analysis of uncertainties in manometric gas-adsorption measurements. i: propagation of...

Analysis of Uncertainties in Manometric Gas-AdsorptionMeasurements. I: Propagation of Uncertainties in BET

Analyses

Alexander Badalyan and Phillip Pendleton*

Center for Molecular and Materials Sciences, University of South Australia,Mawson Lakes, SA 5095, Australia

Received December 30, 2002. In Final Form: June 20, 2003

A detailed analysis and calculation of the uncertainties associated with manometric gas-adsorptionmeasurements are presented for experimental data for nitrogen adsorption at ≈77 K by a traceable standardcarbon black material (004-16820-02). Equipment- and measurement-related uncertainty sources derivefrom the dosing and sampling volumes; temperature control of these volumes; dosing, equilibrium, andbarometric-pressure measurements; liquid nitrogen level control; and sample-mass measurements. Dataprocessing errors derive from ignoring thermal transpiration effects and nonideal gas behavior. Departurefrom ideal gas behavior contributions to the amount adsorbed was accounted for by considering thetemperature relationship of the second virial coefficient of the virial equation of state. Variation in theliquid nitrogen level control is shown to have an enormous impact on the pressure-measurement precisionand, hence, the amount adsorbed. Variation of the liquid nitrogen level by (1 mm results in a variationof the equilibrium pressure from -0.42 to +0.52% and the volume of gaseous nitrogen adsorbed from -8.53to +5.94% when compared with the results obtained during precise level control (within (0.2 mm). Inaddition to these uncertainty sources, reproducibility in the sample-mass measurement is important; adecrease in the mass resolution from 5 × 10-5 to 5 × 10-4 g generates a relative combined standarduncertainty of the volume of nitrogen adsorbed by 10-fold varying from 2.78 to 9.86% over the relativepressure range from 0.0007 to 0.98. For a similar standard mass uncertainty applied to the Brunauer-Emmet-Teller specific surface area analysis, the final area relative combined uncertainty increases from0.63 to 6.19%. The calculated cumulative relative combined uncertainty in the volume adsorbed increasescontinuously with each experimental point from 0.28 (for the first experimental point on the adsorptionbranch of the isotherm) to 9.54% (for the last experimental point on the desorption branch of the isotherm),with subsequent implications for mesopore modeling and analysis accuracy.

1. IntroductionVolumetric or, more correctly, manometric or barometric

gas-adsorption measurements are used extensively forthe characterization of microporous and mesoporouspowdered materials and to obtain the details of gas- orvapor-solid interactions.1 To the best of our knowledge,the last published major international survey of adsorbentspecific surface area (SSA) analysis of a single samplewas presented in 1969.2 In this survey, most, if not all, ofthe adsorption equipment was operated manually. Theadvent of commercial equipment and computer control ofthe adsorption data collection rendered real-time datacollection.3,4 A serious issue regarding data collection withany equipment, which often becomes exacerbated byautomation, is the need for frequent calibration to en-sure accuracy of the various measuring devices in theequipment. Commercial adsorption apparatus manufac-turers and several national standard laboratories pro-vide traceable SSA standards for powders. These powderarea values are quoted with a degree of precision, whichhas been defined statistically from a survey of measure-

ments performed at various international laboratories.Statements can be readily found claiming that thereproducibility of the BET method for SSA evaluation isnot high, typically quoted as >(5%.5,6 Also reported areareas derived from krypton and nitrogen that agreeonly to within (20%5 and for argon and nitrogen towithin (10%6. The magnitude of these variations is mostlydue to differences in gas-solid interfacial interactionsbut may also be partly due to a lack of experimentaluncertainty considerations between different equipmentand calibration details. Rouquerol et al.6 provide abeautiful anthology for any researcher interested in gas-adsorption experimental methods, their intricacies, andtechnique development. They provide the details of manypreviously described apparatus for both manometric andgravimetric adsorption measurements. They also present“Details of the Operational Stages”, a discussion ofcalibration volume determination and of the correctionsrequired for the temperature, pressure, and nonidealbehavior of the bulk gas. In the body of this paper, we willrevisit each of these variables with respect to theircontributions to theoverall combinedstandarduncertainty(CSU) associated with manometric adsorption measure-ments.

Throughout this paper, we used terms for the uncer-tainties as were recommended in NIST7 and EURACHEM8

uncertainty guidelines.

* To whom all correspondence should be addressed. E-mail:[email protected].

(1) Rouquerol, J.; et al. Pure Appl. Chem. 1994, 66 (8), 1739-1758.(2) Sing, K. S. W. In Surface area determination; Everett, D. H.,

Ottewill, R. H., Eds.; Proceedings of the International Symposium;Butterworth: London, 1970.

(3) ASAP 2405 Accelerated Surface Area and Porosimetry Analyser,Operations Manual; Micromeretics: Norcross, GA, 1998 (http://www.micromeritics.com).

(4) Coulter Omnisorp Series Automated Gas Sorption Analysers,Operations Manual; Beckman Coulter: Fullerton, CA, 1998 (http://www.beckman.com).

(5) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity,2nd ed.; Academic Press: Sydney, 1982.

(6) Rouquerol, F.; Rouquerol, J.; Sing, K. Adsorption by Powders andPorous Solids; Academic Press: Sydney, 1999.

7919Langmuir 2003, 19, 7919-7928

10.1021/la020985t CCC: $25.00 © 2003 American Chemical SocietyPublished on Web 08/15/2003

Loebenstein and Deitz9 indicated that the relativecombined standard uncertainties (RCSUs) in the amountof nitrogen adsorbed for materials with SSAs varyingbetween 0.7 and 1720 m2/g were 10.4 and 0.12%, respec-tively. Their analysis only considered the first point ofeach isotherm, commenting on their cumulative natureand on the difficulty in evaluating the CSU of measure-ments for the BET methods. Unfortunately, they did notpresent their calculations. Ross and Olivier,10 Webb andOrr,11 and more recently Robens et al.12 addressed in detailthe issues of apparatus calibration, thermal transpiration,and the ideal and nonideal behavior of gaseous heliumand nitrogen, respectively, at cryogenic temperatures.Calculation of the departure from ideal gas behavior, evenat low pressure, is essential for accurate isothermaladsorption data. Departure from ideal gas behaviorimpresses substantially on the higher relative pressureregime,affectingmesoporeanalyses, in terms of inaccuratepore volume analysis and width distribution data. Ofcourse, these analyses have their own intricacies, whichare outside the intention of the present communication.

