Real Gases.doc

REAL GASES

Submitted By: Date Submitted:

Cesar Ian A. Caermare July 21, 2012

Izel Marion Nicole C. Hernando

Submitted to:

Prof. Sheeva Yahcob-Saddalani Chem 142 Lab Instructor

ABSTRACT

The study was carried out to graphically evaluate the behavior of real gases at changing conditions and the ability of selected real gas equations of state to predict P-V-T relationships of gases.

The pressure exerted by carbon dioxide at 31.1 °C and 100 °C was plotted against molar volume and ideal, van der Waals, and experimental isotherm were evaluated. With these, the absolute accuracy of van der Waals equation isotherm was evaluated by comparing its plot against ideal isotherm. Redlich Kwong equation of state was validated by attempting to predict the P-V-T relationship of CO2.

Van der Waals equation of state is best applicable to carbon dioxide at 31.1 °C as the accuracy of the van der Waals isotherm is in unity with the ideal isotherm at a wide range at this condition. Redlich-Kwong isotherm also best describes CO2 at 100 °C since the graph shows that the deviation of this isotherm is very less to ideal isotherm.

In part two, Redlich-Kwong and Dieterici equations of state were evaluated using the thermodynamics quantities of selected samples of gases and then plotted to better understand the equations being evaluated.

The two equations best describe water vapor and ammonia gases as manifested by the plot produced by these equations. However, the slopes produced by Redlich-Kwong EOS approximate the gas constant, R value compared to Dieterici EOS. Thus, Redlich-Kwong EOS has better applicability to water vapor and ammonia. Other equations of state are recommended for validation using the other gas samples

INTRODUCTION:

Gas is a state of matter that is known to have particles which are very far apart from each other and that they are in constant random motion. A gas can behave as perfect or real gas.

A Perfect Gas is proposed by the Kinetic Molecular Theory as a very minute particle size negligible compared to the distance travelled by the molecule and which aside from being very far apart and in random motion, there is no transfer of energy during collisions. Real Gas system on the other hand differs from a Perfect Gas system because there is transfer of energy in the collision of gas molecules, resulting to molecular interactions. However, real gases approximate perfect gas behavior at very low pressure and significantly high temperature. A real gas deviates increasingly from ideality as it is compressed and cooled to near the point at which it will condense into liquid. The deviation from the ideal gas is particularly important at high pressure and low temperature.

The properties of the real gas can be express in terms of isotherm and compression factor (Z). Because perfect gas has compressibility factor equal to 1 under all conditions, deviation of Z from 1 is a measure of imperfection. At very low pressure all gases are expected to have Z≈1 and are behaving nearly perfectly. At high pressure all gases have Z>1, signifying that they are more difficult to compress than the perfect gas. At intermediate pressure most of the gases have Z<1, indicating that attractive forces are dominant and compression is easy.

PART I.

DATA AND CALCULATION

Table 1.1: tabulates real gas equation state relating pressure (P), volume (V) and temperature (T) of a given quantity of gas.

Real Gas Equations of the State

Van der Waals P= (RT/Vm-b)-(a/Vm2₎

Where a and b are van der Waals constants Berthelot P= (RT/Vm-b)-(a/TVm2₎

Where a and b are Berthelot constants Modified Berthelot P= RT/Vm[1+(9PTc/128PcT)][1-(6Tc2_/T2_]

Where Pc and Tc are critical state pressure and temperature Dieterici P= (RTe-a/RTVm_/Vm-b)

Where a and b are Dieterici constants Clausius [P-(a/T(Vm+c)2_)][Vm-b]=RT

Where a, b, and c are Clausius constants Redlich-Kwong P= (RT/Vm-b)-(a/T 1/2_Vm(Vm+b))

Where a and b are Redlich-Kwong constants General Virial PVm=RT[1+(B/Vm)+(C/Vm2_)+(D/Vm3_)+……+]

B, C, and D are the second, third, fourth……virial coefficients Respectively which are functions of temperature

Table 1.2: Tabulates experimental data on the variation of molar volume of carbon dioxide with the pressure at the critical point, (T=31.1 °C) in the table, alpha correspond to volumes to which 17 molar volume of carbon dioxide measured at 0°C and 1 atm, would be changed by altering the pressure to P and temperature to T.

Table 1.3: Tabulates experimental data for T=100°C, which is way above the critical temperature. The data (Ø) are volumes equal to Vm/VSTP.

