Langmuir 2003, 19, 7919-7928
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Analysis of Uncertainties in Manometric Gas-Adsorption Measurements. 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-adsorption measurements are presented for experimental data for nitrogen adsorption at ≈77 K by a traceable standard carbon black material (004-16820-02). Equipment- and measurement-related uncertainty sources derive from the dosing and sampling volumes; temperature control of these volumes; dosing, equilibrium, and barometric-pressure measurements; liquid nitrogen level control; and sample-mass measurements. Data processing errors derive from ignoring thermal transpiration effects and nonideal gas behavior. Departure from ideal gas behavior contributions to the amount adsorbed was accounted for by considering the temperature relationship of the second virial coefficient of the virial equation of state. Variation in the liquid nitrogen level control is shown to have an enormous impact on the pressure-measurement precision and, hence, the amount adsorbed. Variation of the liquid nitrogen level by (1 mm results in a variation of the equilibrium pressure from -0.42 to +0.52% and the volume of gaseous nitrogen adsorbed from -8.53 to +5.94% when compared with the results obtained during precise level control (within (0.2 mm). In addition to these uncertainty sources, reproducibility in the sample-mass measurement is important; a decrease in the mass resolution from 5 × 10-5 to 5 × 10-4 g generates a relative combined standard uncertainty of the volume of nitrogen adsorbed by 10-fold varying from 2.78 to 9.86% over the relative pressure range from 0.0007 to 0.98. For a similar standard mass uncertainty applied to the BrunauerEmmet-Teller specific surface area analysis, the final area relative combined uncertainty increases from 0.63 to 6.19%. The calculated cumulative relative combined uncertainty in the volume adsorbed increases continuously with each experimental point from 0.28 (for the first experimental point on the adsorption branch 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. Introduction Volumetric or, more correctly, manometric or barometric gas-adsorption measurements are used extensively for the characterization of microporous and mesoporous powdered materials and to obtain the details of gas- or vapor-solid interactions.1 To the best of our knowledge, the last published major international survey of adsorbent specific surface area (SSA) analysis of a single sample was presented in 1969.2 In this survey, most, if not all, of the adsorption equipment was operated manually. The advent of commercial equipment and computer control of the adsorption data collection rendered real-time data collection.3,4 A serious issue regarding data collection with any equipment, which often becomes exacerbated by automation, is the need for frequent calibration to ensure accuracy of the various measuring devices in the equipment. Commercial adsorption apparatus manufacturers and several national standard laboratories provide traceable SSA standards for powders. These powder area values are quoted with a degree of precision, which has been defined statistically from a survey of measure* 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).
ments performed at various international laboratories. Statements can be readily found claiming that the reproducibility of the BET method for SSA evaluation is not high, typically quoted as >(5%.5,6 Also reported are areas derived from krypton and nitrogen that agree only to within (20%5 and for argon and nitrogen to within (10%6. The magnitude of these variations is mostly due to differences in gas-solid interfacial interactions but may also be partly due to a lack of experimental uncertainty considerations between different equipment and calibration details. Rouquerol et al.6 provide a beautiful anthology for any researcher interested in gasadsorption experimental methods, their intricacies, and technique development. They provide the details of many previously described apparatus for both manometric and gravimetric adsorption measurements. They also present “Details of the Operational Stages”, a discussion of calibration volume determination and of the corrections required for the temperature, pressure, and nonideal behavior of the bulk gas. In the body of this paper, we will revisit each of these variables with respect to their contributions to the overall combined standard uncertainty (CSU) associated with manometric adsorption measurements. Throughout this paper, we used terms for the uncertainties as were recommended in NIST7 and EURACHEM8 uncertainty guidelines. (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 and Porous Solids; Academic Press: Sydney, 1999.