The importance of an accurate liquid nitrogen levelcontrol is taught in the patent awarded to Killip et al.13

describing a porous isothermal jacket. Recently, Yanazawaet al.14 presented results indicating the RCSUs in dead-volume measurements using a newly developed constant-volume cell with a surrounding vacuum jacket at theposition for the expected liquid nitrogen level. A carefulanalysis of these publications motivated us to addressquantitatively the issue of uncertainties associated withmanometric gas-adsorption measurements. Recently, wedescribed a manometric gas-adsorption apparatus designfor real-time nitrogen adsorption measurement andanalysis.15 Originally, the equipment was calibrated forSSA analysis against a traceable carbon black standardmaterial no. 004-16820-02. To correctly interpret anycalibration results of a new apparatus design and comparethem with traceable standards, it is of paramountimportance that the data are accurate. In the present work,we address the issues of uncertainties in the evaluationof gaseous nitrogen volume adsorbed and its translationinto SSA evaluation. In particular, we consider themeasurement CSU effects in the dosing and samplevolume calibrations and associated barometric pressures,the gas-manifold and adsorbent sample-holder temper-ature measurement and control, and the precision of theliquid nitrogen level during measurements. We show theimpact of standard uncertainties (SUs) and CSUs associ-ated with these variables on amounts adsorbed (as anadsorption isotherm). These uncertainties are also trans-lated to show their impact on BET-SSA values. A detailed

knowledge of such information provides theorists whosemodels are fitted to adsorption data with a strongerindication of their model’s accuracy. Finally, this workprovides researchers with information on the measuringprocedure and details of the most important parametersthat should be accurately measured and controlled duringmanometric adsorption measurements. Accuracy is mani-festly required in the increasingly “popular” research fieldof supercritical adsorption, especially because subcriticalmeasurements are required to establish detailed, accuratepore volumes and their distributions with the apparentwidth for correct supercritical isotherm analysis. In manycases, these volumes are determined after many datapoints are collected during the adsorption isothermmeasurements. With increasing the number of points, oneneeds to be aware of the cumulative nature of varioustypes of uncertainty in the data.

2. Materials and Methods

2.1. Materials. Earlier, we reported the development of anautomatic gas-adsorption apparatus, which can be used for thecharacterization of powdered materials.15 The operation of thisapparatus was calibrated (in the “usual way”) via manometricnitrogen gas adsorption by several standard powdered ma-terials via calculations of the multipoint BET SSA. The calcu-lated values agreed with those supplied within the quotedreproducibility. In most cases, such reproducibility is regardedas adequate. Typically, the supplier’s quoted values are deter-mined as sample standard deviations (SSDs) of the area dataobtained under repeatability conditions reported by severallaboratories, but in reality, the experimental uncertainties ofsome of the area data provided to these laboratories may beoutside the quoted SSD value. These data are discounted viastatistical “relevance” arguments. Although our measurementswere repeated as four separate samples of the supplied standardmaterials and were within the quoted SSD, the sources of theuncertainties in the data were never obvious. The analysis ofuncertainties in the present work was performed using a standardcarbon black supplied by Micromeritics Corporation (Sydney,Australia), with a recommended multiple-point BET SSA of 113( 5 m2/g.

2.2. Equipment and Measurement Method. Prior to themeasurements, the carbon black sample was heated at ≈2 °C/min up to 200 °C, then held at this temperature for 4 h, as wasrecommended by the manufacturer, achieving a backgroundpressure <1 × 10-4 Pa. After degassing, the sample and itscontainer were surrounded by liquid nitrogen. The level of liquidnitrogen was controlled by a modified liquid nitrogen level controlsystem, shown in Figure 1, which differs from that reportedearlier.15 Liquid nitrogen (1) is supplied to the sample Dewarflask (2) from the liquid nitrogen reservoir Dewar flask (3) viaa stainless-steel tube (4). The type-N thermocouple (5) sensesvariations of the liquid nitrogen level. A stand-alone computer(6) via the ADAM-4017 16-bit analog input module (7) and theRS-232/RS-485 isolated converter (8) monitors the temperaturereadings. The delivery liquid nitrogen system employs gaseousnitrogen from the cylinder (9) to pressurize and rapidly deliverliquid nitrogen to the sample reservoir. Appropriate control ofthe pressurizing system is via the thermocouple sensor feedbackloop providing control with a precision expressed as a SSDobtained under repeatability conditions similar to that describedpreviously15 of <0.2 mm, as is shown in Figure 2. Greater precisionis achievable at the expense of additional gaseous nitrogenconsumption due to increased on/off switching of the solenoidcontrol valves (11 and 12). We typically perform our measure-ments with a level control precision expressed as a SSD obtainedunder repeatability conditions of 0.1 mm. Activation of thesolenoid valves is carried out by computer via a digital signalfrom the PCL-726 D/A card. The check valve (13) provides asafety precaution, preventing excessive pressure developmentin the liquid nitrogen reservoir. The equilibrium was detectedby stability of the equilibrium pressure within the relativestandard uncertainty (RSU) of the differential pressure gageduring a period of about 10 min.

(7) Taylor, B. N.; Kuyatt, Ch. E. Guidelines for evaluating andexpressing the uncertainty of NIST measurement results; NationalInstitute of Standards and Technology Technical Note 1297, 1994edition; U.S. Government Printing Office: Washington, DC, Sept 1997.

(8) Quantifying uncertainties in analytical measurement, 2nd ed.,EURACHEM/CITAC Guide CG4; Ellison, S. L. R., Rosslein, M.,Williams, A., Eds. http://www.eurachem.ul.pt/guides/QUAM2000-1.pdf (accessed 2000).

(9) Loebenstein, W. V.; Deitz, V. R. J. Chem. Phys. 1947, 15, 687-688.(10) Ross, S.; Olivier, J. P. On physical adsorption; Interscience

Publishers: Sydney, 1964.(11) Webb, P. A.; Orr, C.Analytical methods in fine particle technology;

Micromeritics Instrument Corporation: Norcross, GA, 1997.(12) Robens, E.; Keller, J. U.; Massen, C. H.; Staudt, R. J. Therm.

Anal. Calorim. 1999, 55 (2), 383-387.(13) Killip, G. R.; Camp, R. W.; Orr, C., Jr. Re-issued U.S. Patent

33,567, April 9, 1991 (original U.S. Patent 4,693,124, September 15,1987).

(14) Yanazawa, H.; Ohshika, K.; Matsuzawa, T. Adsorption 2000, 6,73-77.

(15) Badalyan, A.; Pendleton, P.; Wu, H. Review of ScientificInstruments 2001, 72 (7), 3038-3045.

