B.4 Thermodynamics (HL)

A fixed mass of ideal gas is compressed from state X to state Y along the straight line shown on the p–V diagram.

The first law of thermodynamics is written Q = ΔU + W. What is the value of W for this change?

Which process is impossible according to the second law of thermodynamics?

A fixed mass of a monatomic ideal gas is in state X. The diagram shows four changes from X, labelled P, Q, R and S. One change is isovolumetric, one is isobaric, one is isothermal and one is adiabatic.

For which change is the thermal energy supplied to the gas equal to the work done by the gas?

In a simple model of a polymer strand inside a drug-delivery capsule, warming the capsule increases the number of microstates available to the strand by a factor of 8. What is the change in entropy of the strand? (k_B is the Boltzmann constant.)

Which of the following processes is forbidden by the second law of thermodynamics?

An ideal gas in a thermally insulated bicycle-pump barrel is compressed rapidly by the piston. The first law is written as Q = ΔU + W, where W is the work done by the gas. Which row describes the compression?

The working gas in a heat engine is taken once round a closed clockwise loop on a pressure–volume diagram. Which statement is correct for one complete cycle?

A fixed mass of a monatomic ideal gas at pressure p and volume V expands at constant pressure to a volume 2V.

What is the thermal energy supplied to the gas?

A monatomic ideal gas expands adiabatically until its volume is 8 times the initial volume.

By what factor does the pressure decrease?

A quantity Q of thermal energy passes by conduction from a large reservoir at absolute temperature 2T to a large reservoir at absolute temperature T. The temperatures of the reservoirs do not change.

What is the total change in entropy of the two reservoirs?

A container is divided into two equal halves by an imaginary plane. It holds N molecules of an ideal gas that move freely through the whole container. Every microstate — every possible specification of which half each molecule is in — is equally probable.

What is the probability that, at a given instant, all N molecules are in the left half of the container?

Two identical samples of a monatomic ideal gas start in the same state. One sample expands isothermally and the other expands adiabatically. Both changes are reversible and both samples end with the same final volume.

Which statements are correct?

I. The final pressure is greater after the adiabatic expansion.
II. More work is done by the gas in the isothermal expansion.
III. The entropy of the gas does not change in the adiabatic expansion.

A fixed mass of ideal gas is taken once round the cycle X → Y → Z → X shown on the p–V diagram.

Which statements about one complete cycle are correct?

I. The change in internal energy of the gas is zero.
II. The net work done by the gas is −200 J.
III. The net thermal energy transferred to the gas is positive.

During one compression stroke of a bicycle pump, 35 J of work is done on the air in the pump and 12 J of thermal energy is transferred from the air to the barrel of the pump.

The first law of thermodynamics is written Q = ΔU + W. Which row gives the values of W and ΔU for the air?

0.250 mol of neon, a monatomic ideal gas, is in a thermally insulated cylinder. The gas expands and does 180 J of work on a piston. R = 8.31 J mol⁻¹ K⁻¹.

What is the change in temperature of the gas?

A box contains ten gas molecules. Each molecule is equally likely to be found in the left half or the right half of the box, and every arrangement (microstate) specifying which half each molecule is in is equally probable. At first 8 molecules are in the left half and 2 in the right half; later 5 are in each half.

Treating the positions only, what is the change in the entropy S = k_B ln Ω of the system of molecules? (k_B = 1.38 × 10⁻²³ J K⁻¹)

An inventor proposes three engines, I, II and III, each working between a hot reservoir at 227 °C and a cold reservoir at 27 °C. In each cycle each engine takes 500 J of thermal energy from the hot reservoir. The table gives the work done by each engine and the thermal energy it transfers to the cold reservoir in each cycle.

Which engines are impossible?

A Carnot engine in an orbiting power unit works between a hot reservoir at 600 K and a radiator at 300 K. A designer can either raise the hot reservoir temperature by 50 K or lower the radiator temperature by 50 K. Which statement is correct?

In a car engine, 1200 J of thermal energy flows from the engine block at 400 K to the coolant at 300 K. Both temperatures stay effectively constant. What is the total entropy change of the block and the coolant?

In an idealised engine, a monatomic ideal gas is compressed adiabatically to half its original volume. What is the ratio of final temperature to initial temperature?