The symbols used are defined as follows:

T= temperature of CO2 sample in degrees Celsius

VSTP= the volume of one mole CO2 at 1 atm and 0° C = 22.264 L Vm= the molar volume at pressure, P, and temperature, T

Pc = 73 atm; Tc = 31.1 °C; Vc = 0.0956 L

Van der Waals constants evaluated from Pc and Tc is: a= 3.60 L2 _{atm/mole and b= 0.0428 L/mole}

T= 31.1 °C; RT = 24.98 L atm/mole

Table 1.5: Theoretical P-V-T relationship for CO2; T=100 °C; RT = 30.65 L.atm/mole

Tabulation of Data

Table 1.2: Experimental P-V-T relationships for carbon dioxide, T=31.1˚C P(atm) alpha Vm 59.77 0.202 0.264548706 61.18 0.1933 0.253154776 62.67 0.1839 0.240844094 64.27 0.173 0.226568941 65.9 0.1632 0.2137344 67.6 0.1542 0.201947576 69.39 0.1409 0.184529271 71.25 0.1282 0.167896753 73.26 0.1122 0.1469424 73.83 0.1087 0.142358635 75.4 0.0609 0.079757506 77.64 0.0513 0.067184894 79.92 0.0494 0.064696565 82.44 0.0479 0.062732094

Table 1.3: Experimental P-V-T relationships for carbon dioxide, T=100 ˚C P(atm) alpha Vm 20.17 0.0654 1.456066 22.37 0.05878 1.308678 24.85 0.05265 1.1722 27.76 0.04684 1.042846 31.06 0.04155 0.925069 34.57 0.03702 0.824213 40.09 0.03153 0.701984 45.99 0.02711 0.603577 53.81 0.02276 0.506729 64.27 0.01855 0.412997 80.25 0.0142 0.316149 105.69 0.00998 0.222195

Table 1.4: Theoretical P-V-T relationships for carbon dioxide

0.08 0.0372 671.5053763 0.0064 562.5 109.0053763 312.25 84.014 0.11 0.0672 371.7261905 0.0121 297.5207 74.20552932 227.0909 80.573 0.1 0.0572 436.7132867 0.01 360 76.71328671 249.8 81.72 0.12 0.0772 323.5751295 0.0144 250 73.57512953 208.1667 79.426 0.14 0.0972 256.9958848 0.0196 183.6735 73.32241539 178.4286 77.132 0.16 0.1172 213.1399317 0.0256 140.625 72.51493174 156.125 74.838 0.18 0.1372 182.0699708 0.0324 111.1111 70.95885973 138.7778 72.544 0.2 0.1572 158.9058524 0.04 90 68.90585242 124.9 70.25 0.25 0.2072 120.5598456 0.0625 57.6 62.95984556 99.92 64.515 0.3 0.2572 97.12286159 0.09 40 57.12286159 83.26667 58.78 0.4 0.3572 69.93281075 0.16 22.5 47.43281075 62.45 47.31 0.5 0.4572 54.63692038 0.25 14.4 40.23692038 49.96 35.84

Table 1.5: Theoretical P-V-T relationships for carbon dioxide at T=100°C; RT=30.65

Vm Vm-b RT/(Vm-b) Vm2 _a/Vm2 _{P(atm) RT/Vm Pexp(atm)}

0.14 0.0972 315.3292181 0.0196 183.6735 131.6557487 218.9286 96.9822 0.16 0.1172 261.5187713 0.0256 140.625 120.8937713 191.5625 95.8368 0.2 0.1572 194.9745547 0.04 90 104.9745547 153.25 93.546 0.3 0.2572 119.1679627 0.09 40 79.16796267 102.1667 87.819 0.4 0.3572 85.806271 0.16 22.5 63.306271 76.625 82.092 0.5 0.4572 67.03849519 0.25 14.4 52.63849519 61.3 76.365 0.75 0.7072 43.33993213 0.5625 6.4 36.93993213 40.86667 62.0475 1 0.9572 32.02047639 1 3.6 28.42047639 30.65 47.73 1.5 1.4572 21.03348888 2.25 1.6 19.43348888 20.43333 19.095 2 1.9572 15.66012671 4 0.9 14.76012671 15.325 -9.54

CALCULATIONS AND EQUATIONS 1. Calculation of experimental volume Vm

Given that VSTP = 22.264 L/mole, the experimental molar volume of CO2 can be computed from the data as follows:

Alpha = [17 Vm/VSTP]

Vm = [alpha VSTP/17] =1.310(alpha) (1) Ø= [Vm/VSTP]

Vm= ØVSTP = 22.264 Ø (2) Compute column 3 of tables 2 and 3 by using this equation 2. Calculation of pressure from the van der Waals equation of state.

