stationary glider M.

The gliders collide. N rebounds with speed v as shown in Figure 14.

Figure 15 shows the variation of the ratio with the ratio^M.

Show that when is 12, N loses about 30% of its initial kinetic energy in the collision.

In a reactor, the speed of a fast-moving neutron is reduced by a series of y random collisions with carbon-12 nuclei.

The final kinetic energy E_f of the neutron is

where E₀ is the initial kinetic energy of the neutron and b = 0.73

A thermal neutron has kinetic energy equivalent to that of the average particle of an ideal gas with a temperature of 350 K.

One neutron has an initial kinetic energy of 1.0 MeV.

Calculate the minimum value of y required so that this neutron becomes a thermal neutron.

Explain, with reference to Figure 15, why elements with a small nucleon number are preferred as moderator materials.

Section B

Each of Questions 08 to 32 is followed by four responses, A, B, C and D.

For each question select the best response.

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.

A fixed volume of an ideal gas is heated.

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

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 gas occupies a volume V. Its particles have a root mean square speed (c_rms) of u. The gas is compressed at constant temperature to a volume 0.5 V.

What is the root mean square speed of the gas particles after compression?

Three particles are travelling in the same plane with velocities as shown in the vector

What is the root mean square speed of the particles?

A sample P of an ideal gas contains 1 mol at an absolute temperature T. A second sample Q of an ideal gas contains mol at an absolute temperature 2T.

The total molecular kinetic energy of P is E.

What is the total molecular kinetic energy of Q ?

An ideal gas is contained in a cubical box of side length a. The gas has N molecules each of mass m.

What is the pressure exerted by the gas on the walls of the box?

Which statement is true about an experiment where Brownian motion is demonstrated using smoke particles in air?

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.

Brownian motion

Which is not an assumption about gas particles in the kinetic theory model for a gas?

The average mass of an air molecule is 4.8 × 10^–26 kg

What is the mean square speed of an air molecule at 750 K ?

A transparent illuminated box contains small smoke particles and air. The smoke particles are observed to move randomly when viewed through a microscope.

What is the cause of this observation of Brownian motion?

The diagram shows a gas particle about to collide elastically with a wall.

Which diagram shows the correct change in momentum mv that occurs during the collision?

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.