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Figure 1 shows a type...

Figure 1 shows a type of apparatus for investigating the speed distribution f (v) of molecules in a gas. Molecules emerge into a vacuum chamber from an oven held at a constant temperature T, and are collimated by slits into a parallel beam directed towards a pair of discs rotating at a common angular speed ω, which may be varied. The discs are a fixed distance L apart (L = 262 mm) and contain narrow notches, with the second notch offset from the first by a fixed angle θ = 30° as shown. If a molecule is moving at the correct speed v, it will pass through the notches in both discs to be collected at the detector, which records the rate of arrival of molecules.

￼￼￼￼￼￼slits

molecule

ω

detector

θ

L

￼￼￼￼￼￼￼￼￼￼Figure 1

vacuum chamber

For use with Question 1.

(a) Show that a molecule that passes the first slit will reach the detector if its

motor

oven

speed is v = ωL/θ. (3 marks)

(b) The oven contains aluminium at 9.00 × 102 °C, which emerges as a monatomic beam. Calculate the speed vmp and the corresponding rotation speed ω at which the largest rate of arrival for aluminium atoms is observed.

(The molar mass of aluminium is 0.0270 kg mol−1.) (10 marks)

(c) The oven remains at 9.00 × 102 °C, but its contents are changed to silver, also emitted as a monatomic beam. Find the rotation speed ω at which the largest rate of arrival of silver atoms is observed.

(The molar mass of silver is 0.108 kg mol−1.) (3 marks)

(d) By observing the rate of arrival of atoms as ω is changed, the form of f (v) can be found. On a single graph, sketch the variation of f (v) versus ω that you would observe for each metal and label the values of ω for the most probable speed vmp in both cases. Can we determine any physical properties of the metals from the area under each curve? (9 marks)

This question is worth 25 marks, which is 50% of the total marks for this assignment and is related to Chapter 3.

Figure 2 is a highly schematic diagram of an electrically powered refrigerator.

￼-ef-ige-ated

compa-tment

comp-esso-

pump

expansion valve

ci-culating fluid

pipes behind

-ef-ige-ato-

￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼Figure 2

Schematic diagram of a refrigerator for Question 2.

In this refrigerator, unlike the one described in Chapter 3 of CPM, the working fluid remains a gas throughout the cycle. Its working may be described in a greatly simplified form as a sequence of four steps as follows:

1

2

3

4 (a)

(b)

(c)

(d)

The fluid passes through the refrigerated compartment, where its temperature rises. During this step its volume remains constant.

The fluid passes through the compressor pump, where it is compressed adiabatically.

The fluid passes through the pipes behind the refrigerator, where its temperature falls. During this step its volume remains constant.

The fluid passes through the expansion valve, where it expands adiabatically. Show these four steps on a P–V diagram, labelling the steps clearly.

(i) In which step or steps is work done on the fluid?

(ii) In which step or steps is heat transferred to the fluid? (iii) In which step or steps is work done by the fluid?

(iv) In which step or steps is heat transferred from the fluid?

(7 marks)

(4 marks)

The temperature inside the refrigerator is 2.0 °C. It stands in a kitchen whose temperature is 22 °C. After a period of an hour, 95 kJ of heat are transferred from the interior of the refrigerator to the kitchen.

As a result of this heat transfer, what is the entropy change of (i) the contents of the refrigerator, and (ii) the kitchen? (Assume that the heat transfer is reversible.) (7 marks)

State the principle of entropy increase. Are your answers to part (c) compatible with this principle? If not, suggest an explanation (one or two

sentences).

(7 marks)

3

￼￼￼￼￼￼slits

molecule

ω

detector

θ

L

￼￼￼￼￼￼￼￼￼￼Figure 1

vacuum chamber

For use with Question 1.

(a) Show that a molecule that passes the first slit will reach the detector if its

motor

oven

speed is v = ωL/θ. (3 marks)

(b) The oven contains aluminium at 9.00 × 102 °C, which emerges as a monatomic beam. Calculate the speed vmp and the corresponding rotation speed ω at which the largest rate of arrival for aluminium atoms is observed.

(The molar mass of aluminium is 0.0270 kg mol−1.) (10 marks)

(c) The oven remains at 9.00 × 102 °C, but its contents are changed to silver, also emitted as a monatomic beam. Find the rotation speed ω at which the largest rate of arrival of silver atoms is observed.

(The molar mass of silver is 0.108 kg mol−1.) (3 marks)

(d) By observing the rate of arrival of atoms as ω is changed, the form of f (v) can be found. On a single graph, sketch the variation of f (v) versus ω that you would observe for each metal and label the values of ω for the most probable speed vmp in both cases. Can we determine any physical properties of the metals from the area under each curve? (9 marks)

This question is worth 25 marks, which is 50% of the total marks for this assignment and is related to Chapter 3.

Figure 2 is a highly schematic diagram of an electrically powered refrigerator.

￼-ef-ige-ated

compa-tment

comp-esso-

pump

expansion valve

ci-culating fluid

pipes behind

-ef-ige-ato-

￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼￼Figure 2

Schematic diagram of a refrigerator for Question 2.

In this refrigerator, unlike the one described in Chapter 3 of CPM, the working fluid remains a gas throughout the cycle. Its working may be described in a greatly simplified form as a sequence of four steps as follows:

1

2

3

4 (a)

(b)

(c)

(d)

The fluid passes through the refrigerated compartment, where its temperature rises. During this step its volume remains constant.

The fluid passes through the compressor pump, where it is compressed adiabatically.

The fluid passes through the pipes behind the refrigerator, where its temperature falls. During this step its volume remains constant.

The fluid passes through the expansion valve, where it expands adiabatically. Show these four steps on a P–V diagram, labelling the steps clearly.

(i) In which step or steps is work done on the fluid?

(ii) In which step or steps is heat transferred to the fluid? (iii) In which step or steps is work done by the fluid?

(iv) In which step or steps is heat transferred from the fluid?

(7 marks)

(4 marks)

The temperature inside the refrigerator is 2.0 °C. It stands in a kitchen whose temperature is 22 °C. After a period of an hour, 95 kJ of heat are transferred from the interior of the refrigerator to the kitchen.

As a result of this heat transfer, what is the entropy change of (i) the contents of the refrigerator, and (ii) the kitchen? (Assume that the heat transfer is reversible.) (7 marks)

State the principle of entropy increase. Are your answers to part (c) compatible with this principle? If not, suggest an explanation (one or two

sentences).

(7 marks)

3

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