Real Gas Behavior: Gravimetric Determination of the Second ...

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Real Gas Behavior: Gravimetric Determination of the Second Virial Coefficient of CO2 CHEM 457, 2 October 2015, Experiment 04

Faith Tran, Douglas Hiban, Torreh Parach, Daniel Borden

Results and Discussion After obtaining the data shown in Table 1, the second virial coefficient was determined from this experiment. The same vessel (mass of empty vessel = 443.743?0.001 g) was used throughout the experiment, therefore volume (0.5612?0.0001 L) was kept constant. The vessel was kept at a constant temperature (21.2?0.1 ?C) as well. Pressure (varied from 9 to 4 bar) was deliberately changed to determine the amount of CO2 in the vessel.

Mass of Vessel and

P (bar gauge) T (?C)

CO2 (g)

9.001?0.005 21.2?0.1 460.327?0.001

7.972?0.004 21.2?0.1 459.188?0.001

7.027?0.004 21.2?0.1 458.140?0.001

5.943?0.003 21.2?0.1 456.924?0.001

4.949?0.002 21.2?0.1 455.827?0.001

3.990?0.002 21.3?0.1 454.786?0.001

Table 1. Results obtained in units measured (pressure in bar gauge, temperature in Celsius, and

mass of vessel and CO2 in grams).

Figure 1 shows the calibration of the mass of the vessel with various balanced masses (integer

masses from 1 to 10 g).

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Predicted Mass Values (g)

Balance Calibration Check for Empty Vessel

462

460

y = 1x + 0.0131

R? = 1

458

456

454

452

450

448

448

450

452

454

456

458

460

462

Obtained Mass Values (g)

Figure 1. Balance Calibration Check for Empty Vessel using various balanced masses. The calibration is checked via the R2 value, which shows how linear data is. A linear plot would mean good calibration. The R2 value is 1, which means that the graph is extremely linear and that the balance is accurate enough for this experiment.

The amount of CO2 in moles was obtained by subtracting the measured mass of the empty vessel

from the measured mass of the vessel and CO2 and dividing by the molar mass, as shown in Eq.

(1).

2

=

- (2)

(1)

2 = 2

=

=

2 = 2

The pressure was measured relative to the pressure of the atmosphere (gauge pressure), therefore

the absolute pressure would be the atmospheric pressure (726?1 torr) added to the measured gauge

pressure. Eq. (2) shows the interpolation of the pressure for the elevation above sea level.

=

1

+

-1 2-1

(2

-

1)

(2)

= 1261 ()

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1, 2 = () = ()

1, 2 = () The interpolation was used as opposed to the reading of the mercury barometer because of the drastic difference in pressure from literature value. According to the literature, the pressure of the room was for an elevation of about 3500 ft above sea level.[1] The experiment was done at 1261 ft above sea level. This elevation leads to an 8.3% difference in pressure relative to the interpolated value, which leads to significant difference in the values obtained. Because of this large difference, it was concluded that the barometer was incorrect. The rest of this discussion will be based on the interpolated value.

Table 2 shows the data obtained in units that are to be used in this discussion.

P (torr absolute) T (K)

mols CO2

7474.3?3.7 294.4?0.1 0.24049?3E-5

6702.4?3.4 294.4?0.1 0.21461?3E-5

5993.6?3.0 294.4?0.1 0.19080?3E-5

5180.6?2.6 294.4?0.1 0.16317?3E-5

4435.0?2.2 294.4?0.1 0.13824?3E-5

3715.7?1.9 294.5?0.1 0.11459?3E-5

Table 2. Results obtained in units used in discussion (pressure in torr absolute, temperature in

Kelvin, and moles of CO2).

A plot of absolute pressure vs. moles of CO2 was then plotted according to Table 2 (shown in

blue). A plot of absolute pressure vs. moles of CO2 was plotted on the same graph according to the

ideal gas law (Eq. 3), as shown in Figure 2.

2

=

(3)

nCO2 is the moles P is the pressure is in atm, Vc is the volume of the vessel in liters, R is the gas

constant in atm L mol-1 K-1, and T is temperature in K.

The ideal gas law assumes that there are no interactions between the molecules and that the molecules have no finite size. These assumptions are not true, but can be more closely achieved at high temperatures, high molar volumes, and low pressures.[2]

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Pressure of Vessel (torr abs)

8000

7000

6000

5000

4000

3000

2000

1000

0

-0.05

0

Pressure vs. Moles

y = 32719x + 1.5311 R? = 1

y = 29795x + 310.38 R? = 1

0.05

0.1

0.15

0.2

0.25

0.3

anount of CO2 (moles)

Figure 2. Plot of pressure of vessel vs. moles of CO2 in the vessel. The blue line shows the data collected from Table 2. The orange line shows the amount of CO2 that would be in the vessel at the given pressure and temperature according to Eq. (3).

The difference in slopes indicate that the data collected deviates from the ideal gas model. The

lower slope (29795?66 torr/mol) of the real data, compared to the slope assuming ideality

(32719?4.5 torr/mol) means that the amount of CO2 does not affect the pressure as much as

ideality. This can be contributed to a compression factor, Z, which is calculated in Eq. (4).[2]

=

(4)

This compression factor accounts for the non-ideality of a real gas, which can be contributed from

interactive forces and finite size of the molecules.[2] Figure 3 shows the Z values for each pressure.

