Thermodynamics Gas Laws Real Gases and Critical Point QUANTITATIVE ANALYSIS OF A REAL GAS AND DETERMINING ITS CRITICAL POINT Obsering sulhur hexafluoride in both the liquid and gaseous states. Plotting isotherms in a -V diagram. Obsering how the behaiour of real gases deiates from that for the ideal gas state. Determining the ical oint. Plotting ressure cures for a saturated aour. UE43 1/18 JS/GH/UD 1.4 1. C 1. 1.1 (5 C).99 (45 C) r.8.88 (4 C).66 (3 C).6 B.4 A.44 ( C). (1 C)...5 1. 1.5..5 3. 3.5 4. Fig. 1: Claeyron hase diagram, isotherms calculated from simlified form of Van der Waals' equation (alues in brackets are material-secific temeratures for sulhur hexafluoride). V r BASIC PRINCIPLES The ical oint of a real gas is characterised by the ical temerature TC, the ical ressure C, and the ical density ρc. Below the ical temerature, the substance is gaseous at large olumes and liquid at small olumes. At intermediate olumes it can exist as a liquid/gas mixture, in which changing the olume under isothermal conditions causes a change of state: the gaseous fraction increases as the olume is increased, while the ressure of the mixture remains constant. As the liquid and the aour hae different densities, they are searated by the graita- 1 / 9
UE43 tional field. As the temerature rises, the density of the liquid decreases and that of the gas increases until the two densities conerge at the alue of the ical density. Aboe the ical temerature, the gas can no longer be liquefied. Howeer, under isothermal conditions the gas does not obey Boyle s Law until the temerature is raised considerably aboe the ical temerature. The ideal gas equation alies for any ideal gas: (1) V n R T. : Pressure] V: Volume n: Moles R = 8.314 J/(mol K): Uniersal gas constant T: Absolute temerature The behaiour of real gases is gien to a good aroximation by the Van-der-Waals equation: n a V n b n R T. V () a: Cohesion ressure (internal ressure) b: Coolume. Cohesion ressure secifically takes into account Van der Waals forces between gas molecules, while the coolume takes account of finite exansion of gas molecules. For the isotherms in the -V diagram, the following equation therefore suffices: (3) V R T a V n b Vn Since there is a oint of inflection at the ical temerature (Fig. 1), the following is true:. constant after a secific alue, een when the olume continues to be reduced. The isotherms thus follow an een, isobaric ath (horizontal lines in Fig. 1). It is along these horizontal lines where the condensation of the gas takes lace. The condition for the osition of the lines is that the areas A and B are required to be equal. The eneloe for the horizontal lines is called a binodal cure and indicates the region where gas and liquid hases coexist. Sulhur hexafluoride is esecially suitable for inestigating the roerties of real gases, as its ical temerature (TC = 319 K) and its ical ressure (C = 37.6 bar) are both relatiely low. It is also non-toxic and is quite safe for use in teaching and in ractical classes. The aaratus for inestigating the ical oint consists of a transarent measurement cell, which has ery thick walls and can withstand high ressures. The internal olume of the cell can be changed by turning a hand-wheel, which allows one to make fine adjustments and can be read with a recision down to 1/1 of the maximum olume. Pressure is alied by a hydraulic system using castor oil of harmacological quality. The hydraulic system is searated from the cell by a conical rubber seal, which rolls u when the olume is changed. This form of construction ensures that the ressure difference between the measurement cell and the oil sace is ractically negligible. Therefore, instead of measuring the gas ressure directly, a manometer measures the oil ressure, which aoids haing a dead olume in the gas sace. The measurement cell is enclosed within a transarent water jacket. During the exeriment a thermostatic water bath maintains a recisely controlled and adjustable constant temerature, which is measured by a digital thermometer. During obserations of the transition from the gaseous to the liquid hase and the reerse rocess, the fact