Perfect Gas Expansion Lab Report

Perfect Gas Expansion Name: Nabila Sofea binti Zawawi (2018680438) Group Members: Muhammad Sharfawi bin Burhanuddin (2018259708) Nur Aqila binti Mohamad (2018660614) Nur Hannani binti Johari (2018801622) Noormin Suraya binti Anuar Zaidi (2018802362)

Abstract—This experiment is carried out with three aims in mind, to determine the relationship between pressure and volume of an ideal gas, the relationship between pressure and temperature of an ideal gas and the ratio of heat capacity. These aims were successfully achieved by operating the Perfect Gas Expansion Apparatus (Model: TH11) on the 21st of October 2019. The first experiment involves the demonstration of Boyle’s Law, or the P-V relations and was repeated three times, from pressurized chamber to atmospheric chamber, from vacuum chamber to atmospheric chamber and from pressurized chamber to vacuum chamber. The second experiment was carried out to demonstrate the Gay-Lussac Law, or the P-T relations, which took three trials. The third experiment was carried out to show the isentropic expansion process where the pressure and temperature of the gas before and after expansion were observed and recorded. The results obtained tallied with the theoretical laws despite a few errors that caused deviations but the experiment were considered a success.

I. INTRODUCTION An ideal gas is a law that can be simply expressed as PV=nRT, in which P and T are the absolute pressure and temperature, and R is the gas constant. It is an approximation made to make it easier to predict the behavior of gas, derived from simpler gas laws such as Boyle’s, Charles’s and Amonto’s Law (LeTran, 2019). In this experiment, a few laws were observed, namely the Boyle’s and Gay-Lussac’s Law. Each of these law states the relationship between pressure and volume, and pressure and temperature of an ideal gas respectively. Boyle’s Law states that the pressure of a given mass of an ideal gas is inversely proportional to its volume at a constant temperature which gives rise to the equation P1V1=P2V2.

Figure 2: Graphical Representation of Gay-Lussac’s Law

Meanwhile, the third experiment aims to demonstrate the isentropic expansion of gas, a process where the entropy of the gas is kept constant. An isentropic process is also easily understood to be adiabatic in which there is no net transfer of either mass or heat during the process. From this process, the specific heat ratio can be calculated. It is essential to be equipped with the knowledge of ideal gases as it is a fundamental law that is utilized by engineers working with gases to approximate their actual behavior. For instance, ideal gas law is used to estimate the volume of the container that needs to be built based on the known parameters such as the pressure and temperature.

II. OBJECTIVES The objectives of this experiment are:i.

To determine the relationship between pressure and volume of an ideal gas.

ii.

To determine the relationship between pressure and temperature of ideal gas.

iii.

To determine the ratio of heat capacity. III. THEORY

Boyle’s Law is derived from the volume of gas that is inversely proportional to pressure at constant temperature. Figure 1: Graphical Representation of Boyle’s Law

Meanwhile, Gay-Lussac’s Law states that the pressure of a given mass of gas varies directly with the absolute temperature of the gas at constant volume, which gives rise to the equation P1/T1=P2/T2. To clearly demonstrate these two laws, it is understood that the parameters in the equation were to be manipulated in a way that the data can be represented in graphical form.

1 𝑃 or 𝑃𝑉 = 𝐾 𝑉∝

∴ 𝑃1 𝑉1 = 𝑃2 𝑉2

Gay-Lussac’s Law is derived from the pressure-temperature proportionality for gas.

For specific heat relations of ideal gas, combining ℎ = 𝑈 + 𝑃𝑉 and 𝑃𝑉 = 𝑅𝑇 gives ℎ = 𝑈 + 𝑅𝑇.

Since 𝑃 ∝ 𝑇 for fixed mass kept at constant volume, 𝑃1 =𝑘 𝑇1 𝑃2 =𝑘 𝑇2 𝑃1 𝑃2 = =𝑘 𝑇1 𝑇2 𝑃1 𝑃2 ∴ = 𝑇1 𝑇2

Differentiating the equation, 𝑑ℎ = 𝑑𝑢 + 𝑅𝑇

Replacing dh with Cp dT and du with Cv dT and diving the resulting expression with dT, 𝐶𝑝 = 𝐶𝑣 + 𝑅 Specific heat ratio, k is also an ideal gas property. It is defined as,

The ideal gas law is derived from three simple gas laws, namely the Boyle’s Law, Charle’s Law and Avogadro’s Law.

𝑘=

From Boyle’s Law, 𝑉∝

1 (1) 𝑃

𝐶𝑝 𝐶𝑣

IV. PROCEDURES

Charles’s Law,

General Start-up Procedures

𝑉 ∝ 𝑇 (2)

1.

The equipment was connected to single phase power supply and was switched on.

Avogadro’s Law,

2.

The valves were fully opened and pressure reading was checked.

3.

All valves were then closed.

4.

The pipe was connected from the compressive port of the pump to pressurized chamber.

5.

The unit was ready for use.