A Stirling engine at the focus of a solar dish takes a fixed mass of ideal gas around a closed clockwise cycle on a pressure–volume diagram. Which statement is correct for one complete cycle?

A monatomic ideal gas in a balloon is heated slowly at constant pressure. What fraction of the energy supplied by heating goes into increasing the internal energy of the gas?

A sealed sample of argon, a monatomic ideal gas, has pressure p. It expands reversibly and adiabatically until its volume has doubled. Take 2^2/3 ≈ 1.6. What is the final pressure?

A fixed mass of helium in a cryocooler undergoes several separate processes. In which process is the entropy change of the helium zero?

A sealed, rigid steel container holds 0.50 mol of helium, a monatomic ideal gas. While standing in a laboratory, it warms by 40 K. Take R ≈ 8.3 J K⁻¹ mol⁻¹. What is the energy transferred to the helium by heating?

In a car airbag test rig, gas expands so that its pressure falls linearly from 4.0 × 10⁵ Pa to 1.0 × 10⁵ Pa. Over the same change, its volume increases from 2.0 × 10⁻³ m³ to 5.0 × 10⁻³ m³. What is the work done by the gas?

A Carnot engine operating between a hot reservoir at absolute temperature T_h and a cold reservoir has efficiency η.

The absolute temperature of the hot reservoir is doubled; the cold reservoir is unchanged. What is the new efficiency?

A monatomic ideal gas is the working substance of an engine that follows the cycle A → B → C → D → A shown. The pressure and volume at A are p₀ and V₀.

What is the efficiency of the engine?

A combined-cycle power station is modelled as two Carnot engines in series. The first engine operates between 800 K and 400 K. All the energy it rejects is absorbed by the second engine, which operates between 400 K and 200 K. What is the overall efficiency of the combination?

A rigid, thermally insulated container is divided into two equal halves. One half holds an ideal gas and the other half is a vacuum. The partition is punctured, and the gas expands to fill the whole container. Which statement is correct?

Two identical samples, X and Y, of a monatomic ideal gas start at the same pressure, volume and temperature. X expands isothermally to twice its volume. Y expands reversibly and adiabatically to twice its volume. Which compares the work done by the gases, W, and their final pressures, p?

A sample of argon, a monatomic ideal gas, is enclosed in a vertical cylinder by a piston of mass 2.00 kg and cross-sectional area 5.00 × 10⁻³ m². Atmospheric pressure is 1.01 × 10⁵ Pa. A heater inside the cylinder supplies thermal energy Q, measured with a joulemeter, and the temperature rise ΔT is recorded. In experiment 1 the piston is clamped (constant volume). In experiment 2 it moves freely without friction (constant pressure); for ΔT = 20 K it rises by 12.6 cm. The graph shows Q against ΔT with the lines of best fit.

A light piston of mass m = 16.5 g fits closely in a vertical glass tube of internal diameter 16.0 mm on top of a flask of volume V filled with argon, a monatomic gas. When displaced, the piston performs simple harmonic oscillations. The compressions are rapid, so the gas is treated as adiabatic, and the period T is then

T = 2π √(mV / (γpA²))

where p = 1.02 × 10⁵ Pa is the gas pressure and A the cross-sectional area of the tube. The student times 20 oscillations with a hand-held stopwatch for flasks of different volume. The graph shows T² against V with the line of best fit.

A small model heat engine runs between a hot reservoir kept at T_h = 368 K and a cold reservoir whose temperature T_c is set with water baths. The engine winds a string onto a drum and raises a load at constant speed. For each value of T_c the student measures the thermal energy taken from the hot reservoir with an electrical heater and joulemeter, and the work done on the load. The graph shows the efficiency η of the engine against T_c with the line of best fit, extended to η = 0.

Two identical funnels are filled with crushed ice at 0 °C and stand in a laboratory whose air is at 20 °C. Funnel H contains an immersed electrical heater of power 60.0 W; funnel C has no heater and acts as a control. The mass m of melt water collected from each funnel is recorded at times t after the heater is switched on. While ice remains, the ice–water mixture in each funnel stays at 273 K, and the laboratory air may be treated as a reservoir at a constant 293 K.

