c° ' • • % /■ •*<£&% y^-> /.-iSfcS. > ** ■^ ^ ...- / v^ f V %*^v v^V V'^'V'^"- ••.V .**,&&%. X-^ii.V \/.jtffcr.V y.-^afe-V /.***& ^^ *U* T +<>0* V'-*> ^*^>* V' f;?; > ^'^V 7 V : —*V V* "\ /tiafcS X/JafeX ***&&>* <* *' . $9* * * o„ ^ .(V . • • • » «*. ^ A" ♦VGKB*'. ' *. * ^ . 0( ^^jg # . "*b^ »0 V v*>^ *^7V»* A *bV v-o^ «u * >^ . o » • . *h ^c? ^\ \ BUREAU OF MINES INFORMATION CIRCUU\R/1989 C3iO 1*1 A Method for the Calibration of Class 2 and Class 4 Standards of Mass By Nabii A. Bibawy UNITED STATES DEPARTMENT OF THE INTERIOR Mission: As the Nation's principal conservation agency, the Department of the Interior has respon- sibility for most of our nationally-owned public lands and natural and cultural resources. This includes fostering wise use of our land and water resources, protecting our fish and wildlife, pre- serving the environmental and cultural values of our national parks and historical places, and pro- viding for the enjoyment of life through outdoor recreation. The Department assesses our energy and mineral resources and works to assure that their development is in the best interests of all our people. The Department also promotes the goals of the Take Pride in America campaign by encouraging stewardship and citizen responsibil- ity for the public lands and promoting citizen par- ticipation in their care. The Department also has a major responsibility for American Indian reser- vation communities and for people who live in Island Territories under U.S. Administration. Information Circular] 9230 A Method for the Calibration of Class 2 and Class 4 Standards of Mass By Nabil A. Bibawy aq^ UNITED STATES DEPARTMENT OF THE INTERIOR Manuel Lujan, Jr., Secretary BUREAU OF MINES T S Ary, Director i\0' ™* CONTENTS Page Abstract 1 Introduction 2 Acknowledgments 2 Balance and weights 2 Equations and designs for weighing comparisons 3 Calibration procedure 4 Calibration of 1-kg weights 4 Calibration within a set of weights 4 Calibration results 5 Calibration of 5- to 1-kg, class 2 weights , 6 Calibration of 1-kg, class 2 weights 6 Calibration of 500- to 100-g, class 2 weights 7 Calibration of 50- to 10-g, class 2 weights 7 Calibration of 5- to 1-g, class 2 weights 8 Calibration of 5- to 1-kg, class 4 weights 8 Calibration of 1-kg, class 4 weights 9 Calibration of 500- to 100-g, class 4 weights 9 Calibration of 50- to 10-g, class 4 weights 10 Calibration of 5- to 1-g, class 4 weights 10 Discussion of results 11 Conclusions 11 Appendix A.-Multipliers of the observations for determining the calculated correction values and deviation . . values for four equal weights X x to X 4 , NIST design A.1.2 12 Appendix B. -Multipliers of the observations for determining the calculated correction values and deviation . . values for weights X 1 to X 6 , less than 1 kg, NIST design C.10.5.2.2.1.1.1 13 Appendix C— Multipliers of the observations for determining the calculated correction values and deviation . values for the calibration of weights X t to X 6 , larger than 1 kg, NIST design C.10.5.2.2.1.1.1 14 ILLUSTRATION 1. Voland model HCE 100G balance, class 2 and class 4 sets of weights 3 TABLES 1. National Institute of Standards and Technology report of calibration for class M weights 2 2. Individual comparisons of class 2 weights with the class M standard 5 3. Individual comparisons of class 4 weights with the class M standard 5 UNIT OF MEASURE ABBREVIATIONS USED IN THIS REPORT g gram mg milligram kg kilogram pet percent L liter A METHOD FOR THE CALIBRATION OF CLASS 2 AND CLASS 4 STANDARDS OF MASS By Nabil A. Bibawy 1 ABSTRACT The U.S. Bureau of Mines has established a method for the calibration of class 2 and class 4 standards of mass using a procedure from designs developed by the National Institute of Standards and Technology (NIST). This method produces a weight calibration that is within 1.0 pet of the balance sensitivity. A set of four 1-kg weights yielded a standard deviation of 3.25 mg compared to ±2.91 mg balance sensitivity. The method employs the use of a high-accuracy balance to measure the difference between a set of weights calibrated by NIST and the set to be calibrated. The error of the balance is included in the error for each weight calibrated. Chemist, Helium Field Operations, U.S. Bureau of Mines, Amarillo, TX. INTRODUCTION The U.S. Bureau of Mines investigates methods to improve helium analysis for the certification of calibration standards used in helium production, purification, liquifi- cation, transportation, and storage operations. Since the ultimate composition of primary standard gas mixtures depends upon the accuracy of the balance weights used, the weights require accurate calibration. Calibrating the weights against existing NIST certified weights provides the best solution because the error of the balance is included in the calibration. The true mass of an object is usually determined on an analytical balance by comparing the object to a group of weights whose individual masses are accurately known. True mass of the balance weight is determined by compar- ing it to known standards of mass and applying the correc- tion value to make the weight agree with the standards. Preparation of primary standard gas mixtures for ana- lytical instruments was described by Miller, Carroll, and Emerson in 1965. 