𝑉 ∝ 𝑛 (3)

Combining (1), (2), and (2), nT P Converting proportionality to equality, V ∝

knT V = P Replacing k with gas constant, R, nRT V = P Rearranging to give the ideal gas law, PV = nRT

Experiment 1: Boyle’s Law Experiment 1.

The general start-up procedures were performed. The valves were made sure to be fully closed.

2.

The compressive pump was switched on and the pressure inside the chamber was increased up to 150 kPa. The pump was then switched off and the hose was removed from the chamber.

3.

The pressure reading inside the chamber was monitored until it stabilized.

4.

The pressure reading for both chambers before expansion were recorded.

5.

V 02 was fully opened and the pressurized air was allowed to flow into the atmospheric chamber.

6.

The pressure reading for both chambers after expansion were recorded.

7.

The experimental procedures were repeated for the following conditions:

Where P = absolute pressure V = volume n = number of moles R = universal gas law T = absolute temperature

a) From atmospheric chamber to vacuum chamber b) From pressurized chamber to vacuum chamber 8.

Experiment 2: Gay-Lussac’s Law Experiment 1.

The general start-up procedures were performed. The valves were made sure to be fully closed.

2.

The hose from compressive pump was connected to pressurized chamber.

3.

Then, valve V 01 was slightly opened to allow the pressurized air to flow out. The temperature reading for every increment of 10 kPa was recorded.

5.

The experiment was stopped when pressure reached atmospheric pressure.

7.

Before expansion

After expansion

PT 1 (kPa abs)

154.5

143.8

PT 2 (kPa abs)

126.1

142.3

Table 2. Vacuum Chamber to Atmospheric Chamber Before expansion

After expansion

PT 1 (kPa abs)

143.2

114.8

PT 2 (kPa abs)

59.8

113.6

Table 3. Pressurized Chamber to Vacuum Chamber Before expansion

After expansion

PT 1 (kPa abs)

151.1

120.5

PT 2 (kPa abs)

61.8

118.9

The graph of pressure versus temperature was plotted.

1.

The general start-up procedures were performed.

2.

The hose from compressive pump was connected to pressurized chamber.

3.

The compressive pump was switched on and the pressure inside the chamber was allowed to increase until about 160 kPa. Then, the pump was switched off and the hose was removed from the chamber.

4.

The pressure reading inside the chamber was monitored until it stabilized. The pressure reading PT 1 and temperature TT 1 were recorded.

6.

Table 1. Pressurized Chamber to Atmospheric Chamber

The experiment was repeated three times to get the average value.

Experiment 3: Determination of ratio of heat capacity

5.

Experiment 1: Boyle’s Law Experiment

The compressive pump was switched on and the temperature for every increment of 10 kPa was recorded. The pump was stopped when pressure PT 1 reached about 160 kPa.

4.

6.

V. RESULTS AND DISCUSSIONS

The PV value were calculted to prove the Boyle’s Law.

The valve V 01 was fully opened and brought back to closed position after a few seconds. The pressure reading PT 1 and TT 1 were monitored until it became stable. The ratio of heat capacity was determined and compared with the theoretical value.

For pressurized chamber to atmospheric chamber, V1 = 0.025 m3 V2 = 0.01237 m3 By using Boyle’s Law, 𝑃1 𝑉1 = 𝑃2 𝑉2 (154.5 × 0.025) + (126.1 × 0.01237) = (143.8 × 0.025) + (142.3 × 0.01237) 5.4224 = 5.3553 The difference between the two values is 0.0671. Or, 𝜀=

5.4224 − 5.3553 × 100% = 1.2% 5.4224

For vacuum chamber to atmospheric chamber, 𝑃1 𝑉1 = 𝑃2 𝑉2 (143.2 × 0.025) + (59.8 × 0.01237) = (114.8 × 0.025) + (113.6 × 0.01237) 4.3197 = 4.2752

General Shut-down Procedures

The difference between the two values is only 0.0445.

1.

The pump was switched off and both pipes were removed from chambers.

Or,

2.

Valves were fully opened to release air inside the chambers.

3.

The main switch and power supply were switched off.

𝜀=

4.3197 − 4.2752 × 100% = 1.03% 4.3197

Table 7. Average Data

For pressurized chamber to vacuum chamber. 𝑃1 𝑉1 = 𝑃2 𝑉2 (151.1 × 0.025) + (61.8 × 0.01237) = (120.5 × 0.025) + (118.9 × 0.01237) 4.542 = 4.4833

Pressure (kPa abs)

Temperature (oC)

110 120 130 140 150 160

32.55 33 33.43333 34 34.65 35.08333

The difference between the two values is only 0.0587. Or, 𝜀=

4.542 − 4.4833 × 100% = 1.3% 4.542 Graph 1. Graph of Pressure vs Temperature

Experiment 2: Gay-Lussac’s Law Experiment Table 4. Trial 1

110 120 130 140 150 160

Pressure (kPa) vs Temperature (ºC)