Tables 4 and 5 has been set up (columns 4-9) so as to compute for the pressure, P, corresponding to selected values of temperature and molar volume Vm, in accordance with the van der Waals equation of state.

The van der Waals equation of state is usually written as [P+ a/Vm2_][V

m- b]= RT

Where T is in Kelvin (3a) It can be rearranged to solve for the pressure P.

P=[RT/(Vm-b)]-[a/Vm2_{] (3b)}

For each value of Vm in column 4, compute for the values of the terms in column 5-8 and the van der Waals pressure in column 9.

3. Calculation of pressure from ideal gas equation of state.

For the selected values Vm and T= 100 °C, compute for the corresponding from ideal gas laws. P= RT/Vm (4)

Tabulate these in column 10 of table 5. TREATMENT OF DATA

1. From the data available in Tables 1.2, 1.3, 1.4 and 1.5, plot the different isotherms, P (atm) vs. Vm (L/mole). Use solid lines for experimental isotherms, dashed lines for van der Waals isotherms, and a dash and dot line for the ideal isotherms. Combine the plots of these 3 isotherms in a single graph, one for T=31.1 °C and one for T= 100 °C.

2. From the experimental isotherm graph, determine the experimental pressure (PEXP) corresponding to the volume Vm in tables 1.4 and 1.5. You may need to extrapolate from the graph. Use column 11 for these purpose.

3. Comment on the use of van der Waals equation as a description of a real gas with respect to its absolute accuracy (experimental vs. theoretical) and in comparison with the ideal gas law. You may do these graphically or you may use tabulated data.

4. Choose another equation of state and evaluate its ability to predict the P-V-T relationship for CO2 for the same P-V-T range and condition. Consult appropriate references to find the corresponding constants for the equation chosen.

GRAPHICAL REPRESENTATIONS AND INTERPRETATION

The graphs below plot the Pressure-Volume relationship of Carbon Dioxide at constant temperature and number of moles using van der Waals isotherm, Ideal Isotherm, and Experimetal Isotherm. Figure 1.1 plots the Pressure-Volume relationship of CO2 at 31.1°C, with pressure as a function of molar volume.

Fig. 1.1: At 31.1°C and below 100 atm, van der Waals isotherm and Experimental Isotherm is

reasonably the same. However, as the molar volume increases indepedently, van der Waals Isotherm, Experimental Isotherm, and Ideal Isotherm also approaches homogeneity in the sketch, signifying the behavior of a real gas and perfect gas approaching unity to each other at this condition.

Fig.1.2 The plot of the experimental isotherm is significantly different from the plot of the van der

Waals Isotherm at 100 °C. However, van der Waals isotherm approaches ideality in the condition of increasing molar volume.

Figures 1.1 and 1.2 shows us that at very low pressure and reasonably high temperature, gases’ behavior approaches ideality. In figure 1.1, the temperature is 31.1°C compared to figure 1.2 with temperature 100 °C. In figure 1.1, ideality is obtained at around 50 atm, while in figure 1.2, ideality is approximated by the van der Waals isotherm above 50 atm. With increasing molar volume and decreasing pressure to zero at constant temperature, a gas behaves ideally. Consequently, a deviation from ideality is significant to high pressure and low molar volume at constant temperature.

Figures 1.3 and 1.4 Shows the absolute accuracy of the van der Waals equation of state graphically

Fig.1.3: Experimental and theoretical van der Waals isotherms at 31.1°C

The curve in figure 1.3 shows that at 31.1°C experimental isotherm is accurate with reference to the theoretical van der Waals isotherm at 0.160≤Vm≤0.443 L/mol. Deviation of real gasfrom a perfect gas is significant at increasing pressure. However, figure 1.4 shows that at 100 °C, experimental isotherm attains unity with the theoretical van der Waals isotherm at approximately 90 atm and 20 atm with molar volumes 0.3 L/mol and 1.5 L/mol, respectively. Thus, determination of gas is best suitable at a condition of temperature equals 31.1 °C as manifested by figure 1.3