that there is ery little dead olume makes it ossible to obsere the formation of the first dro of liquid or the disaearance of the last bubble of gas. (4) d dv and d dv. From this the cohesion ressure and coolume are found to be as follows: (5) a 3 V n and b V 3 n. By substituting into equation (3) and introducing the simlified alues. (6) r, Vr V T and Tr V T This results in the simlified form of the Van der Waals-equation 3 r 3 Vr 1 8 Tr, Vr (7) This is indeendent of material. In the simlified reresentation, the following equation suffices for the isotherms: (8) r V 8T 3. r r 3Vr 1 Vr In the Claeyron diagram grah (Fig. 1) they exhibit a maximum and minimum below the ical ressure. In fact, though, in the case of isothermal comression the ressure remains LIST OF EQUIPMENT 1 Critical Point Aaratus 167 (U141) 1 Immersion/Circulation Thermostat @3V 18654 (U144-3) or 1 Immersion/Circulation Thermostat @115V 18653 (U144-115) 1 Digital Quick Resonse Pocket Thermometer 183 (U11853) 1 K-Tye NiCr-Ni Immersion Sensor -65 55 C 184 (U11854) Silicone Tubing, 6 mm 16 (U1146 Additionally required: Sulhur hexafluoride (SF6) Anti-freeze rotection (e.g. Glysantin G 3 from BASF) to establish calibrated temerature medium Comressor or bicycle um and ale for olume calibration / 9
UE43 SAFETY INSTRUCTIONS Before set-u and oeration of the ical oint aaratus, it is essential for the safety instructions in section of the instruction manual for the aaratus to be read through carefully and comlied with. GENERAL INSTRUCTIONS On deliery, the ical oint aaratus is filled with hydraulic fluid. The test gas is not included. Before filling with the test gas, carry out a olume calibration using air as an aroximation of an ideal gas. Volume calibration and filling with test gas are described under "Procedure" in these exeriment instructions. Instructions for storage after long eriods out of use are to be found in section 9 of the manual for the ical oint aaratus. Owing to the ineitable diffusion of the test gas through the conical seal, it is necessary to degas the hydraulic fluid in the equiment, as described in chater 1. This must be done before the equiment is ut away for storage (after remoing the test gas) or if it has been in use for a long time. The threaded bush in the frame must be lubricated regularly and also insected at lengthier interals. Refer to section 11 for instructions. Maintenance work as described in chater 1 is only required if the rubber comonents get worn out and their functionality is adersely affected. SET-UP Set u the aaratus at a height suitable for a good iew of the measurement cell and align it in such a way that the safety ale is ointing away from any eole or roerty which might sustain damage. Use silicone tubing to connect the outlet of the circulation thermostat to the inlet of the calibrated temerature jacket and the outlet of the jacket to the inlet of the circulation thermostat. Make the calibrating medium from arts of water and 1 art of anti-freeze by olume. Fill the circulation thermostat. EXPERIMENT PROCEDURE Zero oint calibration The zero oint for the olume scale must be determined by conducting a calibration. For this, we take adantage of the fact that in a ressure range of 1-5 bar and in a temerature range of 7-34 K, air acts as a near-ideal gas (the real gas factor has a deiation of less than 1% from 1). Therefore, at a constant temerature (e.g. room temerature) for two iston dislacements s and s1 and for the corresonding ressures and 1 of the traed air, we get: (9) s 1 s1 Substituting (1) s 1 s s 1 s and rearranging gies: 1 s Rough calibration of scales: Oen the regulating ale wide. Loosen the grub screw for the ernier scale by half a turn (it is now ossible to turn the scale easily on the threaded axle without moing the handwheel, although a counterressure acts against this indeendent moement). Wind the handwheel out till you detect a noticeable resistance. Without turning the handwheel, turn the ernier scale on the threaded axle till the. mark is on the to and the fixed scale shows arox. 48 mm. Loosen the knurled screws of the fixed scale and shift the scale to the side till the 48-mm bar is exactly aboe the centre line of the ernier scale (see Fig. ). Tighten the knurled screws again. In doing so, make sure that the fixed scale does not ress against the ernier scale. 1 3 4 5mm 1 9 1 8 1 7 1 6 Fig. : Piston osition reading at 48. mm 1 5 Zero correction: Shut the regulating ale (the ressure in the measuring cell now corresonds to the ambient ressure = 1 bar. To within the accuracy of the measurement, the manometer should dislay an excess ressure of bar). 3 / 9