7920 Langmuir, Vol. 19, No. 19, 2003 Badalyan and Pendleton

Ultrahigh pure helium (CIG, Australia) was used to evaluatethe sample dead volume. Ultrahigh purity gaseous nitrogen (CIG,Australia) was used for the adsorption measurements. After thedead-volume measurements, the sample tube was warmed to≈100 °C and 1 × 10-4 Pa to ensure total helium removal. Theexperimental data recording and all the calculations associatedwith manometric nitrogen adsorption were carried out in realtime using VisiDAQ version 3.11 data acquisition software andcustomized software.15

3. Evaluation of Errors for Calibrating Volumes

3.1. Evaluation of Volumes V1, V2, and V3. Figure 3represents schematically the internal volumes of the gas-adsorption apparatus used in this work.15 The dosingvolume calibration began by measuring accurately thevolume V1, which comprises the volume of the sampletube (1), the internal volume of the adapter (2), and partof the manual, sealed-bellows valve (3).

We determined volume V1 by filling it with deionized,degassed water and evaluating the water mass at a knowntemperature. To raise the precision of this measurement,we evacuated the tube prior to filling it with water andrepeated the filling procedure several times until twoconsecutive measurements of the mass of the tube withwater agreed within the reported SU of the digital massbalance, 0.005 g. Ten separate values of the mass were,thus, obtained. The SU in both the empty and the filledV1 masses become u(mt) ) u(mt+w) ) 0.005 g. The RCSUs

for these measurements (Table 1) were determined via eq1, giving a value of 0.001%.

Well-defined water density data16 were approximatedby a quadratic polynomial as a function of the temperature.The temperature was measured with a SU of u(T) ) 0.05

Figure 1. Modified liquid nitrogen control system.

Figure 2. Liquid nitrogen level fluctuations.

Figure 3. Calibrating volumes.

uc(z)/|z| ) x[u(p)/p]2 + [u(q)/q]2 + ..., when z * 0(1)

Uncertainties in BET Analyses Langmuir, Vol. 19, No. 19, 2003 7921

K. The SU in the water density data calculated from thepolynomial is u(F) ) 0.03 kg/m3. However, the RCSU ofthese density data was listed as 0.03%, yielding a CSU ofapproximately uc(F) ) 0.30 kg/m3. All further calculationsuse this latter value of the CSU for the water density dataand V1 calculations.

A SSD

is an indication of the repeatability/reproducibility ofseveral consecutive measurements of the same parameter.The SSD of a parameter should only be used for consecutivecalculations of the CSU of a function of that parameter,when it exceeds the value of the CSU of that parameter,calculated from the law of propagation of uncertainties.If the CSU of a parameter, calculated from the law ofpropagation of uncertainties, is greater than the corre-sponding SSD of that parameter, one should use or quotethe former in the analysis of the experimental uncertain-ties. This overall approach is applied throughout this work.Uncertainty values are summarized in Table 1.

The water mass and density data were applied to theV1 calculations for each of the 10 tube fillings, resultingin their mean value, V1,mean ) 56.63 × 10-6 m3, anddeviations of each V1 from V1,mean. The SSD s(V1) ) 0.09× 10-6 m3. The CSU for each V1 and subsequent calcula-tions were made via eq 2 given elsewhere,17 resulting inuc(V1) ) 0.02 × 10-6 m3. Consequently, we employ s(V1)for RCSU evaluation, which becomes 0.16%.

Accurate calibration of the volume V1 renders volumesV2 and V3, by nitrogen or helium expansion at room tem-perature to P3 and to P2 from P1, respectively. The dosing

volume V3 includes the volume of the manifold (4, Figure3) and the measuring volume of the MKS Baratron 698capacitancedifferentialpressure transducerequal toabout2 × 10-6 m3. Material balance expressions for these ex-pansions, assuming ideal gas expansion and no nitrogenadsorption by the equipment at room temperature, leadsto

We calculated the values for V2 and V3 for 10 separatesets of gas expansions, giving V2,mean ) 48.37 × 10-6 m3

and V3,mean ) 362.30 × 10-6 m3. The RSU in the pressurereadings was 0.05% of the reading. Table 1 presents thecorresponding uncertainties in V2 and V3.

The ratio between the expansion and the dosing volumesof the manometric adsorption apparatus is equal to 0.29.Ross and Olivier10 suggest that the minimum RCSU inthe measurement of V1 + V2 occurs for this ratio equal to1. Their data indicate that when this ratio <1, it “does notexercise too harmful an effect on the precision of thedetermination”.10

3.2. Evaluation of Volumes V4 and V5. Volume V4(see Figure 3) is the part of the physical volume of thesample tube that is placed in liquid nitrogen. A markeretched on the outside of the tube acts as a reference pointfor the experimental uncertainty analysis and adsorptionmeasurements. This volume was determined using thesame procedure as that for V1, giving V4,mean ) 26.787 ×10-6 m3. In this case, an average of 10 fillings gave uc(V4)av< s(V4), and, therefore, s(V4) was adopted as the CSU inV4. The RCSU was 0.13%.

The relative linear thermal expansion coefficient18 forthe Pyrex sampling tube RT

Pyrex ) 33 × 10-7 K-1. Thesampling tube volume is calculated at a cryogenic tem-perature via V4 at 77.43K ) V4 at 295.39K(1 + 3RT∆T) ) 26.728× 10-6 m3, giving a RCSU of 0.22%. This uncertainty isgreater than the RCSU in the evaluation of V4 at 295.39K(0.13%). However, V4 at 77.43K is not used in any of ourcalculations. Instead, the dead volume, determined byhelium expansion at 77.43 K, is used in all consecutivecalculations. The physical volume V4 at 295.39K is measuredonly for the determination of the volume V5, which is usedfor the evaluation of the amount of nitrogen n′′0. There-fore, the variation of V4 with the temperature is not takeninto account.

V5 is the physical volume between the valve (3) and thelevel of liquid nitrogen (see Figure 3). The liquid nitrogensurface is isolated from ambient conditions by a relativelytight-fitting expanded polystyrene insulating layer. Con-sequently, we assume without a significant error that thispart of the system is kept at 298.15 K. The value of V5 isdetermined via simple arithmetic, giving V5,mean ) 78.22× 10-6 m3, while its RCSU (0.24%) is calculated via thesummation of independent random measurements, as ineq 5 given elsewhere.17

The relative linear thermal expansion coefficient for316 stainless steel,19 RT

316SS, in the temperature range(16) Lide, D. R.; Kehiaian, H. V. CRC Handbook of Thermophysicaland Thermochemical Data; CRC Press: London, 1999.

(17) Taylor, J. R. An Introduction to Error Analysis. The Study ofUncertainties in Physical Measurements; University Science Books,Oxford University Press: Mill Valley, CA, 1982.

(18) Doremus, R. H. Glass science; John Wiley and Sons: Sydney,1973.