Validation of Redlich-Kwong Equation

Table 1.6: Pressures using Redlich-Kwong EOS at T=31.1°C

Vm RT/Vm Vm-b Vm+b T^1/2*Vm RT/Vm-b a/T^1/2* Vm(Vm+b) p (atm) 0.08 312.25 0.05031032 0.109689 68 1.395421083 496.2394187 416.6478 79.59167 0.11 227.0909 0.08031032 0.139689 68 1.91870399 310.8686897 237.9402 72.92851 0.1 249.8 0.07031032 0.129689 68 1.744276 354 355.0824 964 281.915 8 73.166 72 0.12 208.1667 0.09031032 0.149689 68 2.093131625 276.4464122 203.5409 72.90551 0.14 178.4286 0.11031032 0.169689 68 2.441986896 226.3248257 153.901 72.42386 0.16 156.125 0.13031032 0.189689 68 2.790842167 191.58854 120.4651 71.12348 0.18 138.7778 0.15031032 0.209689 68 3.139697438 166.09614 96.86686 69.22928 0.2 124.9 0.17031032 0.229689 68 3.488552709 146.5910225 79.58905 67.00197 0.25 99.92 0.22031032 0.279689 68 4.360690886 113.3218088 52.28876 61.03305 0.3 83.26667 0.27031032 0.329689 68 5.232829 063 92.36038 028 36.9656 4 55.394 74 0.4 62.45 0.37031032 0.429689 68 6.977105417 67.41903372 21.27208 46.14696 0.5 49.96 0.47031032 0.529689 68 8.721381771 53.08402323 13.8049 39.27912

Table 1.7: Pressures using Redlich-Kwong EOS at 100°C

Vm RT/Vm Vm-b Vm+b T^1/2*Vm RT/Vm-B a/T^1/2* Vm(Vm+b ) p (atm) 0.14 218.92 86 0.11031032 0.169689 68 2.704392723 277.5780073 138.968 138.61 0.16 191.56 25 0.13031032 0.189689 68 3.090734541 234.9753942 108.7764 126.199 0.2 153.25 0.17031032 0.229689 68 3.863418 176 179.7878 062 71.866 57 107.921 2

0.3 102.16 67 0.27031032 0.329689 68 5.795127263 113.2761739 33.37888 79.89729 0.4 76.625 0.37031032 0.429689 68 7.726836351 82.68664727 19.20806 63.47859 0.5 61.3 0.47031032 0.529689 68 9.658545439 65.10535174 12.46542 52.63993 0.75 40.866 67 0.72031032 0.779689 68 14.48781816 42.50906583 5.645668 36.8634 1 30.65 0.97031032 1.029689 68 19.31709 088 31.55662 491 3.2062 1 28.3504 1 1.5 20.433 33 1.47031032 1.529689 68 28.97563 632 20.82534 441 1.4388 11 19.3865 3 2 15.325 1.97031032 2.029689 68 38.63418176 15.54055648 0.813277 14.72728

Data in tables 1.6 and 1.7 regarding the validation of Redlich-Kwong equation can be better understood using figures 1.5 and 1.6.

Fig. 1.6: Comparison of modified Berthelot isotherm and ideal isotherm at 100°C

Figures 1.5 and 1.6 illustrates the P-V-T relationship of an perfect gas and ideal gas using Redlich-Kwong equation of state. At 31.1 °C, the Redlich-Kwong isotherm behaves ideally at 50 atm while at 100 °C, Redlich-Kwong behaves almost perfectly as ideal isotherm below 150 atm and shows less deviation from ideality for carbon dioxide compared to the condition of carbon dioxide being treated at 31.1 °C. Thus, Redlich-Kwong equation of state best describes the thermodynamic relationship of P-V-T for carbon dioxide at 100 °C.

PART II

Data and Calculations

The following data for nitrogen, methane water and ammonia are taken from the thermodynamic table. The data are given in terms of volume per kilogram and pressure is in terms of Mpa or kPa. These data will be used to test real gas equation of state and to calculate the compressibility factor at each temperature.

Tables 2.1, 2.2, 2.3 and 2.4 describes the behavior of gases in terms of compressibility factor at increasing temperature.

Table 2.1 shows that nitrogen behaves ideally at 80 K. However, the compressibility factor decreases as temperature and pressure were raised from 80 K to 125 K.