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P(torr abs)

Pressure vs. Compression Factor

8000

7000

6000

5000

4000

y = -88240x + 90953 R? = 0.9773

3000 0.945 0.95 0.955 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995

Compression Factor, Z

Figure 3. Compression factor, Z, with corresponding pressures. From Figure 3, it can be seen that as pressure goes down, Z goes up. This supports the idea that lower pressures move gases towards ideality. At lower pressures the molecules are not as close to each other, so fewer interactions are taking place between the molecules.[2]

=

=

1

+

+

2

+

(5)

The virial equation of state, shown in Eq. (5) accounts for these interactions between molecules. The virial equation is a model for estimating deviations from ideal gas behavior.[2] The first term of the right side denotes ideality, the second term denotes interactions between two molecules, the third term denotes interactions between three molecules, etc. Typically, including for this experiment, the third term and beyond are much smaller than the first two terms, so they can be ignored.[2] Only finding the value of B, the second virial coefficient, is of interest.

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Z-1

Z-1 vs. 1/Vm

0.03

0.02

0.01

0 -0.1 -0.01 0

0.1

0.2

0.3

0.4

0.5

-0.02

-0.03 -0.04 -0.05

y = -0.185141x + 0.025855 R? = 0.976700

-0.06 1/Vm (mol/L)

Figure 4. Plot of Z-1 vs. 1/Vm.

From Figure 4, the slope of the line gives the second virial coefficient, B, -185.1?14.3 cm3/mol. This is different from the calculated second virial coefficient, according to the literature value.

The calculated B value at 0 ?C is -149.7 cm3/mol.[3] The literature B value is less negative than the one determined from the data. At ideality, B would be 0. The temperature in which a particular gas is most ideal is called Boyle's Temperature. In the case of CO2, Boyle's Temperature, TB, is 714.8 K.[3] Above this temperature, B would be positive. Below this temperature, B is negative. This means that the more negative B is, the less ideal it is. Therefore the B value determined from Figure 4 indicates a lesser ideality. This should not be the case if no errors occurred with the experiment because as temperature gets closer to Boyle's temperature, ideality should increase. From the B values, this does not follow the expectation. The value from the data is 23.65% higher than the literature value.

The y-intercept of Figure 4 shows the deviation from ideality because an ideal gas would yield a y-intercept of 0. The y-intercept here is 0.0258?0.00464, which shows a deviation from ideality. If the gas were ideal, then the y-intercept would be 0. The compression factor would be 1, so Z-1 would be 0. This, however, shows that at infinitely large molar volumes, the Z value is above 1.

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This shows that at higher molar volumes, above the ideal molar volume, the repulsive forces dominate.

The more negative B value from the experiment could come from a couple errors. The most likely error is due to a leak in the vessel. A leak would mean that less pressure and less CO2 would be in the vessel at the time of weighing, which would ultimately increase the Z value. This explains why the y-intercept is very positive as opposed to being closer to 1. The difference in the literature value from the experimental value could be attributed to two things. The first is the purging of the vessel and CO2 lines. The second is a leak in the vessel. If the lines were not completely purged correctly or a leak was present, then contaminant molecules, such as oxygen and nitrogen, could cause error. Oxygen and nitrogen have a lower molar mass than CO2, so the calculation for 1/Vm would decrease as well. This is more prone and has a higher impact at higher pressures due to the pressure difference between atmospheric pressure and the pressure of the vessel system. Because it would affect higher pressure more, it would shift the slope of Figure 4 to be more negative than the literature value. The amount of time spent weighing the vessel varied, which could have caused a change in temperature while outside of the water bath.

The significance of this work is that gas ideality can be used to calculate the desired amount of a gas. If, for example, measuring quantities for ballistics, explosions pressure can be more accurately determined. For certain loading pressures, the difference between ideal and non-ideal models are more than 100% different.[4]

The experimental limitations of this experiment are from the inability to measure the temperature inside the vessel (another possible source of error). The temperature just outside the vessel could be different than inside the temperature, which means that the calculations for Z may be inaccurate. Even within the vessel, temperature may differ from spot to spot. This is mostly kept constant with the water bath.

One way to improve this experiment is possibly have a thermocouple placed inside the vessel and sealed. This would make the temperature reading inside the vessel more accurate. Also having the

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balance at the same site as the experiment could minimize slight temperature changes inside water bath and in the room.

Because of the high error in the second virial coefficient, the experiment was not very successful. The balance was calibrated well and the volume and temperature stayed fairly constant, but pressure and the amount of CO2 in the vessel most likely decreased due to the leak in the vessel. The high R2 value for all linear figures meant that there was certainly a trend that was noticeable though.

References [1] Air Pressure and Altitude above Sea Level. Engineering ToolBox. [2] Milosavljevic, B. H., Lab Packet for Chem 457: Experimental Physical Chemistry, 2015, 4.14.7. [3] Atkins, P.; De Paula, J. Atkins' Physical Chemistry 10th ed. W.H. Freeman and Company: New York. 2014. 46-47. [4] Volk, F.; Bathelt, H., Application of the Virial Equation of State in Calculating Interior Ballistics Quantities. Propellants and Explosives. 1976, 1, 7-14.

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