Table 1. Summary of Uncertainty Analysis Results

parameter units CSU RCSU

mt+w kg 0.000 005 0.001mt kg 0.000 005 0.001mw kg 0.000 005 0.01Tw K 0.05 0.02Fw kg/m3 0.30 0.03V1 m3 0.09 × 10-6 0.16P1, P2, P3 Pa 0.05V2 m3 0.19 × 10-6 0.39V3 m3 0.81 × 10-6 0.22V4 m3 0.036 × 10-6 0.13V5 m3 0.20 × 10-6 0.26V4,dead m3 0.14 × 10-6 0.45Tamb K 0.05 ≈0.02Tsample K 0.01 ≈0.06B(Tamb) m3/mol 2.50 × 10-7 5.33B(Tsample) m3/mol 8.00 × 10-7 0.31Nmoles mol 2.58 × 10-7-1.75 × 10-4 0.26-17.76msample kg 5.0 × 10-7 0.28Vads m3/kg 3.39 × 10-5-1.53 × 10-3 0.28-0.93Pe/Patm 4.66 × 10-5-1.87 × 10-4 0.08-0.10Pe/Patm/[Vads(1 -

Pe/Patm)]kg/m3 1.39 × 10-2-6.17 × 10-2 0.58-0.81

a kg/m3 0.245 0.63b kg/m3 0.020 7.33Vads

m m3(STP)/kg 1.59 × 10-4 0.63SBET m2/kg 690 0.63

s(p) ) x{[∑i)1

n

(pi - pj)]/(n - 1)}, where pj ) (∑i)1

n

pi)/n

uc(z) ) x[∂z∂p

u(p)]2+ [∂z

∂qu(q)]2

+ ... (2)

V2 )(P1 - P2)P3

P1(P2 - P3)V1 (3)

V3 )P2P3

P1(P2 - P3)V1 (4)

uc(V5) ) x[uc(V1)]2 + [uc(V2)]

2 + [uc(V3)]2 (5)


273-373 K is 16.0 × 10-6 K-1. The manifold volume V5variation over 298.15 ( 1 K leads to a RSU of 0.005%,which is ,uc(V5). Thus, we ignored the thermal expansioncontributions in the subsequent analyses.

3.3. Evaluation of Volume V4,dead. Prior to nitrogenadsorption measurements, we determined the dead vol-ume around the reference carbon sample, V4,dead, by fillingthe sampling tube with helium gas at ≈77 K (volume V4can be determined similar to volume V4,dead). The samplingtube was immersed in liquid nitrogen to the depthcorresponding to the physical volume V4. Because theadsorbent contains no micropores, we do not encounterpossible helium adsorption problems according to Rou-querol et al.6 Of course, V4,dead is not a physical volume butis the volume that will be occupied by helium gas whenthe sample tube is at the calibration temperature of volumeV4 and at the experimental or equilibrium expansionpressure. The dosing pressures were increased in theinterval 0.67 × 103 Pa ∈ 26.7 × 103 Pa. We calculatedV4,dead for six pairs of dosing and equilibrium pressuresusing eq 6:

where Pdi, Pei-1 and Pei are the dosing and equilibriumpressures. The CSU in V4,dead was again calculated via eq2 leading to s(V4,dead) < uc(V4,dead)av.

In an effort to reduce the influence of any temperaturegradient within the sample tube at the level correspondingto the liquid nitrogen-air interface in the Dewar, Yanaza-wa et al.14 developed a sample tube with a surroundingevacuated double-walled jacket at this interface. Theyreport SSD data for their dead volumes. It is not clearfrom their publication if the analyses of experimentaluncertainties were performed for the dead-volume mea-surements. From Yanazawa et al.’s SSDs, the RSUs inthe dead volumes are 0.21% with the jacket and 0.60%without the jacket. Although an incorrect use, we calcu-lated the dead-volume RSU via the SSD s(V4,dead) insteadof uc(V4,dead)av, giving 0.19%, similar to that reported byYanazawa et al. for their jacketed tube. Considering thatthe dead volume in our adsorption apparatus (V4,dead)av )31.20 × 10-6 m3 is approximately 4 times less that thatreported by Yanazawa et al., a RCSU of 0.44% indicatesthat our liquid nitrogen level control system provides anexcellent level control.

4. Evaluation of Uncertainties for the Amount ofNitrogen Adsorbed

The experimental data obtained during manometricnitrogen adsorption measurements are used to evaluatethe amount of nitrogen adsorbed by an adsorbent, its SSA,and its pore size distribution. Such calculations carryexperimental uncertainties arising from the previouslycalibrated volumes. An analysis of the uncertainties ofthe amount of gas adsorbed must also consider the effectof thermodynamic influences manifested as thermaltranspiration and departure from ideal gas behavior.

4.1. Thermal Transpiration Errors. A thermaltranspiration effect arises from the temperature difference

between the connected parts of an apparatus and occurswhen the mean free path of the adsorptive comparesfavorably with the diameter of the connecting tubing.Typically, this tubing connects the sample and dosingvolume sections of an adsorption apparatus. This effectpredominates at low adsorptive pressure and depends onthe adsorptive gas, on the diameter of the tubing con-necting the two volumes (at widely different tempera-tures), and on the temperature difference. During allmanometric adsorption measurements, the dosing-volumepart of the apparatus is maintained at a constanttemperature, Tamb. In the present design, Tamb is measuredcontinuously during the adsorption and desorption mea-surements. The sample tube containing the adsorbent ismaintained at Tsample or the liquid nitrogen temperature,77 K. Although the latter temperature depends on thelocal atmospheric pressure, the temperature differencebetween the sample and ambient varies around 220 °C.If the tubing diameters are concordant with the meanfree path of the adsorptive, and with such a temperaturedifference, a thermal transpiration effect will exist.6,20,21

Takaishi and Sensui22 investigated the thermal tran-spiration of several gases, suggesting the empiricalrelationship

Here, X ) 2PambDtube/(Tsample + Tamb), Pamb is the gaspressure in the sample tube held at the cryogenictemperatures, and Dtube refers to the tube diameter aboveand around the sample. Takaishi and Sensui report valuesfor A, B, and C for several gases, including nitrogen overthe range 77-190 K. The mean free path for nitrogen isgiven by Atkins:23

where k is the Boltzmann constant and θ ) 0.43 nm3 isthe differential scattering collision cross section of thenitrogen molecule.

In the present equipment, the nitrogen mean free pathcompares with the sample tube diameter Dtube ) 10 mmat Psample ) 0.175 Pa. The lowest equilibrium pressuremeasured for this work was >400 × Psample, giving athermal transpiration contribution of 0.04%. For mea-surements employing 1.33 × 102, × 104, and × 105 Pafull-scale differential manometers, this value is less thanthe precision of either pressure sensor. Thus, for thepresent discussion we do not consider the thermaltranspiration effect errors.