UE43 Wind the handwheel in till an excess ressure of 15 bar has been reached (absolute ressure 1 = 16 bar). Read the iston osition s1 and calculate the dislacement s = s s1. Calculate the zero corrected iston osition s1, corr using equation 1. Adjust the ernier scale to the corrected alue. If required, wind the handwheel out a little and secure the ernier scale with the grub screw. Measurement examle: = 1 bar, 1 = 16 bar, 1 = 15 bar s = 48. mm, s1 = 3.5 mm, s = 44.5 mm Therefore, s1, corr =.97 mm. The ernier scale must therefore be adjusted so that now only.97 mm are shown instead of 3.5 mm. Note: After calibrating the zero oint, it is ossible to obtain qualitatiely accurate measured alues. With regard to temerature T and ressure, it is also ossible to obtain quantitatiely accurate measurements of the isotherms in range around to the ical oint where the two hases exist simultaneously. Howeer, esecially in the liquid hase, the measured isotherms are rather too widely searated. Volume calibration using air as an ideal gas The exact relation between the olume VG in the measuring cell and the scale reading s is deendent on the olume of oil in the oil chamber. The oil chamber also exands marginally in roortion to the ressure as a result of the sring in the manometer tube. Additionally, when the temerature is increased, the castor oil exands to a greater extent than the rest of the equiment. This means that the ressure rises at a slightly greater rate at higher temeratures. All of these henomena can be calculated if aroriate calibration has been effected using air as an ideal gas. The following alies according to the ideal gas equation (1): (11) V n R. T Fig. 3: Exeriment set-u After taking the oerressure reading e, the absolute ressure can be calculated from: (1) e 1bar The absolute temerature is gien by: (13) T where 73.15 C The olume is gien by: (14) V A s where A = 3.14 cm and at the effectie iston osition (15) s se s se: Piston osition reading : Absolute ressure : Temerature in C s, P, : Free arameters Substituting equations (13), (14) and (15) into equation (11) results in the following: (16) se s A n R. If we take seeral readings N at arious temeratures and ressures, we can calculate the term: N i sei s i i A (17) Q n R i1 i 4 / 9
1 9 1 8 1 7 1 6 1 5 UE43 The free arameters s, P, and n should be aroriately selected so that the alue of Q is reduced to a minimum. Connect the lastic tube (3-mm internal diameter) to the 1/8" gas connection fittings. Oen the regulating ale. Wind the handwheel outwards, making the iston moe till it reaches say the 46. mm osition. Use a comressor or a bicycle um to create an excess air ressure of arox. 3-8 bar in the measuring cell. Shut the regulating ale. To record measurements, ary the olume in the measuring cell or the temerature of the thermostat and wait till a stationary equilibrium has been attained. Then take a ressure reading. Use aroriate adjustment software to set the s, P, and n arameters so that the quadratic equation for the errors Q is reduced to a minimum (see equation 17). If you like, you can adjust the ernier scale around s so that this correction is not necessary. With the set arameters, it is ossible to calculate the effectie iston dislacement s from the measured dislacement se using Equation 15 and then to calculate the calibrated measuring cell olume using Equation 14. Filling with test gas Sulhur hexafluoride (SF6) is a non-toxic gas and is absolutely safe for humans. The MAC alue for danger of suffocation on account of oxygen deriation is 1 m. That is equialent to 6 filled measuring cells er 1 m 3 of air. Howeer, SF6 is extremely harmful to the enironment and can gie rise to a greenhouse effect 4, times stronger than CO. Therefore, do not allow large quantities to be released into the enironment. 