Table 2.1: Thermodynamic table of nitrogen

T(K) P(Mpa) V(m3_{/kg) P(atm) V(L/mol) PVm Z}

80 0.13699 0.163744 1.351986 4.584832 6.19863 0.944222 85 0.22903 0.101503 2.26035 2.842084 6.424106 0.921006 90 0.36066 0.066146 3.559437 1.852088 6.592391 0.892625 95 0.54082 0.044792 5.337478 1.254176 6.694137 0.858696 100 0.77881 0.031216 7.686257 0.874048 6.718158 0.818688 105 1.08423 0.022195 10.70052 0.62146 6.649944 0.771786 110 1.46717 0.015952 14.47984 0.446656 6.467508 0.716494 115 1.93875 0.011445 19.13397 0.32046 6.131674 0.649755 120 2.51248 0.007998 24.79625 0.223944 5.552971 0.563914 125 3.20886 0.004883 31.66899 0.136724 4.32991 0.422121

Table 2.2 shows the behavior of methane. At 115 K, compression factor of methane is approximating ideal compression factor. Compression factor for methane continuously decrease when temperature is raised up to 155 K. At 160 K however, the compression factor for methane drastically increased from 0.787 to stunning 1.799.

Table 2.2: Thermodynamic table for methane

T(K) P(Mpa) V(m3_{/kg) P(atm) V(L/mol) PVm Z} 115 0.13232 0.43048 1.305897 6.88768 8.9946 0.9531 61 120 0.19158 0.30615 1.890748 4.8984 9.261638 0.9405 65 125 0.26896 0.22359 2.654429 3.57744 9.49606 0.9257 97 130 0.3676 0.16702 3.62793 2.67232 9.69499 0.9088 38 135 0.49072 0.12717 4.84303 2.03472 9.85421 0.8895 5 140 0.64165 0.09839 6.332593 1.57424 9.969021 0.8677 74 145 0.82379 0.07716 8.130175 1.23456 10.03719 0.8435 8 150 1.04065 0.06117 10.27042 0.97872 10.05186 0.8166

53 155 1.2958 0.04892 12.78855 0.78272 10.00986 0.7870 07 160 1.59296 0.03935 15.72129 1.5024 23.61967 1.7990 2

Table 2.3 shows the behavior of water vapor as a function of temperature.378.15K, the temperature at which the gas behaves more likely as an ideal gas

Table 2.3: Thermodynamic table of water vapor

T(K) P(Mpa) V(m3_{/kg) P(atm) V(L/mol) PVm Z}

378.15 0.12082 1.4194 1.192401 25.5492 30.46488 0.981757 383.15 0.14328 1.2102 1.414064 21.7836 30.8034 0.979712 388.15 0.16906 1.0366 1.668492 18.6588 31.13207 0.97741 393.15 0.19853 0.8919 1.959339 16.0542 31.45562 0.975009 398.15 0.2321 0.77059 2.290649 13.87062 31.77272 0.97247 403.15 0.2701 0.6685 2.66568 12.033 32.07612 0.96958 408.15 0.313 0.58217 3.08907 10.47906 32.37055 0.966493 413.15 0.3163 0.50885 3.121638 9.1593 28.59202 0.843346 418.15 0.4154 0.44632 4.099679 8.03376 32.93584 0.959854 423.15 0.4759 0.39278 4.696768 7.07004 33.20634 0.956302

Table 2.4 shows that at 223.15K, the ammonia gas behaves almost the same with that of the ideal gas. As the temperature and pressure increases, ammonia gas deviates from ideality to imperfection as manifested by compression factor, Z, decreasing at increasing temperature and pressure.

Table 2.4: Thermodynamic table of ammonia

T(K) P(kPa) Vm3_{/kg) P(atm) V(L/mol) PVm Z}

223.15 40.86 2.62667 0.403257 44.65339 18.00679 0.983349 225.15 45.94 2.3544 0.453393 40.0248 18.14695 0.9822 227.15 51.52 2.11503 0.508463 35.95551 18.28204 0.9808 229.15 57.66 1.90406 0.56906 32.36902 18.41991 0.979572 231.15 64.38 1.71769 0.635381 29.20073 18.55359 0.978144 233.15 71.72 1.55269 0.707821 26.39573 18.68346 0.976541 235.15 79.74 1.40627 0.786973 23.90659 18.81383 0.974991 237.15 88.48 1.27607 0.87323 21.69319 18.94314 0.973413 239.15 97.98 1.16004 0.966987 19.72068 19.06965 0.971719

The data in tables 2.1-2.4 were plotted in figure 2.1-2.4 to better understand the data obtained from calculations.

Relating compressibility factor to pressure, theoretically, the gas are expected to be more compressible at moderate pressure because in this case forces are helping to draw the molecules together. At high pressure, the gas can be expected to be less compressible due to the force that help

to drive molecules apart and at low pressure intermolecular forces play no significant role and thus, gas behaves perfectly. Since, ideal gas has a Z equal to 1 at all conditions; the deviation of the compressibility factor from 1 is the measure of imperfection.

