Expansion Processes of a Perfect Gas

Assignment:

1. Attend the laboratory class and complete the lab tests according to the supervisor’s requirement. Students will not allowed to submit their report if they do not come to the lab class.

2. Write one about 2000-2500 word report and submit it by the due time.

In the report, you need to do the following analysis.

Test A

• Draw the capacity ratio v.s. pressure graph. Discuss how well the results obtained from the lab compared with the expected results (you should find the exact heat capacity ratio yourself from library, internet or some other sources), by putting the test results and the expected results in the same graph. Explain the reasons for any difference.

The exact heat capacity ratio varies between 1.399 and 1.401 for temperature in the range of –40° to 50°. You may use 1.4 as the exact heat capacity ratio in your report.

• Discuss why the initial expansion process can be considered adiabatic?

Test B

Discuss

• How well the result obtained compared to the expected result? Give possible reasons for any difference. The exact volumes of the large and the small vessels are 22.4 and 9.1 litres, respectively.

• Comment on how the rate of change affects the temperature of the air inside the vessels.

The report will be assessed based on the following criteria.

Introduction and conclusion (10%)

This section outlines the aims of the laboratory tests and the main outcomes of the report. The purpose of the report needs to be clearly stated. In the conclusion section, the main outcomes of the laboratory test need to be summarized.

Description of the methodology (20%)

A student needs to present the methodology, describe the test procedure and explain the basic theory using their own words. It is not acceptable to copy from the lab note.

Results (40%)

This is the main part of the report. It will be assessed based on whether all relevant data are presented, how well the data are organized, how well charts and tables are presented, whether the methods for generating the data are correct and if a student show a good understanding of the fundamental theory. If a student makes fundamental mistakes, he/she will lose a significant percentage of marks.

Discussion (20%)

Based on the results obtained from the test, a student needs to discuss what are the outcomes of the tests.

Writing (10%)

The writing needs to be logical, clear, concise.

The equipment includes

A common base plate (4)

Top plate (2)

Large vessel (3) Small vessel (6)

Air pump (9)

Pressure sensors (P & V)

Sensor P measures the pressure inside the larger pressure vessel (3)

Sensor V measures the vacuum inside the smaller vacuum vessel (6) (Positive values for vacuum pressure)

Temperature sensors T1 and T2 inside each vessel

Valves

V1 – the large vessel to atmosphere

V2 – the large vessel to the small vessel (via a large bore pipe)

V3 – the small vessel to atmosphere

V4 – pump to the large vessel (pressurize the large vessel)

V5 – needle valve connecting the large vessel to the small vessel

V6 – the large vessel to the small vessel (via needle valve V5)

V7 – pump to the small vessel (the pump draws air from the small vessel)

Test A – Determination of Heat Capacity ratio (Specific heat ratio)

Theory

The heat capacity ratio can be determined experimentally using a two-step process.

1. An adiabatic reversible expansion from the initial pressure Ps to an intermediate pressure Pi.

{Ps, Vs, Ts} ? {Pi, Vi, Ti} (1)

2. A return of the temperature of its original value Ts at constant volume Vi.

{Pi, Vi, Ti} ? {Pf, Vi, Ts} (2)

The heat capacity ratio in the two-step process is

cp ? lnPs ? lnPi (3) cv lnPs ? lnPf

The cp/cv can be approximated to be the heat capacity ratio at the temperature of Ts and the pressure of (Ps+Pf)/2. Please refer Appendix A for the derivation of Eq. (3).

Procedure

1. Measure and record Patm using a barometer.

2. Close ball valves V1 and V3 and open valve V4.

3. Pressurize the large vessel by switching on the air pump. When P reaches approximately 30 kN/m2, switch off the air pump and close valve V4.

4. Wait until pressure P in the large vessel has stabilized (P will fall slightly as the vessel contents cools to room temperature).

5. Record the starting pressure Ps.

6. Open the close valve V1 very rapidly with a snap action to allow a small amount of air to escape from the vessel.

7. Record Pi. (accurate instantaneous value)

8. Allow the vessel contents to return to ambient temperature then record the final pressure Pf.

9. By repeating steps 6 to 8, the exercise can be repeated at different initial pressures in the vessel (Pf becoming Ps for the subsequent run).

Following table can be used to record the pressures.