4.2. Uncertainties in Calculating the Amount ofNitrogen AdsorbedsDeparture from Ideal Gas Be-havior. During the nitrogen adsorption measurements,volumes V3 and V5 are maintained at 25 °C, Tamb. Theremaining volume of the sample tube, V4, is maintainedat ≈77 K, Tsample. Tsample depends on the atmosphericpressure and is calculated using PS - TS reference data.24

The sample “dead-volume”, V4,dead, was determined in theusual manner via helium expansion from V3 to V1.

(19) Engineering properties of steel; Harvey, P. D., Ed. AmericanSociety for Metals: Metal Park, Ohio, 1982.

(20) Knudsen, M. Ann. Physik. 1930, 31 (210), 633-640.(21) Young, D. M.; Crowell, A. D. Physical Adsorption of Gases;

Butterworth: London, 1962.(22) Takaishi, T.; Sensui, Y. Trans. Faraday Soc. 1963, 59, 2503-

2514.(23) Atkins, P. W. Physical Chemistry; 5th ed.; Oxford University

Press: Melbourne, 1994.(24) Agnus, S.; de Reuck, K. M.; Armstrong, B. International

thermodynamic tables of the fluid state. 6: Nitrogen; Pergamon Press:Sydney, 1979.

V4,deadi)

PdiV3Tsample

(Pei- Pei-1

)Tamb+

PeiV3Tsample

(Pei- Pei-1

)Tamb-

V3Tsample

Tamb(6)

Psample

Pamb) 1 +

x(Tsample/Tamb) - 1

AX2 + BX + CxX + 1(7)

λ ) kT/(x2θP) (8)


Adsorption measurements were also performed in theusual manner, with the appropriate nitrogen expansionfrom V3 into V1.

At the start of the nitrogen adsorption measurements,the total amount of nitrogen in the volume V3 is given asn1 ) Pd1V3/(RTamb). Volumes V4,dead and V5 were evacuatedto the background pressure of 10-5 Pa, regarded as 0, and,thus, n′0 ) Pe0V4,dead/(RTsample) ≈ 0 and n′′0 ) Pe0V5/(RTamb)≈ 0. At a pressure of 10-5 Pa, the amount of residual(helium) gas is ≈1.4 × 10-8 mol, again regarded as ≈0relative to the amount of gas at the lowest measurablepressure in the dosing section of the apparatus. After thefirst equilibration, the amount of nitrogen occupyingvolumes V4,dead, V5, and V3 are respectively n′1 ) Pe1V4,dead/(RTsample), n′′1 ) Pe1V5/(RTamb), and n′′′1 ) Pe1V3/(RTamb).Thus, the amount of nitrogen adsorbed by the sampleafter the first equilibration is determined via the materialbalance

In the similar manner, with increasing pressure, wecan determine the incremental amount of nitrogen ad-sorbed during each consecutive dosing/equilibration pro-cedure from the expression

This equation is developed on the basis of ideal gasbehavior. Webb and Orr11 pointed out that nitrogen doesnot exhibit ideal gas behavior at 77 K. Loebenstein25

introduced a correction factor, R, giving

and

We analyzed data on the second virial coefficient ofnitrogen B(T) reported elsewhere.24,26,27 An equationdescribing B(T) in thermodynamic tables for nitrogen24 isbased on a critical evaluation of pressure-density-temperature data for nitrogen in the vapor and the gasstates, as was reported by various researchers. Nowak etal.27 presented a new set of experimental pressure-density-temperature data for nitrogen in the range 66-340 K encompassing the liquid, gas, and supercriticalregions. The agreement between the B(T) data calculatedaccording to thermodynamic tables fornitrogen24 andthoseof Nowak el al.27 agree within 0.02 and 2.76% when thetemperature varies from 98 to 300 K. These thermody-namic tables do not supply values of the CSU in B(T);therefore, using the good agreement between these twosets of data we adopted values of the RSU in the secondvirial coefficient for nitrogen at cryogenic and ambienttemperatures as 0.31 and 5.33%, respectively. In our work,

we calculate the ideal gas departure via the virial equationof state24 truncated at the second term coefficient B(T),giving the general pressure expression as

Truncation errors for the second virial coefficient arenegligible at 77 K and pressures below 101 kPa. Ourcalculations show that an increasingly significant differ-ence develops between the amounts adsorbed for idealand nonideal gas property contributions. Nonideal prop-erties provide a higher amout adsorbed. Equation 13transforms to a quadratic function in the amount adsorbed,ni, which is solved numerically. A similar approachproduces values of n′i, n′′i and n′′′i.

4.3. Calculating Uncertainties in the Amount ofNitrogen Adsorbed. For each dosing/equilibration pro-cedure, eq 9 gives the amount of nitrogen adsorbed. Theexpressions for the amount of nitrogen in each volume,via eq 13, lead to CSUs for the calculation of the amountof nitrogen contained in the corresponding volumes of theadsorption apparatus. Because all parameters in theseequations are independent and provide random errors,the CSUs for ni may be determined via

A similar approach is applied to calculate uc(n′i-1),uc(n′′i-1), uc(n′i), and uc(n′′i ).

Let us consider the variation of the CSUs for the amountof nitrogen adsorbed as the relative pressure, Pe/Patm,increases from 7.0 × 10-4 to 0.98. For this pressure range,the cumulative CSU for the amount of nitrogen adsorbedvaries from 2.58 × 10-7 to 1.75 × 10-4 mol, correspondingto a range of cumulative RCSUs from 0.26 to 17.76%. Thissignificant increase in the RCSU indicates that the RCSUin all the previous calculations of the amount of nitrogenadsorbed contributes to similar uncertainties in theconsecutive amount of nitrogen adsorbed.

Using experimental data, we calculated the volume ofnitrogen adsorbed at standard temperature and pressurefor each dosing/equilibration procedure in the usualmanner. A CSU for the volume of gaseous nitrogenadsorbed can be determined from eq 15:

Over the same relative pressure range, the cumulativeCSU, uc

cumul(Vads), varies from 3.39 × 10-5 to 1.53 × 10-3

m3/kg, with the corresponding cumulative RCSU varyingfrom 0.28 to 0.93%.

Figure 4 shows the nitrogen adsorption/desorptionisotherm for the standard carbon black. Here, we showthe cumulative CSU, uc

cumul(Vads), corresponding to se-lected experimental points for the adsorption and de-sorption branches of the isotherm. Clearly, as the numberof data points in the isotherm increases, the cumulative

(25) Loebenstein, W. V. J. Colloid Interface Sci. 1971, 36 (3), 397-400.

(26) Dymond, J. H.; Smith, E. B. The Virial Coefficient of Pure Gasesand Mixtures: A Critical Compilation; Clarendon Press: Oxford, 1980.

