Ideal gases and the gas laws

A hot air balloon consists of a fabric envelope, heaters and a basket, as shown.

When the balloon is set up, the envelope is partly filled with air at 20°C. The air is then heated to 120°C and expands to fill the envelope and becomes less dense.

The air pressure inside the envelope is always equal to the air pressure outside the envelope because the envelope is open at the bottom.

The balloon takes off when the upthrust is more than the total weight of the balloon, the air in the envelope and the passengers.

Deduce whether the balloon can take off.

volume of air at 120°C in inflated envelope = 2800 m³ density of air at 20°C = 1.2 kg m^–3 mass of balloon = 380 kg mass of passengers = 340 kg upthrust when the envelope is full = 33 000 N

State one assumption of the kinetic theory of gases.

Derive an equation to show that, for a gas at temperature T, the mean kinetic energy of the molecules = kT

Calculate the root-mean-square speed of nitrogen molecules at a temperature of 120°C.

mass of nitrogen molecule = 28 u

Helium is available in small metal canisters. The helium can be used to fill party balloons.

A full canister contains helium gas at a temperature of 18.5C and a pressure of 1.65×10⁷Pa. The canister is approximately spherical, with a radius of 4.3610^–2m.

Calculate the mass of helium gas in the full canister.

mass of 6.02×10²³ atoms of helium = 4.00×10^–3kg

Student X and Student Y discuss the weight of the canister and its contents after a number of balloons have been filled.

Student X suggests that the weight will have increased, because the upthrust exerted on the canister by the helium will be reduced.

Student Y suggests that the weight will have decreased, because helium has been released from the canister.

Assess which student’s suggestion is correct.

The fuel used in a camping stove is butane, which is stored in a canister as shown.

Some of the butane in the canister is in a liquid state, and some is a gas.

When the stove uses the butane gas, some of the liquid butane evaporates.

Explain why the temperature of the canister decreases when the stove is used.

The pressure inside the canister is 220 kPa and the temperature of the gas is 21C.

The canister is in the shape of a cylinder of length 0.23 m and radius 0.11 m.

Calculate the number of molecules of butane gas in the canister.

Assume the volume of liquid butane inside the canister is negligible.

Calculate the r.m.s. speed of the molecules of butane gas.

mass of butane molecule = 9.6×10^–26kg

A fine-beam tube is used for investigating properties of electrons.

An electron beam is produced inside a spherical glass bulb. The bulb contains neon gas at a very low pressure.

The neon gas is at a pressure of 1.25 Pa and a temperature of 25 °C.

Calculate the number N of neon atoms inside the bulb.

bulb diameter = 16.0 cm

*(b) Interactions between electrons and the neon atoms in the tube make the beam visible. Part of the spectrum of visible light produced by these interactions is shown.

Explain the process that results in the emission of this spectrum. Your answer should include reference to energy levels in atoms.

In 2011, a tsunami was caused by a massive earthquake centred some distance off the coast of Japan. The tsunami caused a cooling system failure at the Fukushima Nuclear Power Plant. This resulted in a nuclear meltdown and radioactive materials were released into the surroundings.

A reservoir beside one of the reactor buildings contained a large volume of water. In 2013, this water was found to have an extremely high concentration of caesium-137. Caesium-137 is a radioactive isotope of caesium.

Complete the nuclear equation for the decay of caesium-137.

An activity of 2.35 × 10¹² Bq per m³ of water in the reservoir was measured. It is suggested that a safe level for the activity of all water in the reservoir would be 100 Bq.

Calculate the time in years for the caesium-137 to decay to a safe level.

volume of water in reservoir = 5000 m³ half-life of caesium-137 = 30 years

The most common radionuclide amongst the fission products in the fuel was iodine-131, which decays with a half-life of 8.0 days to form a stable isotope of the gas xenon.

Deduce whether enough xenon would have collected in 32 days to exert a pressure of 1.0 10⁵ Pa in a volume of 450 m³. Assume that no gas escapes.

temperature = 20 °C initial number of iodine nuclei = 1.25 × 10²⁸

Buildings in nearby Tohoku University suffered structural damage during the 2011 earthquake.

The graph shows how the acceleration of one of the buildings, measured on the 9th floor, varied with time during the earthquake.

(Source: https://www.sciencedirect.com/science/article/pii/S0038080612001035)

At the time it was reported that during the earthquake the 9th floor of the building displaced by more than 30 cm from its normal position.

Assess the accuracy of this report.

The photographs show a wooden pop gun before and after the cork is popped.

The diagram shows a cross-section through the pop gun.