A geothermal binary-cycle power plant takes thermal energy from underground brine at 180 °C. It rejects energy to air-cooled condensers at 25 °C. The plant delivers a useful output power of 2.0 MW with an actual efficiency of 0.21.

An office chair contains a sealed gas spring. When a person sits down, the gas is compressed. During this compression, 45 J of work is done on the gas, and 12 J is transferred from the gas to the surroundings by heating. Assume the gas behaves as an ideal gas.

The graph is the indicator diagram (pressure p against volume V) of the gas in one cylinder of a car engine during one cycle. The cycle is traversed clockwise.

A rigid, thermally insulated container holds 0.500 mol of argon, a monatomic ideal gas, at 290 K. The volume of the container is 1.20 × 10⁻² m³. A paddle wheel inside the container is turned by a cord attached to a 2.0 kg mass, which falls at a small constant speed through a height of 0.90 m. The mass is wound back up and the fall is repeated until it has fallen 12 times. Neglect friction in the pulley and the heat capacity of the container and paddle.

A 350 W immersion heater warms 0.50 kg of water in a laboratory sterilising bath. Assume all the energy from the heater goes into the water. The specific heat capacity of water is 4200 J kg⁻¹ K⁻¹. The table shows the temperature of the water at intervals.

time / s: 0, 60, 120, 180, 240
temperature / °C: 20.0, 30.0, 40.0, 50.0, 60.0

A rubber band is modelled as a chain of 10 identical links. Each link points either left or right along the band. The length of the band is proportional to (number of links pointing right − number of links pointing left). The band has its maximum length when all 10 links point right. Its length is zero when 5 links point right and 5 point left. It can be shown that the zero-length macrostate has 252 microstates.
(k_B = 1.38×10⁻²³ J K⁻¹)

During resurfacing, an ice rink's refrigeration plant freezes 1500 kg of water at 0 °C. The plant rejects energy to the outside air at 288 K. The specific latent heat of fusion of water is 3.34×10⁵ J kg⁻¹.

A coastal power station delivers 500 MW of electrical power with an overall efficiency of 0.38. Steam enters its turbines at 820 K, and waste energy is rejected to sea water at 290 K. Treat both as reservoirs at constant temperature.

The bellows of a hospital ventilator compress a fixed mass of air slowly. The air can be treated as an ideal gas. The table gives measurements taken during the compression.

V / 10⁻⁴ m³: 5.0, 4.0, 3.0, 2.0
p / kPa: 100, 125, 167, 250

In a hypersonic wind-tunnel facility, a free piston rapidly compresses helium, a monatomic ideal gas. The compression can be treated as adiabatic. Initially, the helium has pressure 1.0 × 10⁵ Pa, volume 2.4 × 10⁻³ m³ and temperature 300 K. It is compressed to a volume of 3.0 × 10⁻⁴ m³.

Engineers testing the thermal control system of a satellite use a cylinder containing 0.80 mol of argon, a monatomic ideal gas. A frictionless piston keeps the gas at constant pressure when required. Assume that all the energy from any heater used is transferred to the gas.

In a closed-cycle power unit for a spacecraft, a fixed amount of helium, a monatomic ideal gas, is taken round the cycle ABCDA shown on the graph.

A → B: adiabatic compression; B → C: heating at a constant pressure of 8.0 × 10⁵ Pa; C → D: adiabatic expansion; D → A: cooling at a constant pressure of 2.0 × 10⁵ Pa. The temperature at A is 300 K and the temperature at C is 1100 K.

An ocean thermal energy conversion (OTEC) plant runs a heat engine between warm surface sea water and cold sea water pumped up from a depth of 1000 m. The graph shows how the temperature θ of the sea varies with depth z at the site. Specific heat capacity of sea water = 4.2 × 10³ J kg⁻¹ K⁻¹.

In a light-gas gun used to test materials, 2.0 × 10⁻⁴ m³ of helium, a monatomic ideal gas, at a pressure of 5.0 × 10⁶ Pa and a temperature of 300 K is released behind a projectile of mass 10 g. The projectile starts from rest in a barrel of cross-sectional area 1.0 × 10⁻⁴ m² and length 1.20 m. The expansion of the helium is so rapid that it can be treated as adiabatic. Friction is negligible and atmospheric pressure is 1.01 × 10⁵ Pa.