(27) Nowak, P.; Kleinrahm, R.; Wagner, W. J. Chem. Thermodynam.1997, 29, 1137-1156.

n1ads) n1 + n′0 + n′′0 - n′1 - n′′1 - n′′′1 (9)

niads)

(Pdi- Pei

)V3

RTamb- (Pei

- Pei-1)( V4,dead

RTsample+

V5

RTamb)

(10)

R )Tc

T1

10.24Pc[2.16

Tc

T (Tc

T+ 1) - 1] (11)

n ) PVRT

(1 + RP) (12)

Pi ) RTV

ni[1 +B(Tamb)

Vni] (13)

uc(ni) ) {[ ∂ni

∂Tambu(Tamb)]2

+ [∂ni

∂V3uc(V3)]2

+

[ ∂ni

∂B(Tamb)u(B(Tamb))]2

+ [ ∂ni

∂Pdi

u(Pdi)]2}1/2

(14)

uc(Vadsi) )

x[ Vmol

Msampleuc(Nmolesi

ads )]2

+ [-VmolNmolesi

ads

Msample2

u(Msample)]2

(15)


CSU in each point increases as a result of the cumulativenature of the uncertainty of the measurements.

5. Evaluation of Uncertainties for the BET-SSA5.1. Analysis of Parameters in the BET Equation.

This equation is the most widely used to evaluate the SSAof adsorbent materials, written in the commonly appliedlinear form as eq 16

Equation 16 is usually linear over the relative pressurerange 0.05-0.35. Figure 5 shows the BET plot for thestandard carbon black over the supplier-recommendedrelative pressure range 0.04-0.20. The slope and interceptof the linear plot were defined via a least-squares analysis.The SSA is calculated via eq 17, where “a” and “b” are theslope and intercept, and σm is the cross-sectional area ofthe adsorbate. Clearly, the CSU in the BET SSA, uc(SBET),

is a function of the “slope” and “intercept”. The CSU inthese coefficients is a function of the CSUs in the abscissaand ordinate values.17 The CSUs in the x-y values aredefined via

An SSU for atmospheric pressure measurements istaken as u(Patm) ) 0.1 kPa. The values of uc(x) vary from4.66 × 10-5 to 18.7 × 10-5 and those of uc(y) from 1.39 ×10-2 to 6.12 × 10-2 kg/m3. The corresponding RCSUs varyfrom 0.08 to 0.10% and from 0.58 to 0.81%, respectively.As in the adsorption isotherm, each consecutive point hasa greater uncertainty than the previous point. The CSUin Vads

m is calculated via eq 20:

5.2. Calculation of the Uncertainty in SBET. Fromeq 18, the values of uc(xi) are negligible compared withthose of uc(yi), which have various (increasing) values.Therefore, we may apply the method of weighted leastsquares to evaluate the CSUs in the slope and intercept.Taylor17 recommends this method when uc(yi) are known.Aweighting functionof the form wi ) 1/[uc(yi)]2 is employed.Consequently, the slope and intercept are given as

Using these equations, we obtain a slope of 39.041, anintercept of 0.276, and R2 ) 0.9996. The uc(y) can bedetermined via eq 24:

Because uc(yi) are not constant, the corresponding CSUsin the slope, uc(a), and intercept, uc(b), can be calculatedas weighted CSUs according to Bevington.28 From Figure5, Vads

m ) 2.54 × 10-2 m3(STP)/kg with uc(Vadsm ) ) 1.59 ×

10-4 m3(STP)/kg, leading to SBET ) 110.71 m2/g anduc(SBET) ) 0.69 m2/g (see Table 2). Consequently, the RCSUin SBET is 0.62%.

(28) Bevington, P. R. Data Reduction and Error Analysis for thePhysical Sciences; McGraw-Hill Book Company: Sydney, 1969.

Figure 4. Adsorption isotherm.

Figure 5. BET plot for standard carbon black.

Pe/Patm

Vads[1 - (Pe/Patm)]) 1

Vadsm c

+ c - 1Vads

m c

Pe

Patm(16)

SBET )σmNA

(a + b)Vmol(17)

uc(xi) ) x[ 1Patm

u(Pei)]2

+ [ -Pei

Patm2u(Patm)]2

(18)

uc(yi) ) {[ Patm

Vadsi(Patm - Pei

)2u(Pei

)]2

+

( -Pei

Vadsi(Patm - Pei

)2u(Patm)]2

+

[ Pei

(Patm - Pei)Vadsi

2uc(Vadsi

)]2}1/2

(19)

uc(Vadsm ) ) x[ -1

(a + b)2uc(a)]2

+ [ -1(a + b)2

uc(b)]2(20)

a )

(∑i)1

n

wixi2)(∑

i)1

n

wiyi) - (∑i)1

n

wixi)(∑i)1

n

wixiyi)

∆(21)

b )

(∑i)1

n

wi)(∑i)1

n

wixiyi) - (∑i)1

n

wixi)(∑i)1

n

wiyi)

∆(22)

∆ ) (∑i)1

n

wi)(∑i)1

n

wixi2) - (∑

i)1

n

wixi)2 (23)

uc(y) ) x[uc(y)]2 ) x 1

n - 2∑i)1

n

(yi - a - bxi)2 (24)


5.3. Calculation of the Uncertainty in SBET usingthe Formula for the Propagation of Uncertainties.Equation 17 represents SBET as a nonlinear function of“a” and “b”. According to Box et al,29 it is possible torepresent this function as a plane in the close vicinity ofany point SBET0. As the result of a local linearization of afunction SBET(a, b), the latter may be presented in theform of a surface

where θa and θb are the gradients in the directions of axes“a” and “b” for the local linearization of eq 17 around SBET0.We carried out calculations of gradients for this equationover the (3uc(a) and (3uc(b) ranges of the CSU. The CSUfor SBET can be evaluated using the formula for thepropagation of uncertainties which, when applied to eq25, has the following form:29

The results of these calculations are summarized in Table3, showing a slight nonlinearity of about 0.036% for theSBET function around the point SBET0. The RCSU in SBETis similar to that obtained via Bevington’s method.Although a good agreement exists between these methods,this approach should only be used for the estimation ofthe RCSU in SBET because in eq 26 we assume that “a”and “b” are independent variables, which of course is notthe case for the “slope” and the “intercept” in the BETequation.

5.4. Reproducibility of the BET-SSA Data. Adetermination of the reproducibility of the measurementsand BET-SSA values using the same equipment may bemade via one of two different methods. In both cases, thematerial should be prepared following exactly the sup-plier’s recommended preparation conditions. Method onemay employ a series of measurements where the sampleis exposed to nitrogen in the usual manner, then reheatedat the end of the measurements, followed by a repeat ofthe previous measurements. This procedure should beperformed several times. Such a technique suffers fromnot accounting for random errors.