Initially the piston is at the right-hand end of the cylinder, as shown. Then the cork is pushed into the other end of the cylinder.

When the handle is pushed in, the pressure of the air in the cylinder increases. This exerts an additional force on the cork.

Once the additional force is sufficient to overcome the frictional force between the cork and the cylinder, the cork is pushed out.

Show that the pressure of the air in the cylinder must be about 2 10⁵Pa in order to push the cork out.

maximum frictional force = 8.8 N cross-sectional area of cork = 9.2 10⁻⁵m² atmospheric pressure = 1.0 10⁵Pa

Calculate the temperature of the gas in the cylinder at the instant the cork is expelled.

volume of air in the cylinder with the handle pulled out = 1.1 10⁻⁵m³ volume of air in the cylinder at the moment the cork is pushed out = 6.7 10⁻⁶m³ atmospheric pressure = 1.0 10⁵Pa initial temperature of air = 19 °C

The formulae sheet for this paper includes the equation

Calculate the root mean square speed of the molecules of air in the cylinder before the handle is pushed in.

average mass of molecule of air = 4.8 10⁻²⁶kg temperature of air = 19 °C

At the Culham Centre for Fusion Energy (CCFE) experiments are carried out to investigate nuclear fusion and the properties of plasmas. A plasma consists of ionised gas, containing positive ions and electrons.

In a fusion experiment at CCFE, ions of two isotopes of hydrogen fuse to produce helium ions and fast-moving neutrons.

Show that a single fusion reaction releases about 3 10⁻¹²J of energy.

mass of₁²H = 2.013553 u mass of₁³H = 3.015501 u mass of⁴₂He = 4.001506 u mass of₀¹n = 1.008665 u

Fusion occurs naturally in the core of stars.

Explain why very high densities of matter and very high temperatures are needed to bring about and maintain nuclear fusion in stars.

In a plasma experiment 5.0 mg of deuterium, an isotope of hydrogen, occupies a volume of 98 m³. The temperature of deuterium is raised to 1.3 10⁸K. In this experiment, the deuterium behaves as an ideal gas.

Calculate the pressure due to the deuterium ions.

mass of deuterium ion = 3.3 10⁻²⁷kg

Calculate the root mean square speed of the deuterium ions at this temperature.

The temperature of the plasma is monitored using the Doppler effect. Light from a laser is directed into the plasma and the wavelength of the light reflected is measured.

The Doppler shift observed when light is reflected by a deuterium ion is twice the Doppler shift that would be observed for a source of light moving at the same speed as the deuterium ion.

Calculate the maximum wavelength of light that would be detected after reflection from a deuterium ion moving at 1.5 × 10⁶m s⁻¹.

wavelength of laser light = 1064 nm

At the Culham Centre for Fusion Energy (CCFE) experiments are carried out to investigate nuclear fusion and the properties of plasmas. A plasma consists of ionised gas, containing positive ions and electrons.

Fusion occurs naturally in the core of stars.

Explain why very high densities of matter and very high temperatures are needed to bring about and maintain nuclear fusion in stars.

In a plasma experiment 5.0 mg of deuterium, an isotope of hydrogen, occupies a volume of 98 m³. The temperature of deuterium is raised to 1.3 10⁸K. In this experiment, the deuterium behaves as an ideal gas.

Calculate the pressure due to the deuterium ions.

mass of deuterium ion = 3.3 10⁻²⁷kg

Calculate the root mean square speed of the deuterium ions at this temperature.

The temperature of the plasma is monitored using the Doppler effect. Light from a laser is directed into the plasma and the wavelength of the light reflected is measured.

The Doppler shift observed when light is reflected by a deuterium ion is twice the Doppler shift that would be observed for a source of light moving at the same speed as the deuterium ion.

Calculate the maximum wavelength of light that would be detected after reflection from a deuterium ion moving at 1.5 × 10⁶m s⁻¹.

wavelength of laser light = 1064 nm

A student is investigating a ‘Cartesian diver’.

The diver is made from a plastic pipette. When placed in a plastic bottle full of water the diver rises to the top and touches the lid.

Show that the downward force of the lid on the diver is about 0.02 N.

volume of diver = 8.010^–6m³

mass of diver = 0.0059 kg

density of water = 1.010³kg m^–3

When the pressure is increased by squeezing the bottle, water is forced into the diver increasing its weight. The diver sinks, then remains at rest in the position shown.

The volume of air in the empty pipette in part (a) was 8.010^–6m³.

Show that the volume now occupied by the air is about 610^–6m³.

The pressure of the air in the empty pipette in part (a) was 1.0110⁵Pa.

Calculate the pressure of the air in part (b).