p-V diagrams

State what is meant by an isothermal change.

Figure 6 shows a p – V diagram in which a fixed mass of an ideal gas is taken through a cycle of three processes.

A → B: adiabatic change B → C: isothermal change C → A: reduction in volume at constant pressure

The maximum temperature of the gas in the cycle is 536 K and the minimum temperature is 291 K.

Calculate, in mol, the amount of gas.

Show that the work done in process C → A is about 100 J.

Complete Table 2 for each process and for the whole cycle.

Two criteria for a heat engine to achieve the maximum theoretical efficiency are:

1. the gas must always be at the same temperature as the hot source when energy is being transferred to the gas by heat transfer

2. the gas must always be at the same temperature as the cold sink when energy is being transferred from the gas by heat transfer.

The temperatures of the source and sink do not change.

Deduce whether the cycle in Figure 6 satisfies either or both of these criteria.

The overall efficiency of a heat engine is defined as

An engineer suggests designing a real engine based on the cycle shown in Figure 6.

Discuss the engineer’s suggestion.

In your answer you should:

• use Table 2 to determine the overall efficiency of a theoretical engine that uses the cycle

• compare this overall efficiency with the efficiency of an ideal engine working between the same temperatures as the cycle

• discuss with reasons the problems that would need to be overcome in making a real engine work on this cycle

• explain whether the engineer’s suggestion would lead to a useful engine.

Figure 5 shows a tool for driving nails into wood. Only part of the tool is shown.

Fuel is mixed with air in the combustion chamber and is ignited by a spark. The gas expands rapidly and drives the piston along the cylinder. The plunger attached to the piston drives the nail into the wood.

Table 2 shows the average force needed to drive nails of various lengths completely into a particular type of wood.

Figure 6 shows the variation of pressure p with volume V as the gas expands on the right-hand side of the piston when the correct nail is used.

The combustion chamber has a volume of 20 × 10⁻⁶ m³ and the piston moves through

The work done by the expanding gas is just enough to drive the correct nail completely into the wood.

Deduce which nail in Table 2 is the correct one to use in the tool.

After a nail has been used, another nail takes its place automatically. The tool can drive up to 180 nails per minute.

Discuss why the expansion cannot be isothermal.

Which row in Table 3 shows

• Process 1 in which work done is zero, and

• Process 2 in which the change in internal energy is zero?

Tick () one box.

When irregular particles are packed, air gaps are left between the particles. The true volume of a quantity of irregular particles must be determined using a method that does not include the volume of the air spaces between them.

The apparatus shown in Figure 5 is used by an agricultural engineer to measure the true volume of some grains.

The volume of air in the syringe is 1.00 × 10⁻⁴ m³. The volume of the empty container and connecting pipe is 2.80 × 10⁻⁴ m³. Grains of total true volume V are now placed in the container and the lid is screwed on. The pressure inside both the syringe and the container is 1.01 × 10⁵ Pa. The plunger is slowly pushed fully into the cylinder of the syringe, compressing the air isothermally.

Determine V.

Figure 6 shows how the pressure in the container and syringe varies with volume as the plunger is pushed in fully very slowly.

Sketch on Figure 6 the variation of pressure with volume when the plunger is pushed in fully very quickly and then left for several seconds. Assume no leakage past the plunger.

Explain why the compression of a gas can be considered to be an isothermal change when the gas is compressed very slowly.

In an ideal heat-engine cycle a fixed mass of air is taken through the following four processes.

A → B isothermal compression from an initial pressure of 1.0 × 10⁵ Pa and a volume of 9.0 × 10^–2 m³ to a pressure of 2.2 × 10⁵ Pa. The work done on the air is 7100 J.

B → C increase in volume at constant pressure to a volume of 5.9 × 10⁻² m³.

C → D isothermal expansion to a pressure of 1.0 × 10⁵ Pa and a volume of 13 × 10⁻² m³. The work done by the air is 10 300 J.

D → A reduction in volume at constant pressure to the original volume.

Figure 3 shows the cycle.

Show, by calculation, that the volume at B is 4.1 × 10⁻² m³.

The temperature of the air between A and B is 295 K.

Show that the temperature of the air at C is about 420 K.

An ideal engine based on this cycle uses a device called an economiser. The economiser stores all the energy transferred in the cooling process D → A and gives up all this energy to the air in process B → C. This means that an external source supplies energy to the air by heating only in process C → D.

Complete Table 1 to show the values of work done W and energy transfer Q in each of the four processes. Use the space below Table 1 for any calculations. Use a consistent sign convention.

Explain why W is equal to Q in process A → B and in process C → D.

It is claimed that the efficiency of this engine cycle is the same as the maximum theoretical efficiency of a heat engine operating between the same temperatures.

Deduce whether this claim is true.

Discuss one problem that would be faced by an engineer designing a real engine based on this cycle.

Figure 3 shows the p – V diagram for an idealised diesel engine cycle. In this cycle a fixed mass of air is taken through four processes 1 2 3 4 1.

Which statement about this cycle is true? Tick (✓) the correct answer.

The cycle in Figure 3 may be modified by allowing the air to continue to expand adiabatically from state 4 until it is at atmospheric pressure at state 5. Figure 4 shows the modified cycle.

The expansion stroke 3 5 is now longer than the compression stroke 1 2. Process 5 1 takes place at constant pressure.

It has been claimed that, compared to the cycle in Figure 3, the modified cycle of Figure 4 gives

A an increase in work done per cycle of 130 J B an increase in efficiency of more than 15%

Deduce whether these claims are true.

Claim A

Claim B

For the air in process 5 1 in Figure 4, ΔU = –374 J

Calculate the energy that must be removed by cooling for process 5 1.

0.060 mol of air is taken through the cycle.

Determine the maximum temperature in the cycle.

A fixed mass of nitrogen gas is sealed into the cylinder of the strut.

The gas is initially at a pressure of 1.2 × 10⁶ Pa, a volume of 9.0 × 10^–5 m³ and a temperature of 290 K.

When the lid is closed quickly the gas is compressed rapidly to a final volume of

Calculate the pressure and temperature of the gas at the end of the compression assuming the compression to be an adiabatic process.

adiabatic index γ for nitrogen = 1.4

Explain why the rapid compression of the gas can be assumed to be an adiabatic process.

When the lid is closed slowly, the compression can be assumed to be isothermal.

The gas can be compressed either isothermally or adiabatically from the same initial conditions to the same final volume.

Compare without calculation the work done in each process.