A second method, performed in this work, is to performthe first evaluation, remove the sample, and prepare thesecond and subsequent measurements as though each isan independent evaluation. This method has the advan-tage of taking into account the random errors in V4,dead,sample-tube liquid nitrogen level repeatability, and liquidnitrogen level control during the measurements. Weperformed this second method four times, giving a SSDin SBET of 0.14%, well within the corresponding RCSU,implying high reproducibility of the equipment, technique,and liquid nitrogen level control.

5.5. Effect of Fluctuations in the Liquid NitrogenLevel on the Equilibrium Pressure and Volume ofNitrogen Adsorbed. During manometric gas-adsorptionmeasurements at cryogenic temperatures, much careshould be given to the accurate control of the liquidnitrogen level. This issue was first addressed in thedetailed description of a newly developed liquid nitrogenlevel control system.15 We have performed numerousmanometric gas-adsorption experiments and found thatthe liquid nitrogen level can be controlled readily within(0.2 mm (Figure 2). Under these experimental conditionsand depending on the actual pressure, Figure 6 indicateshow fluctuations in equilibrium pressures may be withinthe claimed SU of the differential pressure gages usedduring our experimental measurements, that is, 0.05%.To demonstrate the impact of liquid nitrogen level controlon the data precision, we deliberately adjusted the liquidnitrogen level surrounding the sampling tube and mea-sured the resulting equilibrium pressures. How suchvariations affect the pressure readings and the volumeadsorbed is summarized Figures 6 and 7.

Figure 6 summarizes the influence of the liquid nitrogenlevel control on the pressure precision. We show the SUin the Pe/Patm determination as 0.10% (see Table 1) in thelimits of (0.2 mm level control. The area of the plot outsidethese coordinates represents an error we classify as beingdue to (potentially) random error, or at least one that isoutside the bounds of systematic errors, especially as weassume that the level control during an experiment is“constant” within the above variation (Figure 2). We mayalso conclude from Figure 6 that low equilibrium pressuresare more sensitive to fluctuations in the liquid nitrogenlevel control. This observation has important ramificationsfor micropore filling investigations compared with me-sopore analyses. In Figure 7, we present the effect of thevariation in the liquid nitrogen level on the volume ofnitrogen adsorbed, with these amounts adsorbed calcu-

(29) Box, G. E. P.; Hunter, W. G.; Hunter, J. S. Statistics forExperimenters. An Iintroduction to Design, Data Analysis, and ModelBuilding; John Wiley and Sons: Brisbane, 1978.

Table 2. Results of BET-SSA Calculations for CarbonBlacka

msample, g SBET, m2/g uc(SBET), m2/g uc(SBET)/SBET × 100, %

0.2087 110.25 0.51 0.460.1796 110.71 0.69 0.620.2096 110.26 0.58 0.530.2189 110.37 0.56 0.510.1903 109.97 0.58 0.53

a (SBET)mean is 110.31 m2/g and the SSD is 0.27 m2/g.

Table 3. Results for the Calculation of uc(SBET)a

values of“a” and “b”

SBET,m2/g

difference,m2/g gradient

a + 3uc(a) 0.3371 110.54 -0.17a 0.2763 110.71 -2.8159a - 3uc(a) 0.2155 110.89 -0.18b + 3uc(b) 39.7769 108.68 -2.03b 39.0412 110.71 -2.8169b - 3uc(b) 38.3055 112.83 -2.12

a uc(SBET) is 0.693 m2/g and uc(SBET)/SBET × 100 is 0.63%.

SBET ) SBET0+ θa(a1 - a2) + θb(b1 - b2) (25)

uc(SBET) ) x[θauc(a)]2 + [θbuc(b)]2 (26)

Figure 6. Influence of liquid nitrogen level control on pressuremeasurement precision: b, 6.0; [, 26.8; 9, 53.2; and 2, 79.0kPa.


lated from the pressure measured at the optimized liquidlevel in Figure 6 and at (1.0 mm. In contrast with Figure6, we see that the slope of the error in the volume adsorbedwith the liquid level is greatest for the highest pressure.Although we suggest that the pressure measurements atlow-pressureadsorptionconditions (in themicropore fillingregions of the adsorption isotherm) are sensitive to theliquid nitrogen level control, the volumes adsorbed athigher pressures (in the mesopore filling regions of theadsorption isotherm) are sensitive to the liquid nitrogenlevel control. Consequently, the whole range of relativepressures requires good liquid nitrogen level control foraccurate isotherm measurements. It is important toappreciate that the data in Figures 6 and 7 are not“universal” errors because they are dependent upon theamount and type of adsorbent (making these errorssystem-specific) and include fixed contributions from thetubing in the equipment but, nonetheless, point out theimportance of liquid nitrogen level control.

Overall, deviations of the equilibrium pressures fromthe control values (corresponding to the liquid nitrogenlevel control within (0.2 mm) extend well beyond theclaimed SU of the differential pressure gages, convertingto an 8.53% deviation in the volume of the nitrogen gasadsorbed. These induced deviations demonstrate theimportance of liquid nitrogen level control. These datacontrast with the RCSUs determined as a part of our level-controlled analyses summarized in Table 1, where theRCSU in the volume of adsorbed nitrogen is 0.93%,considerably less than the observed deviations.

The calculation of the nitrogen amount adsorbed alsorequires control of the ambient temperature, or at leastthe volume surrounding the equipment, eq 11, becausefluctuations in the temperature generate a poor equilib-rium pressure stability. Enclosing the whole dosing partof the apparatus in a temperature-controlled unit canreduce the temperature fluctuations. In our presentequipment, if the temperature of the dosing volume shouldgradually change during the isotherm measurements, theassociated data acquisition software monitors this tem-perature and introduces corresponding corrections intoour calculations.

6. DiscussionHistorically, SSA measurements have been awarded a

SU of 5%.5,6 Loebenstein and Deitz9 have suggested thatthe RCSU in the calculated amount adsorbed by a materialdepends on the final SSA, quoting a RCSU varying from

10.4-0.12%, depending on the adsorbent. More recently,with the advent of computer-controlled commercial devicesand accepted values for selected reference standards, theRCSUs of SSA evaluation have slightly improved. Partlyresponsible for this improvement is the introduction ofliquid nitrogen level control via isothermal jackets13 andvacuum jackets.14 Sparse evidence exists for the detailedanalysis of uncertainties during manometric gas-adsorp-tion measurements. To the best of our knowledge, thispresent work provides the first detailed experimentaluncertainties source identification and analysis and thefirst attempt to provide a measure of the relativeimportance of these uncertainty sources. Calibrationvolumes must be known accurately because these areprominent in the material balance calculations. Althoughequipment temperature control and thermal transpirationeffects are important, our analyses suggest that liquidnitrogen level control is the principal source of uncertainty,and not accounting for nonideal gas behavior contributessignificantly to amounts adsorbed in the high relativepressure range of the adsorption measurements.