1 3 4 5mm Filling begins with seeral flush cycles in which the air is flushed out of the ie. The number of cycles required to flush out the air deends on the length of the ie (more recisely, on the ratio of the ie length to the olume of the measuring cell). In the rocess, care should be taken that the quantity of the greenhouse gas SF6 released in the enironment is reduced to a minimum. If necessary, ull out the rotection for the gas connection and loosen the ale nut (11 mm) to remoe the 1/8" gas connection fittings. Connect the ie (if necessary with adaters) to the gas fitting. Beginning with the ale nut, slide the sulied screw joints onto the tubing. (See Fig. 4: follow the sequence and alignment secified along with the cable binder) Insert the ie into the regulating ale and tighten the ale nut till the oint is reached where it is no longer ossible to moe the ie any further using only your fingers. Hold the regulating ale still with an oen-end sanner (13 mm) and tighten the ale nut by a further 7. Now, the connection is gas-tight. When loosening the ale nut afterwards, the regulating ale also needs to be held still with a sanner. Use the handwheel to set the iston osition to 1 mm. Slowly oen the regulating ale and let in the SF6 till a ressure of arox. 1 bar has been attained. Shut the regulating ale. Oen the flush ale slightly till the ressure has droed to almost bar. Shut the flush ale. After at least four flush cycles, oen the regulating ale till the ressure attained is once again 1 bar. Shut the regulating ale. Turn the handwheel in the reerse direction till the iston reaches a osition of say 46 mm. Slowly oen the regulating ale and shut it again when a ressure of 1 bar has been attained. If the equiment is used only occasionally, it is more ractical to draw the test gas from a MINICAN gas container (e.g. from the comany Westfalen (www.westfalen-ag.de)). The gas connection of a MINICAN container is similar in design to a commercial sray can, i.e. it oens when the MINICAN container is ressed directly onto the gas connection fittings. Here too, filling begins with seeral rinsing cycles for flushing out the air. a Fig. 4: Connecting a fixed ie (a) flush ale, (b) regulating ale According to the rinciles of good laboratory ractice, it is recommended to utilise a gas suly (e.g. SH ILB gas cylinder and Y11 L15DLB18 regulating ale from Airgas (www.airgas.com)) ia fixed ies (outer diameter of 1/8" and, if necessary, adaters, e.g. from Swagelok (www.swagelok.com)), esecially if the equiment is regularly in oeration. b If necessary, ull off the rotection for the gas connection. Use the handwheel to set the iston osition to 1 mm. After remoing the rotectie ca, osition the MINICAN container with SF6 onto the gas connection fittings (Fig. 5). Press the MINICAN container onto the gas connection fittings, slowly oen regulating ale (b) and let in SF6 till a ressure of arox. 1 bar has been attained. Shut the regulating ale. Oen the flush ale slightly till the ressure has droed to almost bar. Shut the flush ale. 5 / 9
1 9 1 8 1 7 1 6 1 5 UE43 After at least four flush cycles, ress the MINICAN gas container against the gas connection fittings. Slowly oen the regulating ale and let in SF6 till a ressure of arox. 1 bar has been attained. Shut the regulating ale. Wind the handwheel in the oosite direction till the iston reaches a osition of say 46 mm. Press the Minican gas container against the gas connection fittings, slowly oen the regulating ale and shut it again when a ressure of 1 bar has been attained. SF 6 1 3 4 5mm With the olume at its maximum, set u temeratures = 1 C, C, 3 C, 4 C, 45 C and 5 C on the circulation thermostat. For each of these temeratures, decrease the olume in the measurement cell ste by ste until the iston is at the 1 mm osition. For each iston osition wait until stable equilibrium is reached then read off the iston osition se and the ressure e oer atmosheric, making a note of both these. Next, starting with the lowest olume