Runs Ps Pi Pf

1

2

3

4

5

6

7

Test B – determination of ratio of volume using an isothermal process

Method

One vessel is initially pressurized and allowed to stabilise at ambient temperature. Then air is allowed to leak very slowly from the pressurized vessel into another vessel of different size via a needle valve. This process is isothermal.

Observation of the pressure before and after the process enables the ratio of the volumes of the vessels to be calculated.

Theory

Based on the ideal gas equation of state, the volume ratio of the vessels can be determined by

V1 ? P2s ? Pf (4)

V2 Pf ? P1s

where

V1 – volume of the large vessel

V2 – volume of the small vessel

P1s – initial pressure in the large vessel

P2s – initial pressure in the small vessel

Pf – final pressure in the two vessels

All the pressures are absolute pressure. The detail for obtaining Eq. (4) can be found in Appendix B.

Procedure

Measure and record Patm using a barometer.

Close ball valves V1 and V3 and Valve V5. Open valve V4.

Pressurise the large vessel by switching on the air pump. When P reaches approximately 30 kN/m2, switch off the air pump and close valve V4.

Wait until pressure P in the large vessel has stabilised (Pressure will fall slightly as the vessel contents cools to room temperature).

Record the starting pressure Ps.

Ensure that needle valve V5 is fully closed then open isolating valve V6. Open needle valve V5 very slightly to allow air to leak from the large vessel to the small vessel. Adjust V5 to so that P falls slowly with no change in T1 or T2 (if the flow of air is too fast then T1 and T2 will change and the exercise must be repeated).

As the pressure P falls in the large vessel and the pressure rises in the small vessel, valve V5 can be opened slightly to reduce the duration of the exercise.

Allow the contents of both vessels to stabilise in pressure and temperature then record the final pressure Pf.

Repeat the exercise at four different initial pressures in the large vessel. The following table can be used to record the pressures.

Runs T Patm P1s P2s Pf

1

2

3

4

5

6

Appendix A – theory for Test-A

For a perfect gas

cp ?cv ?R (A1)

where cp is the molar heat capacity at constant pressure and cv is the molar heat capacity at constant volume.

The heat capacity ratio can be determined experimentally using a two-step process.

1. An adiabatic reversible expansion from the initial pressure Ps to an intermediate pressure Pi.

{Ps, Vs, Ts} ? {Pi, Vi, Ti} (A2)

2. A return of the temperature of its original value Ts at constant volume Vi.

{Pi, Vi, Ti} ? {Pf, Vi, Ts} (A3)

For a reversible adiabatic expansion, no heat goes through the boundary of the system dq? 0 (A4)

From the first law of thermodynamics and Eq. (A4)

dU ?dq?dW or dU ??pdV (A5)

At constant volume, the heat capacity relates the change in temperature to the change in internal energy

dU ?cvdT (A6)

Based Eqs. (A5) and (A6), we obtain

cvdT ??pdV (A7)

Based on Eq. (A7) and the ideal gas law, we obtain

cv ln???? TTfi ??????Rln????V Vif ???? (A8)

For an ideal gas

Ti ? PiVi (A9)

Ts PsVs

Therefore

?? Pi ln Vi ?????Rln???VVsi ???? (A10)

cv??ln Ps ? Vs ? ?

According to Eq. (A1) and Eq. (A10)

ln ??

PPi ccp ln????VVsi ???? (A10)

s v

During the return of the temperature to the starting value

Vi ? Ps (A11)

Vs Pf

Then, the heat capacity ratio in the two-step process is

cp ? lnPs ? lnPi (A12) cv lnPs ? lnPf

Appendix B – theory for Test-B

The theory for this experiment makes the assumption that air behaves as a perfect gas. The final equilibrium pressure Pf can be determined from the ideal gas equation of state:

Pf ?mRT ? (m1 ?m2)RT (B1)

V V1 ?V2

where m is the sum of the mass in the two vessels, V is the total volume of the two vessels and T is the equilibrium temperature. Both vessels are at room temperature before the valve is opened. As the process is isothermal, the initial temperature will be the same as the final temperature (T=T1s=T2s=T1f=T2f). Taking the ideal gas equation of state once again gives:

V1P1s , m2 ?V2P2s (B2) m1 ?

RT RT

Substituting Eq. (B2) into Eq. (B1) yields

??V1P1s ?V2P2s ??RT ? RT RT ? VP

Pf ?? 1 1s ?V2P2s (B3)

V1 ?V2 V1 ?V2

Eq. (B3) can be written as

V1 ? P2s ?Pf (B4)

V2 Pf ?P1s

Assignment:

1. Attend the laboratory class and complete the lab tests according to the supervisor’s requirement. Students will not allowed to submit their report if they do not come to the lab class.