Temperature (oC)

Pressurize vessel 32.5 32.6 33.2 34.1 35.0 35.6

Depressurize vessel 35.3 33.7 33.2 32.4 32.0 31.6

Pressure (kPa)

Pressure (kPa abs)

180 160 140 120 100 80 60 40 20 0 32

Table 5. Trial 2 Pressure (kPa abs)

110 120 130 140 150 160

32.5

Temperature ( C)

Pressurize vessel 31.2 32.6 32.2 33.3 34.3 35.3

Depressurize vessel 32.5 33.9 35.0 35.3 36.0 36.1

Table 6. Trial 3

33.5

o

TT 1 ( C)

Depressurize vessel 32.1 33.2 34.4 35.4 36.1 36.4

35

35.5

Before expansion

After expansion

161.6

110.3

36.0

33.0

Temperature (oC)

Pressurize vessel 31.7 32.0 32.6 33.5 34.5 35.5

34.5

Table 8. Isentropic Expansion Result

For isentropic process, 110 120 130 140 150 160

34

Experiment 3: Isentropic Expansion Process

PT 1 (kPa abs)

Pressure (kPa abs)

33

Temperature (ºC)

o

(

𝑘−1

𝑇2 𝑃2 𝑘 = 𝑇1 𝑃1

)

𝑘−1 ( )

33.0 110.3 𝑘 = 36.0 161.6

0.9167 = 0.6825 ln 0.9167 = (

𝑘−1 ) 𝑘

𝑘−1 ) ln 0.6825 𝑘

𝑘−1 = 0.2277 𝑘 𝑘 = 1.2948

(

For the first experiment with the objective of demonstrating Boyle’s Law, it is first understood that the law states that the pressure of a given mass of an ideal gas is inversely proportional to its volume at a constant temperature, or simply expressed as the equation 𝑃1 𝑉1 = 𝑃2 𝑉2 . Therefore, the data gathered for this specific purpose is the pressure of the gas before and after expansion at constant volume in each tank. The experiment is repeated in three different conditions to observe the precision of the results, that is pressurized chamber to atmospheric chamber, vacuum chamber to pressurized chamber and vacuum chamber to atmospheric chamber. For all conditions, it can be observed that the difference of values using pressure before and after expansion calculated using Boyle’s Law have very small differences of approximately 1%. From this pattern, it can be deducted that Boyle’s Law is verified. For the second experiment with the objective of demonstrating Gay-Lussac’s Law, it is first understood that the law states the pressure of a given mass of gas varies directly with the absolute temperature of the gas at constant volume, which gives rise to the equation P1/T1=P2/T2. In order to achieve this, the equipment is run and the data gathered is the temperature at pressurized and depressurized vessel at different pressure. The data is then presented in the form of graph. From Graph 1, it can be observed that the trendline goes upwards, representing the increasing temperature as the pressure also increases. This is the similar trend suggested by Gay-Lussac’s Law. Therefore, it can be deducted that Gay-Lussac’s Law had successfully been demonstrated. The third experiment was conducted to obtain the ratio of specific heat capacity. It was achieved by gathering the absolute pressure and temperature of the gas before and after expansion. From the data, the ratio of heat capacity was calculated using the equation for isentropic process, which is 𝑇2 𝑇1

=

𝑘−1 ( ) 𝑃2 𝑘

𝑃1

. The value of k calculated is 1.2948. Therefore it can

also be understood that 𝑘 =

𝐶𝑝 𝐶𝑣

= 1.2948. k, or the ratio of

specific heat capacity, also known as Laplace’s coefficient plays an important role in thermodynamics, especially in adiabatic processes.

CONCLUSION

The objectives of the experiment were successfully achieved, with the Boyle’s Law verified, with the difference in values of P1V1=P2V2 approximately 1%. Gay-Lussac’s Law was also verified with the graph showing that the pressure of gas varies directly with the temperature at constant volume. The ratio of heat capacity was also calculated, giving the value of 1.2948. RECOMMENDATIONS

Various recommendations could be made in order to obtain more accurate results, and that is to ensure that the instrument was properly calibrated before it is run in an experiment. This is to detect any abnormalities in the instrument itself so it can be fixed to give more reliable data. Other than that, it is important to consult the lab assistant for guidance so no confusion arises during the experiment is run. Lastly, it is very recommended to calculate the standard deviation for each data obtained as this reflects its accuracy and the need for a repetition of the experiment for more reliable results. REFERENCES [1] [2] [3] [4]

A. Esposito (1997), Fluid Mechanics with Applications, 4th Revised US Edition, Prentice-Hall R. C. Binder (1960), Fluid Mechanics, 3rd Edition, Prentice Hall R. L. Mott (2015), Applied Fluid Mechanics, 7th Edition, Prentice-Hall R. L. Street (1995), Elementary Fluid Mechanics, 7th Edition, John Wiley & Sons Inc.

APPENDIX

Figure 1: Raw Data Recorded on the Day of the Experiment