A space-station module of volume 12.0 m³ contains air at 1.00 × 10⁵ Pa and 290 K. It is joined through a closed valve to an empty (evacuated) module of volume 6.0 m³. Both modules have rigid, thermally insulating walls. The valve is opened and the air spreads into both modules. Treat the air as an ideal gas.

In a simple model of the microstates, the space available to the gas is divided into a very large number of identical small cells. Each molecule can occupy any cell, independently of the other molecules, and all arrangements are equally probable.

A monatomic ideal gas is enclosed in a horizontal cylinder by a frictionless piston of area A = 2.0 × 10⁻³ m². The piston is attached to a spring whose other end is fixed; at A the spring has its natural length. The atmosphere outside the piston is at p₀ = 1.0 × 10⁵ Pa.

At A the gas has volume V₀ = 1.0 × 10⁻³ m³, pressure p₀ and temperature 300 K. The gas is heated slowly until it reaches state B. The graph shows the pressure p against the volume V during the heating.

A radioisotope power source on a space probe contains plutonium-238, which decays by alpha emission. Each decay releases 5.59 MeV, all of which becomes thermal energy in the source. The graph shows how the thermal power P of the source is predicted to vary over the first 40 years. The mass of a plutonium-238 atom is 238 u.

A copper bar of length 0.250 m and cross-sectional area 4.0 × 10⁻⁴ m² is lagged along its length. One end is in boiling water at 373 K and the other end is in a mixture of ice and water at 273 K. After a short time the temperature gradient along the bar is uniform and steady. The boiling water and the ice–water mixture stay at constant temperatures and may be treated as thermal reservoirs. Thermal conductivity of copper = 390 W m⁻¹ K⁻¹; specific latent heat of fusion of ice = 3.34 × 10⁵ J kg⁻¹.

An ideal Stirling engine uses 0.150 mol of helium, a monatomic ideal gas. Its cycle ABCDA has four stages:

A → B: heating at a constant volume of 1.00 × 10⁻³ m³ from 320 K to 720 K;
B → C: isothermal expansion at 720 K to 2.50 × 10⁻³ m³;
C → D: cooling at constant volume to 320 K;
D → A: isothermal compression at 320 K back to 1.00 × 10⁻³ m³.

The work done by an ideal gas in an isothermal change is W = nRT ln(V_f/V_i). R = 8.31 J mol⁻¹ K⁻¹.

In an expansion cloud chamber, argon mixed with a small amount of alcohol vapour is suddenly expanded by a piston. The argon cools, the vapour becomes supersaturated, and charged particles leave visible tracks. The argon is initially at 1.00 × 10⁵ Pa and 293 K and occupies 8.00 × 10⁻⁴ m³. It is expanded rapidly to 1.00 × 10⁻³ m³. Treat the argon as a monatomic ideal gas and ignore the vapour.

An optical table of mass 48 kg rests on four identical gas springs that isolate it from vibrations of the floor. Each spring is a vertical cylinder, closed at the bottom, containing argon (a monatomic ideal gas) beneath a frictionless piston of area 3.0 × 10⁻³ m² and negligible mass. The table rests on the four pistons, and the atmosphere, at 1.01 × 10⁵ Pa, acts on the top of each piston.

In equilibrium the column of argon in each cylinder is 0.080 m high and the argon is at 293 K.

A prototype engine for a Mars rover takes a fixed mass of monatomic ideal gas through a cycle of four stages:
A → B: expansion at a constant pressure of 3.0×10⁵ Pa, from 2.0×10⁻³ m³ to 5.0×10⁻³ m³
B → C: cooling at constant volume until the pressure is 1.0×10⁵ Pa
C → D: compression at a constant pressure of 1.0×10⁵ Pa, back to 2.0×10⁻³ m³
D → A: heating at constant volume, back to state A

A deep-space probe uses a Carnot engine with a monatomic ideal gas as its working substance. A radioisotope source holds the hot reservoir at 900 K, and a radiator holds the cold reservoir at 300 K. In each cycle the gas absorbs 1800 J during its isothermal expansion.