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1 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). 2 Balance Calibration Check for Empty Vessel Predicted Mass Values (g) 462 y = 1x + 0.0131 R² = 1 460 458 456 454 452 450 448 448 450 452 454 456 458 460 462 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). πππ βππ ππΆπ2 = (ππ πΆπ2 ) (1) ππΆπ2 = πππππ ππ πΆπ2 πππ = πππ π ππ ππ¦ππππππ πππ πππ ππ = πππ π ππ ππ¦ππππππ πππΆπ2 = ππππππ’πππ π€πππβπ‘ ππ πΆπ2 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. βββ1 ππ = π1 + β 2 ββ1 (π2 β π1 ) ππ = ππππ π π’ππ ππ ππππ ππ‘ 1261 ππ‘ (π‘πππ) (2) 3 π1 , π2 = πΌππ‘πππππππ‘πππ πππππππππ ππππ π π’πππ (π‘πππ) β = ππππ£ππ‘πππ ππππ£π π ππ πππ£ππ (ππππ‘) β1 , β2 = πΌππ‘πππππππ‘πππ πππππππππ ππππ£ππ‘ππππ (ππππ‘) 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. ππΆπ2 = πππ π π (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] 4 Pressure vs. Moles Pressure of Vessel (torr abs) 8000 y = 32719x + 1.5311 R² = 1 7000 6000 y = 29795x + 310.38 R² = 1 5000 4000 3000 2000 1000 0 -0.05 0 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] π= πππ π π (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. 5 Pressure vs. Compression Factor 8000 P(torr abs) 7000 6000 5000 y = -88240x + 90953 R² = 0.9773 4000 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] πππ π π =π =1+ π΅ ππ + πΆ π 2π +β― (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. 6 Z-1 vs. 1/Vm 0.03 0.02 0.01 0 Z-1 -0.1 -0.01 0 0.1 0.2 0.3 0.4 0.5 -0.02 -0.03 -0.04 y = -0.185141x + 0.025855 R² = 0.976700 -0.05 -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 Boyleβs Temperature. In the case of CO2, Boyleβs 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 Boyleβs 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. 7 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 8 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. Atkinsβ 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. 9 Appendix Calculations and Error Analysis Regression for Calibration of Vessel Mass Interpolation for Pressure of Room ππ = π1 + β β β1 (π β π1 ) β2 β β1 2 ππ = ππππ π π’ππ ππ ππππ ππ‘ 1261 ππ‘ (π‘πππ) π1 , π2 = πΌππ‘πππππππ‘πππ πππππππππ ππππ π π’πππ (π‘πππ) β = ππππ£ππ‘πππ ππππ£π π ππ πππ£ππ (ππππ‘) β1 , β2 = πΌππ‘πππππππ‘πππ πππππππππ ππππ£ππ‘ππππ (ππππ‘) ππ = 733 π‘πππ + 1261 ππ‘ β 1000 ππ‘ (720 β 733 π‘πππ) 1500 ππ‘ β 1000ππ‘ ππ = 726.214 = 726 π‘πππ Uncertainty of Interpolation πππ = 1 π‘πππ Uncertainty of Temperature ππ = 0.1 °πΆ ππ 0.1 πΎ Uncertainty of Balance ππ = 0.001 π 10 Uncertainty of Pressure of Vessel for 9 bar ππ = 9.001 × 0.005 = 0.045 πππ ππ = 0.045 πππ × 750.061683 π‘πππ = 33.76 = 30 π‘πππ 1 πππ Conversion of Pressure of Vessel (bar gauge to torr absolute) for 9 bar π = ππ + ππ π = ππππ π π’ππ ππ π£ππ π ππ (πππ πππ’π‘π) ππ = ππππ π π’ππ ππ π£ππ π ππ (πππ’ππ) π = 9.001 πππ π × 100,000 ππ 0.0075006 π‘πππ × + 723 π‘πππ = 7474.29 = 7474 π‘πππ πππ . 1 πππ 1 ππ Moles of CO2 for 9 bar ππΆπ2 = πππ β ππ (πππΆπ2 ) ππΆπ2 = πππππ ππ πΆπ2 πππ = πππ π ππ ππ¦ππππππ πππ πππ ππ = πππ π ππ ππ¦ππππππ πππΆπ2 = ππππππ’πππ π€πππβπ‘ ππ πΆπ2 ππΆπ2 = 460.327 β 449.743 π π 44.0095 πππ ππΆπ2 = 0.24049 = 0.240 πππ Uncertainty for moles of CO2 βππππ 2 + πππ 2 πππΆπ2 = πππΆπ2 = (πππΆπ2 ) β(0.001 π)2 + (0.001 π)2 π 44.0095 πππ πππΆπ2 = 3.213 × 10β5 = 3 × 10β5 πππ Uncertainty of Volume of Vessel πππ = 0.0001 πΏ Molar Volume of Vessel for 9 bar 11 ππ = ππ ππΆπ2 ππ = πππππ π£πππ’ππ ππ = π£πππ’ππ ππ ππ¦ππππππ ππ = 0.5612 πΏ 0.24049 πππ ππ = 2.3336 πΏ πππ Uncertainty of Molar Volume for 9 bar πππ = ππ × β( πππ 2 πππΆπ2 2 ) +( ) ππ ππΆπ2 2 πΏ 0.0001 πΏ 2 3 × 10β5 πππ πππ = 2.3336 × β( ) +( ) πππ 0.5612 πΏ 0.240 πππ πππ = 5.079 × 10β4 = 