1

Engines and Carnot Cycle

1a. [1 mark]

The P–V diagram of the Carnot cycle for a monatomic ideal gas is shown.

State what is meant by an adiabatic process.

1b. [1 mark]

Identify the two isothermal processes.

2

1c. [2 marks]

The system consists of 0.150 mol of a gas initially at A. The pressure at A is 512 k Pa and the volume is

1.20 × 10–3 m3.

Determine the temperature of the gas at A.

1d. [2 marks]

The volume at B is 2.30 × 10–3 m3. Determine the pressure at B.

1e. [1 mark]

At C the volume is VC and the temperature is TC.

Show that

3

1f. [2 marks]

The volume at C is 2.90 × 10–3 m3. Calculate the temperature at C.

1g. [1 mark]

State a reason why a Carnot cycle is of little use for a practical heat engine.

2a. [2 marks]

The pressure–volume (pV) diagram shows a cycle ABCA of a heat engine. The working substance of the

engine is 0.221 mol of ideal monatomic gas.

4

At A the temperature of the gas is 295 K and the pressure of the gas is 1.10 × 105 Pa. The process from A

to B is adiabatic.

Show that the pressure at B is about 5 × 105 Pa.

2b. [1 mark]

For the process BC, calculate, in J, the work done by the gas.

5

2c. [1 mark]

For the process BC, calculate, in J, the change in the internal energy of the gas.

2d. [1 mark]

For the process BC, calculate, in J, the thermal energy transferred to the gas.

2e. [2 marks]

The process from B to C is replaced by an isothermal process in which the initial state is the same and

the final volume is 5.00 × 10–3 m3.

Explain, without any calculation, why the pressure after this change would belower if the process was

isothermal.

6

2f. [2 marks]

Determine, without any calculation, whether the net work done by the engine during one full cycle

would increase or decrease.

2g. [1 mark]

Outline why an efficiency calculation is important for an engineer designing a heat engine.

3a. [1 mark]

7

Part 2 Gas in an engine

A fixed mass of an ideal gas is used as the working substance in an engine. The graph shows the

variation with volume V of the pressure P of the fluid.

For the cycle identify, with the letter I, an isochoric (isovolumetric) change.

3b. [2 marks]

The temperature at point X is 310 K. Calculate the temperature at point Y.

3c. [5 marks]

The shaded area WXYZ is 610 J. The total thermal energy transferred out of the gas in one cycle is 1.3 kJ.

8

(i) State what is represented by the shaded area WXYZ.

(ii) Determine the efficiency of cycle WXYZ.

(iii) Explain why the total thermal energy transferred out of the gas is degraded.

3d. [2 marks]

The work done on the gas during the adiabatic compression XY is 210 J. Determine the change in

internal energy during the change from X to Y.

4a. [3 marks]

Part 2 Thermodynamic cycles

A gas undergoes a thermodynamic cycle. The P–V diagram for the cycle is shown below.

9

In the changes of state B to C and D to A, the gas behaves as an ideal gas and the changes in state are

adiabatic.

(i) State the circumstances in which the behaviour of a gas approximates to ideal gas behaviour.

(ii) State what is meant by an adiabatic change of state.

10

4b. [4 marks]

With reference to the first law of thermodynamics, explain for the change of state A to B, why energy is

transferred from the surroundings to the gas.

4c. [3 marks]

Estimate the total work done in the cycle.

11

5a. [4 marks]

Part 2 A heat engine

The piston of an engine contains a fixed mass of an ideal gas. During one cycle of the engine, the gas

undergoes the thermodynamic processes shown below.

12

(i) State what is meant by an isothermal process.

(ii) Show that process AB is isothermal.

13

5b. [1 mark]

State the nature of process BC.

5c. [2 marks]

During the cycle ABCD, the net work done by the gas is 550J. Calculate the net thermal energy absorbed

by the gas.

14

5d. [3 marks]

Explain why it is not possible for this engine, operating in this cycle, to be 100% efficient.

6a. [2 marks]

A monatomic ideal gas is confined to a cylinder with volume 2.0 x 10–3 m3. The initial pressure of the

gas is 100 kPa. The gas undergoes a three-step cycle. First, the gas pressure increases by a factor of five

under constant volume. Then, the gas expands adiabatically to its initial pressure. Finally it is

compressed at constant pressure to its initial volume.

Show that the volume of the gas at the end of the adiabatic expansion is approximately 5.3 x 10–3 m3.

15

6b. [2 marks]

Using the axes, sketch the three-step cycle.

6c. [2 marks]

The initial temperature of the gas is 290 K. Calculate the temperature of the gas at the start of the

adiabatic expansion.

16

6d. [2 marks]

Using your sketched graph in (b), identify the feature that shows that net work is done by the gas in this

three-step cycle.

7a. [1 mark]

A heat engine operates on the cycle shown in the pressure–volume diagram. The cycle consists of an

isothermal expansion AB, an isovolumetric change BC and an adiabatic compression CA. The volume at

B is double the volume at A. The gas is an ideal monatomic gas.

17

At A the pressure of the gas is 4.00 x 106 Pa, the temperature is 612 K and the volume is 1.50 x 10–4 m3.

The work done by the gas during the isothermal expansion is 416 J.

Justify why the thermal energy supplied during the expansion AB is 416 J.

7b. [2 marks]

Show that the temperature of the gas at C is 386 K.

18

7c. [2 marks]

Show that the thermal energy removed from the gas for the change BC is approximately 330 J.

7d. [2 marks]

Determine the efficiency of the heat engine.

19

7e. [3 marks]

State and explain at which point in the cycle ABCA the entropy of the gas is the largest.

Printed for Skyline High School

© International Baccalaureate Organization 2019

International Baccalaureate® - Baccalauréat International® - Bachillerato Internacional®