2. Write one about 2000-2500 word report and submit it by the due time.

In the report, you need to do the following analysis.

Test A

• Draw the capacity ratio v.s. pressure graph. Discuss how well the results obtained from the lab compared with the expected results (you should find the exact heat capacity ratio yourself from library, internet or some other sources), by putting the test results and the expected results in the same graph. Explain the reasons for any difference.

The exact heat capacity ratio varies between 1.399 and 1.401 for temperature in the range of –40° to 50°. You may use 1.4 as the exact heat capacity ratio in your report.

• Discuss why the initial expansion process can be considered adiabatic?

Test B

Discuss

• How well the result obtained compared to the expected result? Give possible reasons for any difference. The exact volumes of the large and the small vessels are 22.4 and 9.1 litres, respectively.

• Comment on how the rate of change affects the temperature of the air inside the vessels.

The report will be assessed based on the following criteria.

Introduction and conclusion (10%)

This section outlines the aims of the laboratory tests and the main outcomes of the report. The purpose of the report needs to be clearly stated. In the conclusion section, the main outcomes of the laboratory test need to be summarized.

Description of the methodology (20%)

A student needs to present the methodology, describe the test procedure and explain the basic theory using their own words. It is not acceptable to copy from the lab note.

Results (40%)

This is the main part of the report. It will be assessed based on whether all relevant data are presented, how well the data are organized, how well charts and tables are presented, whether the methods for generating the data are correct and if a student show a good understanding of the fundamental theory. If a student makes fundamental mistakes, he/she will lose a significant percentage of marks.

Discussion (20%)

Based on the results obtained from the test, a student needs to discuss what are the outcomes of the tests.

Writing (10%)

The writing needs to be logical, clear, concise.

The equipment includes

A common base plate (4)

Top plate (2)

Large vessel (3) Small vessel (6)

Air pump (9)

Pressure sensors (P & V)

Sensor P measures the pressure inside the larger pressure vessel (3)

Sensor V measures the vacuum inside the smaller vacuum vessel (6) (Positive values for vacuum pressure)

Temperature sensors T1 and T2 inside each vessel

Valves

V1 – the large vessel to atmosphere

V2 – the large vessel to the small vessel (via a large bore pipe)

V3 – the small vessel to atmosphere

V4 – pump to the large vessel (pressurize the large vessel)

V5 – needle valve connecting the large vessel to the small vessel

V6 – the large vessel to the small vessel (via needle valve V5)

V7 – pump to the small vessel (the pump draws air from the small vessel)

Test A – Determination of Heat Capacity ratio (Specific heat ratio)

Theory

The heat capacity ratio can be determined experimentally using a two-step process.

1. An adiabatic reversible expansion from the initial pressure Ps to an intermediate pressure Pi.

{Ps, Vs, Ts} ? {Pi, Vi, Ti} (1)

2. A return of the temperature of its original value Ts at constant volume Vi.

{Pi, Vi, Ti} ? {Pf, Vi, Ts} (2)

The heat capacity ratio in the two-step process is

cp ? lnPs ? lnPi (3) cv lnPs ? lnPf

The cp/cv can be approximated to be the heat capacity ratio at the temperature of Ts and the pressure of (Ps+Pf)/2. Please refer Appendix A for the derivation of Eq. (3).

Procedure

1. Measure and record Patm using a barometer.

2. Close ball valves V1 and V3 and open valve V4.

3. Pressurize the large vessel by switching on the air pump. When P reaches approximately 30 kN/m2, switch off the air pump and close valve V4.

4. Wait until pressure P in the large vessel has stabilized (P will fall slightly as the vessel contents cools to room temperature).

5. Record the starting pressure Ps.

6. Open the close valve V1 very rapidly with a snap action to allow a small amount of air to escape from the vessel.

7. Record Pi. (accurate instantaneous value)

8. Allow the vessel contents to return to ambient temperature then record the final pressure Pf.

9. By repeating steps 6 to 8, the exercise can be repeated at different initial pressures in the vessel (Pf becoming Ps for the subsequent run).

Following table can be used to record the pressures.

Runs Ps Pi Pf

1

2

3

4

5

6

7

Test B – determination of ratio of volume using an isothermal process

Method

One vessel is initially pressurized and allowed to stabilise at ambient temperature. Then air is allowed to leak very slowly from the pressurized vessel into another vessel of different size via a needle valve. This process is isothermal.