B. Treatment of Data

1. Calculate the compressibility factor at each temperature for all gases. Consult appropriate references for the equations and critical constants. Evaluate your results. What do your calculations indicate regarding the behavior of the gases with increasing temperature?

2. Evaluate the validity of the ideal gas law by testing the linearity of the graph of PVm vs T. the slope should be equal to R if the ideal gas law is valid.

3. Choose two real gas equation of state and evaluate their applicability to the four gases. Which equation of state best describes each gas?

GRAPHIAL REPRESENTATION PART II:

Fig. 2.2: Methane gas

Fig. 2.4: Ammonia gas

Figures 2.1-2.4 illustrates the validity of the ideal gas law with the argument stating that ideal gas is true for the sketch with slope, m equals to R value or 0.0820574. Among the figures, only figure 2.4 accounting for ammonia has slope nearest to 0.0820574, while others have significantly far value for m with reference to the R value which is a gas constant. With these, ideal gas law is appropriate only to ammonia gas.

Validation of Redlich-Kwong equation of state

Table 2.5: Thermodynamic table of nitrogen

T(K) RT Vm(L/mol) Vm-b Vm(Vm+b) a/T1/2 Vm(vm+b) P (atm) PVm Z 80 6.5648 4.584832 4.555032 21.15731 0.04761 1.39360 6.389464 0.973291 85 6.9751 2.842084 2.812284 8.162136 0.11972 2.36049 6.708736 0.961812 90 7.3854 1.852088 1.822288 3.485422 0.27247 3.78034 7.001524 0.948022 95 7.7957 1.254176 1.224376 1.610332 0.57402 5.79305 7.265515 0.93199 100 8.206 0.874048 0.844248 0.790007 1.140446 8.57944 7.498848 0.913825 105 8.6163 0.62146 0.59166 0.404732 2.17241 12.39050 7.700205 0.893679 110 9.0266 0.446656 0.416856 0.212812 4.03657 17.61742 7.868928 0.871749 115 9.4369 0.32046 0.29066 0.112244 7.48500 24.98213 8.005775 0.848348 120 9.8472 0.223944 0.194144 0.056824 14.47370 36.24741 8.11739 0.824335 125 10.2575 0.136724 0.106924 0.022768 35.39394 60.53868 8.277091 0.806931

Table 2.6: Thermodynamic table for methane

T(K) RT Vm (L/mol) Vm-b Vm(Vm+b) a/T1/2 Vm(Vm+b) P (atm) PVm Z 115 9.4369 6.88768 6.85468 47.66743 0.02637 1.35033 9.300655 0.985563 120 9.8472 4.8984 4.8654 24.15597 0.05095 1.97296 9.664395 0.981436 125 10.2575 3.57744 3.54444 12.91613 0.09337 2.80059 10.01897 0.976746 130 10.6678 2.67232 2.63932 7.229481 0.16357 3.87829 10.36406 0.971527 135 11.0781 2.03472 2.00172 4.207231 0.27582 5.25846 10.69951 0.965825 140 11.4884 1.57424 1.54124 2.530181 0.45038 7.00361 11.02537 0.959696 145 11.8987 1.23456 1.20156 1.564879 0.71553 9.18717 11.34212 0.953223 150 12.309 0.97872 0.94572 0.990191 1.11181 11.90366 11.65036 0.946491 155 12.7193 0.78272 0.74972 0.63848 1.69622 15.26917 11.95149 0.939634 160 13.1296 1.5024 1.4694 2.306785 0.46209 8.47325 12.73022 0.969582

Table 2.7: Thermodynamic table for water vapor

T(K) RT VM (L/mol) Vm-b Vm(Vm+b) a/T1/2_Vm(V m+b) P(atm) PVm Z 378.15 31.03099 25.5492 25.5305 653.2394 0.00287 1.21256 30.98019 0.998363 383.15 31.44129 21.7836 21.7649 474.9326 0.00393 1.44065 31.38264 0.998135 388.15 31.85159 18.6588 18.6401 348.4997 0.00532 1.70344 31.78419 0.997884 393.15 32.26189 16.0542 16.0355 258.0376 0.00714 2.00475 32.18479 0.99761 398.15 32.67219 13.87062 13.85192 192.6535 0.00951 2.34916 32.58438 0.997312 403.15 33.08249 12.033 12.0143 145.0181 0.01255 2.74103 32.9829 0.99699 408.15 33.49279 10.47906 10.46036 110.0067 0.01645 3.18542 33.38028 0.996641 413.15 33.90309 9.1593 9.1406 84.06406 0.02139 3.68766 33.77647 0.996265 418.15 34.31339 8.03376 8.01506 64.69153 0.02763 4.25347 34.17142 0.995863 423.15 34.72369 7.07004 7.05134 50.11768 0.03546 4.88894 34.56506 0.995432