5 × 10β4 πΏ πππ Inverse Molar Volume for 9 bar 1 1 πππ = = 0.4285 = 0.429 ππ 2.3336 πΏ πΏ πππ Uncertainty of Inverse Molar Volume for 9 bar π 1 1 πππ = × |π| ππ ππ ππ π = πππ£πππ π ππ₯ππππππ‘ (β1) πΏ 5 × 10β4 1 πππ πππ π = 0.429 × |β1| πΏ ππ πΏ 2.3336 πππ π 1 πππ = 9.1918 × 10β5 ππ πΏ Compression Factor, Z, of CO2 at 9 bar 12 π= πππ π π π = πππππππ π πππ ππππ‘ππ π = πππ ππππ π‘πππ‘ = 0.0821 ππ‘π πΏ πππ πΎ π = π‘πππππππ‘π’ππ (πΎ) π = 7474 π‘πππ × 1 ππ‘π πΏ 1 1 × 0.429 × × ππ‘π πΏ 760 π‘πππ πππ 0.0821 298 πΎ πππ πΎ π = 0.9497 = 0.950 Uncertainty of Compression Factor, Z, CO2 for 9 bar ππ 2 πππ 2 ππ 2 β ππ = π × ( ) + ( ) +( ) π ππ π 2 β4 πΏ 5 × 10 3.73 π‘πππ 0.1 πΎ 2 πππ ππ = π × β( ) +( ) +( ) πΏ 7474.3 π‘πππ 294.35 πΎ 0.2405 πππ 2 ππ = 0.00206 = 0.002 Regression for Pressure (torr) vs. moles of CO2 (from Data) 13 Regression for Pressure (torr) vs. moles of CO2 (from Ideal Gas Assumption) Regression for Z-1 vs. 1/Vm Calculated virial coefficient π π΅ =πβ( ) π π π = πππππππ‘πππ πππ πππ‘ππππππππ’πππ ππππππ ( ππ‘π πΏ2 ) πππ 2 πΏ π = πππππππ‘πππ πππ πππππ‘π ππππππ’πππ π ππ§π ( ) πππ 14 π΅ = π πππππ π£πππππ πππππππππππ‘ π΅ = 4.29 × 10β2 3.610 ππ‘π πΏ2 πππ 2 πΏ β πππ (0.0821 ππ‘π πΏ ) (273.15 + 21.2 πΎ) πππ πΎ π΅ = β0.10648 πΏ ππ3 = β106.48 πππ πππ Error in literature value π= π=| |ππ₯ππππππππ‘ππ β πππ‘π’ππ| × 100% πππ‘π’ππ β185.1 β (β149.7) | × 100% = 23.65% β149.7 Report Questions 1. It is important to account for atmospheric pressure when completing data analysis because the gas it effected by the absolute pressure within the vessel, not just the pressure relative to the atmosphere. If the outside pressure is not accounted for, then the compression factor calculated as pressure goes down goes down as well, which is not what the expectation is. Less pressure would mean less interaction and would be more ideal. 2. The literature value states that the second virial coefficient is -149.7 cm3/mol. The calculated second virial coefficient from this experiment was calculated to be -185.14 cm3/mol. The calculated coefficient from the experiment is lower than the calculated literature value. The second virial coefficient is temperature dependent and the literature shows this value at 273 K. The experiment was conducted at 21.2 °C. This, however, does not account for the more negative value. A temperature closer to Boyleβs temperature should denote a B value that is closer to zero. Since this is further away from zero, which is where Boyleβs temperature would be and where a gas is most ideal,[1] then this means that this is less ideal than expected. This may be due to errors in the experiment. There was a leak in the vessel, which would make the mass, and therefore calculated amount of the CO2 lower than what it should be. This would cause the compression factor and second 15 virial coefficient to go down. Another possible source of error could come from having a lower pressure in the room measured than what the actual pressure is. There was concern with the pressure reading on the mercury barometer due to the large difference from 1 atmosphere. Based on literature value, it would mean that the experiment was completed at about 3000 ft above sea level, but the experiment was completed 1261 ft above sea level. Because of this, the elevation was taken and interpolated based on literature. This is still only an estimate, so it still may be inaccurate. 3. Ideal behavior assumes that there are no interactions between the molecules. This does not occur with a real gas, however. A real gas will interact with other molecules because molecules have interactive forces and a finite size. These two factors are taken into account when using van der Waals equation.