ossible, start increasing the olume ste by ste until the iston osition is 1 mm. For each iston osition wait until stable equilibrium is reached then read off the iston osition se and the ressure e oer atmosheric, making a note of both these. During the exeriment kee obsering the liquid and gas states, the dynamic state at the hase transition and the deeloment of transition oints at arious temeratures. e = 8-1 bar a b e = 6 bar max. mm 1 mm 5 mm s e Fig. 5: Filling with test gas from a MINICAN gas container (a) flush ale, (b) regulating ale One gas filling can remain in the measuring cell for seeral days. If no exeriments are being conducted, wind the handwheel back till the iston is in a osition where it is subjected to the lowest ossible ressure say, for instance, 46 mm. If ossible the aaratus should always be ket filled with the thermal medium. Recording of isotherms Due to the bubbles of aour which deelo throughout the liquid, the interhase boundary surface in the case of the hase transition from liquid to gas is much greater than that for the transition from gas to liquid where the interhase boundary area is limited to the surface of the liquid itself. In order for equilibrium to settle down as quickly as ossible during each recording, the following rocedure is recom-mended (Fig. 6): When the ressure of SF6 at maximum olume, i.e. when the handwheel is wound all the way out, is 8 1 bars, the iston osition is set u till there is 1 mm between the lowest and highest ressure settings, the highest being when the wheel is wound all the way in. For small olumes where se < 1 mm, the iston osition is set for 1 mm between the highest ressure and the lowest ressure, i.e. when the handwheel is wound all the way out. The setting for equilibrium takes about 1-5 min using this method, whereby recording the measurements at the edge of the region where both hases are resent takes u the greatest amount of the time. Fig. 6: Setting olume as a function of ressure Determining the mass of gas mm 1 mm 5 mm If necessary, remoe the gas suly ie and attach gas connection fittings. Wind out the handwheel, say to 46 mm. Oen the regulating ale a little and release the gas through the gas connection fittings into a gas-tight lastic bag. Shut the regulating ale. Determine the mass of the released gas. In doing this, take into consideration the emty weight of the bag and the buoyancy of air. Reduce the olume of the measuring cell till the ressure in the measuring cell has reached its original alue. Calculate the original mass of gas from the olume difference before and after emtying the measuring cell and the olume which is still resent in the measuring cell. Alternatiely the mass of gas can be determined by comarison with alues quoted in literature (e.g. Clegg et al. in: Landolt-Börnstein Coefficient alues and functions, olume II, art 1, Sringer-Verlag, Berlin, 1971). s e 6 / 9
UE43 SAMPLE MEASUREMENT AND EVALUA- TION Volume calibration using air as an ideal gas Minimising the alue of Q in equation (17) gies the following alues of the arameters with the measurements in Table 1: s.19mm.3mm/bar (18).34mm/ C n.88mol Note: These arameters aly to the aaratus used for this examle. Volume calibration needs to be carried out secifically for eery aaratus and may also need to be checked. Determining the mass of gas Mass of gas m: 1.5 g Tab. 1: Measurements for olume calibration i se / mm / bar 1 4.. C 6.6.. C 1.4 3 1.. C 3.3 4 5.. C 41.8 5 3.5. C 53.9 6 5.. C 41.8 Recording of isotherms For each of the configured temeratures, conert the readings for ressure oer atmosheric into absolute ressures using equation (1) with atmosheric res-sure = 1 bar and enter these into Table using the unit (1 = 1 bars). Calculate the olumes V using equation (14) with the arameters (18) and use them to obtain the secific olumes = V/m, entering them into Table for each of the temeratures using the unit ml/g (1 cm 3 = 1 ml). Plot of isotherms as a -V diagram (Claeyron diagram) Plot the absolute ressures against the secific ol-umes for each temerature (Table, Fig. 7). Read off the arameters for the ical temerature from the - diagram in Fig. 7: = 45 C = 3.74 = 1 / =.74 g/ml The alues agree ery well with those quoted in literature for sulhur hexafluoride = 45.5 C, = 3.76 and = 1/ =.74 g/ml. Below the ical temerature the isotherms in Fig. 7 differ markedly from the hyerbolic cure for ideal gases but a-roximate to the hyerbola aboe that temerature. Sulhur hexafluoride acts like a real gas, which een aboe the ical temerature does not comletely reach the state of an ideal gas. The deiation from the state for an ideal gas is also eident in the Amegat diagram, i.e. a lot of isotherms in a grah of V against. 