Observation of the pressure before and after the process enables the ratio of the volumes of the vessels to be calculated.

Theory

Based on the ideal gas equation of state, the volume ratio of the vessels can be determined by

V1 ? P2s ? Pf (4)

V2 Pf ? P1s

where

V1 – volume of the large vessel

V2 – volume of the small vessel

P1s – initial pressure in the large vessel

P2s – initial pressure in the small vessel

Pf – final pressure in the two vessels

All the pressures are absolute pressure. The detail for obtaining Eq. (4) can be found in Appendix B.

Procedure

Measure and record Patm using a barometer.

Close ball valves V1 and V3 and Valve V5. Open valve V4.

Pressurise the large vessel by switching on the air pump. When P reaches approximately 30 kN/m2, switch off the air pump and close valve V4.

Wait until pressure P in the large vessel has stabilised (Pressure will fall slightly as the vessel contents cools to room temperature).

Record the starting pressure Ps.

Ensure that needle valve V5 is fully closed then open isolating valve V6. Open needle valve V5 very slightly to allow air to leak from the large vessel to the small vessel. Adjust V5 to so that P falls slowly with no change in T1 or T2 (if the flow of air is too fast then T1 and T2 will change and the exercise must be repeated).

As the pressure P falls in the large vessel and the pressure rises in the small vessel, valve V5 can be opened slightly to reduce the duration of the exercise.

Allow the contents of both vessels to stabilise in pressure and temperature then record the final pressure Pf.

Repeat the exercise at four different initial pressures in the large vessel. The following table can be used to record the pressures.

Runs T Patm P1s P2s Pf

1

2

3

4

5

6

Appendix A – theory for Test-A

For a perfect gas

cp ?cv ?R (A1)

where cp is the molar heat capacity at constant pressure and cv is the molar heat capacity at constant volume.

The heat capacity ratio can be determined experimentally using a two-step process.

1. An adiabatic reversible expansion from the initial pressure Ps to an intermediate pressure Pi.

{Ps, Vs, Ts} ? {Pi, Vi, Ti} (A2)

2. A return of the temperature of its original value Ts at constant volume Vi.

{Pi, Vi, Ti} ? {Pf, Vi, Ts} (A3)

For a reversible adiabatic expansion, no heat goes through the boundary of the system dq? 0 (A4)

From the first law of thermodynamics and Eq. (A4)

dU ?dq?dW or dU ??pdV (A5)

At constant volume, the heat capacity relates the change in temperature to the change in internal energy

dU ?cvdT (A6)

Based Eqs. (A5) and (A6), we obtain

cvdT ??pdV (A7)

Based on Eq. (A7) and the ideal gas law, we obtain

cv ln???? TTfi ??????Rln????V Vif ???? (A8)

For an ideal gas

Ti ? PiVi (A9)

Ts PsVs

Therefore

?? Pi ln Vi ?????Rln???VVsi ???? (A10)

cv??ln Ps ? Vs ? ?

According to Eq. (A1) and Eq. (A10)

ln ??

PPi ccp ln????VVsi ???? (A10)

s v

During the return of the temperature to the starting value

Vi ? Ps (A11)

Vs Pf

Then, the heat capacity ratio in the two-step process is

cp ? lnPs ? lnPi (A12) cv lnPs ? lnPf

Appendix B – theory for Test-B

The theory for this experiment makes the assumption that air behaves as a perfect gas. The final equilibrium pressure Pf can be determined from the ideal gas equation of state:

Pf ?mRT ? (m1 ?m2)RT (B1)

V V1 ?V2

where m is the sum of the mass in the two vessels, V is the total volume of the two vessels and T is the equilibrium temperature. Both vessels are at room temperature before the valve is opened. As the process is isothermal, the initial temperature will be the same as the final temperature (T=T1s=T2s=T1f=T2f). Taking the ideal gas equation of state once again gives:

V1P1s , m2 ?V2P2s (B2) m1 ?

RT RT

Substituting Eq. (B2) into Eq. (B1) yields

??V1P1s ?V2P2s ??RT ? RT RT ? VP

Pf ?? 1 1s ?V2P2s (B3)

V1 ?V2 V1 ?V2

Eq. (B3) can be written as

V1 ? P2s ?Pf (B4)

V2 Pf ?P1s

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