Table 2.8: Thermodynamic table for ammonia T(K) RT Vm (L/mol) Vm-b Vm(Vm+b) a/T 1/2_Vm(V m+b) P(atm) PVm Z 223.15 18.31169 44.65339 44.62919 1993.925 0.00081 0.40949 18.28525 0.998556 225.15 18.47581 40.0248 40.0006 1601.984 0.00100 0.46087 18.4466 0.998419 227.15 18.63993 35.95551 35.93131 1292.798 0.00124 0.51752 18.60772 0.998272 229.15 18.80405 32.36902 32.34482 1047.753 0.00152 0.57983 18.76861 0.998116 231.15 18.96817 29.20073 29.17653 852.682 0.00187 0.64824 18.92927 0.997949 233.15 19.13229 26.39573 26.37153 696.734 0.00228 0.72321 19.08966 0.997772 235.15 19.29641 23.90659 23.88239 571.5245 0.00276 0.80520 19.2498 0.997584 237.15 19.46053 21.69319 21.66899 470.5939 0.00334 0.89473 19.40965 0.997386 239.15 19.62465 19.72068 19.69648 388.9046 0.00403 0.99231 19.56922 0.997176

Data in tables 2.5-2.8 can be better understood using Figures 2.5-2.8.

Fig. 2.6: Methane

Fig. 2.8: Ammonia

The graphs from figures 2.5 through 2.8 illustrate the validity of Redlich-Kwong Real Gas Equation of State. Theoretically, the equation of state must have a slope, m equal to the gas constant, R. Ammonia gas and Water Vapor’s slopes show only a very small deviation from R value with mwater vapor = 0.079 and mammonia = 0.80. With these, Redlich-Kwong equation of state describes the behavior of Water Vapor and Ammonia more accurately with reference to the deviation of the slope to the R value obtained by plotting the pVm of gas samples as a function of their temperature in Kelvins (K).

Validation of Dieterici equation of state

Table 2.9 Thermodynamic Table for Nitorgen

T(K) Vm (L/mol) RT Vm-b (-a/RT*Vm) p (atm) pVm Z

80 4.584832 6.564592 4.543013439 -0.057599 53 1.364107371 6.2542 03 0.952718 85 2.842084 6.974879 2.800265439 -0.087453 37 2.282217116 6.4862 53 0.929945 90 1.852088 7.385166 1.810269439 -0.126744 25 3.593955619 6.6563 22 0.90131 95 1.254176 7.795453 1.212357439 -0.177316 98 5.385213042 6.7540 05 0.8664 03 100 0.874048 8.20574 0.832229439 -0.241711 40 7.742848278 6.7676 21 0.824742 105 0.62146 8.616027 0.579641439 - 10.75321324 6.6826 0.7756

0.323765 03 92 12 110 0.446656 9.026314 0.404837439 -0.429998 12 14.50387186 6.4782 41 0.717706 115 0.32046 9.436601 0.278641439 -0.573272 03 19.08978374 6.1175 12 0.648275 120 0.223944 9.846888 0.182125439 -0.786161 45 24.63217752 5.5162 28 0.5602 125 0.136724 10.257175 0.094905439 -1.236168 44 31.39608727 4.2925 99 0.418497

Table 2.10 Thermodynamic Table for Methane

T(K) V(L/mol) RT Vm-b (-a/RT*Vm) p (atm) pVm Z

115 6.88768 9.436601 6.841261955 -0.044678 53 1.319093958 9.0854 97 0.962793 120 4.8984 9.846888 4.851981955 -0.060205 23 1.910878267 9.3602 46 0.950579 125 3.57744 10.257175 3.531021955 -0.079138 41 2.683848332 9.6013 06 0.936058 130 2.67232 10.667462 2.625901955 -0.101868 03 3.668950681 9.8046 1 0.919114 135 2.03472 11.077749 1.988301955 -0.128834 24 4.897981954 9.9660 22 0.8996 43 140 1.57424 11.488036 1.527821955 -0.160572 34 6.403794071 10.081 11 0.877531 145 1.23456 11.898323 1.188141955 -0.197692 19 8.217898876 10.145 49 0.852682 150 0.97872 12.30861 0.932301955 -0.241057 14 10.3743914 10.153 62 0.8249 2 155 0.78272 12.718897 0.736301955 -0.291696 75 12.9036093 10.099 91 0.794087 160 1.5024 13.129184 1.455981955 -0.147219 10 7.782968945 11.693 13 0.890621