In general, the nitrogen adsorption measurements aremade to evaluate the SSAs of powdered materials. Withthe prevalence of spreadsheets and “trend-line” applica-tions, one may easily ignore contributions to uncertaintiesin the analysis of the linear form of the BET plot. Wemade calculations of the BET SSA (for a standard carbonblack adsorbent) using values of the coefficients “a” and“b”, but included their CSUs and the CSUs in “x” and in“y”. In the latter case, the SUs in the pressure measure-ment (assuming precise level control) render uc(x) neg-ligible; however, the so-called BET function, uc(y), maydevelop various uncertainty-value magnitudes. We usedthe method of weighted least squares to evaluate theparameters “a” and “b”. If in our calculations we use astandard mean-least-squares method (which assumes thatthe CSUs in “x” and “y” have negligible values of similarmagnitudes), we obtain the results shown in Table 4. Theagreement between the corresponding BET-SSA valuesexceeds the reported RCSU in the SBET determination.We recommend using the weighted least-squares methodto evaluate the BET SSA from manometric gas-adsorptiondata.

It was recently suggested30 that high-accuracy and high-precision adsorption isotherms do not necessarily givehigh-accuracy BET-SSA evaluations because of adsorbentstructure anomalies and the lack of availability of anaccurate multilayer adsorption model. Neither of thesecriticisms addresses systematic nor random errors in thedata collection. Nonetheless, the BET analysis is expectedto provide details for an SSA evaluation over a particularrelative pressure range, prior to statistical multilayeradsorption processes. The determination of the BET SSAin the relative pressure range 0.05-0.35 does suffer froma degree of uncertainty; over what maximum relative

(30) Aranovich, G.; Donohue, M. J. Colloid Interface Sci. 1997, 194,293-397.

Figure 7. Influence of liquid nitrogen level control on amountof nitrogen adsorbed: b, 23.47; [, 30.62; 9, 41.62; and 2, 56.23cm3(STP)/g.

Table 4. BET SSA Calculated According to a WeightedLeast-Squares and a Standard Least-Squares Method

SBETa, m2/g SBET

b, m2/g relative error, %

110.25 110.54 0.26110.72 112.35 1.47110.26 110.83 0.51110.37 110.69 0.28109.97 111.34 1.25

a Weighted least-squares method. b Standard least-squaresmethod.


pressure range should a BET-SSA evaluation be con-ducted?Wecarriedoutcalculationsusing theexperimentaladsorption data for various intervals of the relativepressure of 0.05-0.35. These results, together with theerrors and regression coefficients, R2, and values for theBET SSA are given in Table 5. The use of the R2 valuesas a guide is misleading because, even if the BET plot islinear over the relative pressure range 0.05-0.35 (withan R2 close to unity), the calculated SSA values over theranges 0.05-0.30 and 0.05-0.35p0, for example, deviatefrom those evaluated according to the recommendationsof the supplier (0.05-0.20p0) by values greatly exceedingthe RCSUs of our results reported in Tables 1 and 5.

A third common source of uncertainties in the adsorptionmeasurements derives from incorrect balance selectionfor measuring the adsorbent mass. A reduction in theresolution of a balance from 5×10-5 to 5×10-4 g increasesthe CSU in the SSA uc(SBET) to 6.82 m2/g, correspondingto a RCSU of 6.19% compared with 0.63%. A fourth sourceof uncertainty is the imprecision in the dead-volumemeasurements, V4,dead, which will contribute to the RCSUin SBET of ≈1.85%, approximately three times greater thanthe corresponding RCSU reported in Table 1.

The CSU in each data point in the isotherm in Figure4 shows an interesting problem. The CSUs in theadsorption data are cumulative for each point and, as therelative pressure increases, the CSU increases. Clearly,we see that in the low relative pressure range, of interestto micropore filling investigators, the CSUs in the amountsof nitrogen adsorbed are (relatively) small. With increasingpressure, the CSU increases; we include a line of best fitthrough the adsorption points, but as can be seen in Figure4 with the increasing CSU in the data, we have severaloptions of where to draw in the desorption branch of theisotherm. The ordinate axis is amplified to show theamount adsorbed CSU in the low-pressure range. Thedesorption-point CSU at 0.9p0 is less than the adsorption-point CSU at 0.95p0 because 10 additional amount-adsorbed points are excluded from Figure 4. The hysteresisloop, which appears in Figure 4, is only due to thecumulative nature of a CSU during the evaluation of the

amount of nitrogen adsorbed on the standard carbon black.The sample is supplied as a nonporous reference material,and, thus, one would be averse to suggesting a hysteresisloop. But if this material was an unknown adsorbent, theCSU in each data point suggests the presence of mesoporesextending to micropore hysteresis. The recourse forwhether to consider hysteresis is to compare the CSUs foreach point and, if adsorption and desorption CSUs overlap,assume no hysteresis exists. This observation has im-portant ramifications for mesopore condensation inves-tigators regarding the precision they wish to achieve inmodeling adsorption processes in this range of theisotherm. As the number of points increases, the CSU ineach data point increases, suggesting that high-precisiondata collection should be made by measuring the amountadsorbed in different relative pressure regions and thenplotting the data on the same chart. But of course, thisaction introduces additional random errors into theadsorption isotherm evaluation but is probably the bestmethodology for high-precision data in the relative pres-sure regions usually ascribed to mesopore condensationinvestigations.

7. Conclusion

In the present BET-SSA measurements, we were ableto ignore thermal transpiration effect contributions to theRCSU; however, these effects should always be consideredwhenever the difference between the measured and thecorrected pressures is greater than the SU of the measuredpressure. Provided that the dosing and measuring mani-folds are maintained at the calibration temperature,thermal expansion contributions can be neglected. If alarge temperature difference exists, they should beconsidered. To decrease the SU in equilibrium pressuremeasurement and the CSU in the amount of gas adsorbed,it is crucial that the liquid nitrogen level be controlledprecisely. The uncertainties in the outgassed sample mass,the liquid nitrogen level control, and the dead-volumemeasurements and a poor selection of the linear range ofthe BET data are the major contributors to the RCSU inthe BET-SSA evaluation. This latter uncertainty iscumulative. As a result of the above actions, the repro-ducibility between lab-to-lab measurements will be sig-nificantly improved.

Acknowledgment. We thank the University of SouthAustralia for the provision of research support for thiswork.

LA020985T

Table 5. BET SSA Calculated for Different “Linear”Relative Pressure Ranges

p/p0 range SBET, m2/g R2 relative error, %

0.05-0.15 109.82 0.999 95 -0.3970.05-0.20 110.26 0.999 97 0.0000.05-0.25 109.76 0.999 97 -0.4510.05-0.30 109.04 0.999 95 -1.1100.05-0.35 107.91 0.999 85 -2.129


analysis of uncertainties in manometric gas-adsorption measurements. i: propagation of...

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