7 5. 1. C 38.9 8 5. 3. C 45.3 9 5. 4. C 49. 1 5. 5. C 53.5 7 / 9
UE43 Tab. : Recording of isotherms: Secific olumes calculated from effectie iston osition and determined mass of gas as well as the absolute ressure conerted from readings of ressure oer atmosheric = 1 C = C = 3 C = 4 C = 45 C = 5 C.651 5.4.678 5.37.74 5.8.79 5.14.841 5.3.913 5.8.65 4.75.68 4.91.79 4.4.811 4.48.853 4.93.947 4.9.657 3.88.683 4..741 3.6.839 3.9.883 4.44 1.17 4.53.659.96.69 3.15.757.9.874 3.54.9 4.9 1.69 4.38.666.7.716.13.81.68.93 3.37.969 3.89 1.3 4.4.668 1.65.775.1.86.67.983 3.35 1. 3.77 1.35 4.16.696 1.64.95.13 1.9.66 1.11 3.35 1.78 3.74 1.498 4.9.995 1.63 1.1.11 1.158.65 1.396 3.33 1.139 3.7 1.64 4.6 1.93 1.6 1.816.1 1.754.63 1.696 3.31 1.196 3.71 1.789 4.1 1.889 1.61.411.9.349.63 1.993 3.3 1.54 3.71 1.937 3.97.485 1.6 3.7.9.348.63.9 3.3 1.315 3.7.79 3.93 3.79 1.6 3.63.9.944.63.9 3.3 1.373 3.71.9 3.87 3.674 1.61 4.199.9 3.541.6.587 3.7 1.434 3.7.371 3.8 4.71 1.61 4.795.9 3.838.6.88 3. 1.49 3.7.517 3.73 4.867 1.6 5.9.6 4.13.5 3.463.98 1.553 3.7.66 3.66 5.461 1.59 5.388.3 4.716.36 4.45.77 1.611 3.7.84 3.6 5.463 1.6 5.976 1.91 5.889.6 4.67.57 1.67 3.68 3.95 3.44 6.57 1.6 7.155 1.7 7.66 1.8 5.797.3 1.671 3.69 3.386 3.3 6.59 1.61 8.336 1.5 8.47 1.6 6.97 1.95 1.73 3.67 3.386 3.3 6.654 1.59 9.519 1.38 9.49 1.46 6.974 1.95 1.789 3.67 4.544.8 7.48 1.56 1.73 1.6 1.613 1.33 8.15 1.73 1.849 3.67 5.71.41 7.5 1.58 11.89 1.15 11.797 1.1 9.33 1.55 1.99 3.66 6.884.9 8.431 1.4 1.515 1.41 1.966 3.66 8.63 1.85 8.43 1.4 11.697 1.8.114 3.64 9.39 1.65 9.615 1.3.6 3.6 1.4 1.49 1.83 1.18.555 3.5 11.64 1.35 11.985 1.9.844 3.38 11.69 1.35 11.988 1.9 3.45 3.14 4.8.9 4.588.68 5.756.31 6.93. 8.17 1.79 9.87 1.6 1.467 1.45 11.651 1.3 11.66 1.3 8 / 9
UE43 / 5 4 5 C 45 C 4 C 3 C C 1 C The standard sublimation temerature for SF6 is T = 9.5 K ( = -63.9 C). The molar heat of aorisation at the standard sublimation oint is deried from the gradient a: () J kj H a R 136 K 8.314 18. K mol mol The alue is about % different from the alue quoted in literature H = kj/mol (Messer AG's data sheet: 153. kj/kg 146.5 g/mol =.37 kj/mol bei -63.9 C). This is because the selected fixed oint lies on the sublimation cure. Also, the molar heat of aorisation is temerature-deendent and not constant as was assumed for the fitting of the cure. 3 1 4 6 8 1 1 V / ml g -1 Fig. 7: -V diagram of sulhur hexafluoride Fig. 8: Pressure cure for saturated aour of sulhur hexafluoride Pressure cure for saturated aour Calculate absolute temeratures T in Kelin from the temeratures in degrees Celsius using equation (3). Read off the constant absolute ressures inside the binodal cures drawn in Fig. 7 and lot them against absolute temerature T (Fig. 8). The measurements can be ery well described using the integrated form of the Clausius-Claeyron equation (solid line in Fig. 8): ΔH 1 1 1 1 a R T T T T (19) e e. H: Molar heat of aorisation T: Temerature at ressure The molar heat of aorisation can be determined by lotting ln(/) against 1/T - 1/T and fitting a straight line to the oints (Fig. 9). Usually the fixed oint (T, ) is quoted as the standard boiling oint, i.e. the temerature at which the substance boils under standard ressure = 113.5 hpa. Since for SF6 standard ressure is lower than the ressure at the trile oint, there is no standard boiling oint but a standard sublimation oint instead. Fig. 9: Simle logarithmic lot of measurements to determine molar heat of aorisation from a straight line fitted to the oints 3B Scientific GmbH, Rudorffweg 8, 131 Hamburg, Germany, www.3bscientific.com Coyright 18 3B Scientific GmbH