10.1021/la020985t CCC: $25.00 © 2003 American Chemical Society Published on Web 08/15/2003
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Langmuir, Vol. 19, No. 19, 2003
Loebenstein and Deitz9 indicated that the relative combined standard uncertainties (RCSUs) in the amount of nitrogen adsorbed for materials with SSAs varying between 0.7 and 1720 m2/g were 10.4 and 0.12%, respectively. Their analysis only considered the first point of each isotherm, commenting on their cumulative nature and on the difficulty in evaluating the CSU of measurements for the BET methods. Unfortunately, they did not present their calculations. Ross and Olivier,10 Webb and Orr,11 and more recently Robens et al.12 addressed in detail the issues of apparatus calibration, thermal transpiration, and the ideal and nonideal behavior of gaseous helium and nitrogen, respectively, at cryogenic temperatures. Calculation of the departure from ideal gas behavior, even at low pressure, is essential for accurate isothermal adsorption data. Departure from ideal gas behavior impresses substantially on the higher relative pressure regime, affecting mesopore analyses, in terms of inaccurate pore volume analysis and width distribution data. Of course, these analyses have their own intricacies, which are outside the intention of the present communication. The importance of an accurate liquid nitrogen level control is taught in the patent awarded to Killip et al.13 describing a porous isothermal jacket. Recently, Yanazawa et al.14 presented results indicating the RCSUs in deadvolume measurements using a newly developed constantvolume cell with a surrounding vacuum jacket at the position for the expected liquid nitrogen level. A careful analysis of these publications motivated us to address quantitatively the issue of uncertainties associated with manometric gas-adsorption measurements. Recently, we described a manometric gas-adsorption apparatus design for real-time nitrogen adsorption measurement and analysis.15 Originally, the equipment was calibrated for SSA analysis against a traceable carbon black standard material no. 004-16820-02. To correctly interpret any calibration results of a new apparatus design and compare them with traceable standards, it is of paramount importance that the data are accurate. In the present work, we address the issues of uncertainties in the evaluation of gaseous nitrogen volume adsorbed and its translation into SSA evaluation. In particular, we consider the measurement CSU effects in the dosing and sample volume calibrations and associated barometric pressures, the gas-manifold and adsorbent sample-holder temperature measurement and control, and the precision of the liquid nitrogen level during measurements. We show the impact of standard uncertainties (SUs) and CSUs associated with these variables on amounts adsorbed (as an adsorption isotherm). These uncertainties are also translated to show their impact on BET-SSA values. A detailed (7) Taylor, B. N.; Kuyatt, Ch. E. Guidelines for evaluating and expressing the uncertainty of NIST measurement results; National Institute of Standards and Technology Technical Note 1297, 1994 edition; 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 Scientific Instruments 2001, 72 (7), 3038-3045.
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knowledge of such information provides theorists whose models are fitted to adsorption data with a stronger indication of their model’s accuracy. Finally, this work provides researchers with information on the measuring procedure and details of the most important parameters that should be accurately measured and controlled during manometric adsorption measurements. Accuracy is manifestly required in the increasingly “popular” research field of supercritical adsorption, especially because subcritical measurements are required to establish detailed, accurate pore volumes and their distributions with the apparent width for correct supercritical isotherm analysis. In many cases, these volumes are determined after many data points are collected during the adsorption isotherm measurements. With increasing the number of points, one needs to be aware of the cumulative nature of various types of uncertainty in the data. 2. Materials and Methods 2.1. Materials. Earlier, we reported the development of an automatic gas-adsorption apparatus, which can be used for the characterization of powdered materials.15 The operation of this apparatus was calibrated (in the “usual way”) via manometric nitrogen gas adsorption by several standard powdered materials via calculations of the multipoint BET SSA. The calculated values agreed with those supplied within the quoted reproducibility. In most cases, such reproducibility is regarded as adequate. Typically, the supplier’s quoted values are determined as sample standard deviations (SSDs) of the area data obtained under repeatability conditions reported by several laboratories, but in reality, the experimental uncertainties of some of the area data provided to these laboratories may be outside the quoted SSD value. These data are discounted via statistical “relevance” arguments. Although our measurements were repeated as four separate samples of the supplied standard materials and were within the quoted SSD, the sources of the uncertainties in the data were never obvious. The analysis of uncertainties in the present work was performed using a standard carbon 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 the measurements, the carbon black sample was heated at ≈2 °C/ min up to 200 °C, then held at this temperature for 4 h, as was recommended by the manufacturer, achieving a background pressure