Table 2.11 Thermodynamic Table for Water Vapor

T(K) V(L/mol) RT Vm-b (-a/RT*Vm) p (atm) pVm z

378.15 25.5492 31.03000581 25.51626575 -0.008827 52 1.205399465 30.796 99 0.992491 383.15 21.7836 31.44029281 21.75066575 -0.010218 37 1.430791141 31.167 78 0.991332 388.15 18.6588 31.85057981 18.62586575 -0.011775 98 1.689999711 31.533 37 0.9900 41 393.15 16.0542 32.26086681 16.02126575 -0.013512 43 1.986601841 31.893 3 0.988607 398.15 13.87062 32.67115381 13.83768575 - 2.324845537 32.247 0.9870

0.015443 21 05 19 403.15 12.033 33.08144081 12.00006575 -0.017580 84 2.708728796 32.594 13 0.985269 408.15 10.47906 33.49172781 10.44612575 -0.019940 60 3.142839546 32.934 0.9833 47 413.15 9.1593 33.90201481 9.126365752 -0.022537 74 3.631948007 33.266 1 0.981243 418.15 8.03376 34.31230181 8.000825752 -0.025388 05 4.181086454 33.589 85 0.978945 423.15 7.07004 34.72258881 7.037105752 -0.028507 83 4.795536766 33.904 64 0.976443

Table 2.12 Thermodynamic Table for Ammonia

T(K) V(L/mol) RT Vm-b (-a/RT*Vm) p (atm) pVm z

223.15 44.65339 18.31110881 44.61293013 -0.006586 03 0.407749669 18.207 4 0.994337 225.15 40.0248 18.47522361 39.98434013 -0.007282 39 0.458708794 18.359 73 0.9937 49 227.15 35.95551 18.63933841 35.91505013 -0.008035 20 0.514830621 18.511 0.9931 15 229.15 32.36902 18.80345321 32.32856013 -0.008847 60 0.576512574 18.661 15 0.992432 231.15 29.20073 18.96756801 29.16027013 -0.009722 71 0.644165682 18.810 11 0.9916 98 233.15 26.39573 19.13168281 26.35527013 -0.010663 65 0.7182151 18.957 81 0.990912 235.15 23.90659 19.29579761 23.86613013 -0.011673 81 0.7991179 19.104 18 0.99007 237.15 21.69319 19.45991241 21.65273013 -0.012756 41 0.887336151 19.249 15 0.9891 69 239.15 19.72068 19.62402721 19.68022013 -0.013914 99 0.983365528 19.392 64 0.988209

The tables above relates the thermodynamic quantities of selected gas samples using Dieterici equation of states. Figures 2.9-2.12 plots pVm as a function temperature in Kelvin.

Fig 2.9 Nitrogen Gas

Fig. 2.11 Water Vapor

The graphs from figures 2.9 through 2.12 illustrate the validity of Dieterici Real Gas Equation of State. The plot of water vapor and ammonia gas are linear compared to the plots produced by methane and nitrogen gas. However, the slopes of water vapor and ammonia are 0.069 and 0.074, respectively. These slopes deviate significantly from the condition to which an equation must be valid, that is, when the slope approximates or equals the real gas constant, R, having the value 0.08206. Thus, Dieterici equation of state has poor applicability to nitrogen gas, methane gas, water vapor, and ammonia. However, the equation may be better applicable to other gas sample(s).

With these, ammonia and water vapor are the gases best described by the two equations of state. However, between the two equations, Redlich-Kwong equation has better applicability to the given gas samples compared to Dieterici equation.

CONCLUSION:

The objectives of this experiment is to evaluate the P-V-T relationship of carbon dioxide using van der Waals and other equation of state as well as the applicability of two other equations of state to other given gas samples.

With the calculated and plotted thermodynamics data for carbon dioxide and other gases, this paper illustrates that real gases approximate ideality at very low pressures and significantly high temperature as illustrated by the z compression factors of gas samples. Van der Waals isotherm approximates ideal isotherm at pressure approaching zero and very high temperatures.

Redlich-Kwong and Dieterici equation of state were tested for their applicability with nitrogen gas, methane, water vapor, and ammonia. With the plots produced by these equations, Redlich-Kwong equation of state has better applicability to water vapor and ammonia compared to Dieterici equstion of state. Other equations are recommended for testing their applicability to other gases.