2 The nominal mass of the balance weight is the quantity of weight actually observed on the balance from comparison to the weight(s) of a weight set and the balance chain scale. The correction value is added to or subtracted from the nominal weight to obtain the true mass. Designs for the calibration of standards of mass were discussed by Cameron, Croarkin, and Raybold in 1911? This collection of designs for the intercomparisons of sets of weights was used as a resource for determining the cor- rection values for balance weights used in the preparation of primary standard gas mixtures. The designs used are A.1.2 for the calibration of four equal weights (appendix A), and C.10.5.2.2.1.1.1 for the calibration of weights less than 1 kg (appendix B), and larger than 1 kg (appendix C). These designs provide for a check standard weight, which is treated as an additional unknown weight and is used for monitoring the performance of the calibration process. Because the weights calibrated in this study are com- pared against weights of the same metal, the buoyancy corrections are not required. ACKNOWLEDGMENTS The author thanks Dr. Richard S. Davis, physicist, Mass Division, Center for Basic Standards, National Institute of Standards and Technology, Gaithersburg, MD, for his technical assistance. BALANCE AND WEIGHTS The Voland model HCE 100G 4 analytical balance is a double-pan balance used for the comparison of weights against standards (fig. 1). The high precision and large capacity of this balance make it suitable for the prepara- tion of weighed gaseous mixtures in standard-size cylin- ders. An analysis using a double-substitution procedure to determine the reproducibility of the Voland balance yielded a precision of ±2.91 mg. Class 2 and class 4 sets of stainless steel weights with individual masses ranging from 1 g to 5 kg were supplied with the balance. These sets were calibrated using the procedures described in this report. The weights used as the standard and check standard are of a class M set with 1-g to 1-kg masses of stainless steel, 30- to 500-mg masses of tantalum, and 1- to 20-mg masses of aluminum. This set was made by Wm. Ainsworth and Sons, Inc., and calibrated by NIST (table 1). 2 Miller, J. E., A. J. Carroll, and D. E. Emerson. Preparation of Primary Standard Gas Mixtures for Analytical Instruments. BuMines RI 6674, 1965, 10 pp. 3 Cameron, J. M., M. C. Croarkin, and R. C. Raybold. Designs for the Calibration of Standards of Mass. NBS Tech Note 952, June 1977, 64 pp.; NTIS PB 268499. Reference to specific products does not imply endorsement by the U.S. Bureau of Mines. Table 1. -National Institute of Standards and Technology report of calibration for class M weights Item Mass Correction, mq 1 kg 500 g 300 g 200 g 100 g 50g 30g 20g 10 g 5g 3g 2g ig 500 mg 300 mg 200 mg 100 mg 50 mg 30 mg 20 mg 10 mg 5mg 3mg 2mg 1 mg Apparent True Uncertainty, mg 2.285 1.722 .923 .627 -.3111 .0665 -.1964 .0473 .0166 -.0239 -.0228 -.0184 .0044 -.0052 -.0063 -.00886 -.00425 -.00197 .00216 -.00275 .00384 -.00434 .00424 -.00204 .00353 11.039 6.292 3.646 2.443 .6357 .5205 .0761 .2290 .1074 .0215 .0045 -.0002 .0135 -.0406 -.0275 -.02300 -.01132 -.00551 .00004 .00327 .00686 -.00283 .00515 -.00144 .00384 0.069 .040 .034 .027 .0075 .0056 .0058 .0050 .0064 .0048 .0033 .0024 .0023 .0017 .0011 .00083 .00078 .00055 .00054 .00046 .00059 .00049 .00052 .00045 .00059 Volume at 20° C, cm 3 126.4716 63.3962 38.0233 25.3489 12.70656 6.33720 3.80229 2.53488 1.26744 .63372 .38023 .25349 .12674 .030118 .018070 .012046 .006023 .00301 1 .001807 .007408 .003706 .001850 .001113 .000740 .000371 Figure 1.-Voland model HCE 100G balance, class 2 and class 4 sets of weights. EQUATIONS AND DESIGNS FOR WEIGHING COMPARISONS The equations and methods of comparing weights used in this report were performed in accordance with NBS Technical Note 952 5 and are called designs. To calibrate a set of weights, multiple comparison must be made of one weight or set of weights against another. Using the first comparison as an example, Y x is the weight that must be added to the pan carrying X 2 in order to balance X^ If weight must be added to X 1 in order to balance X 2 , then that weight with a negative sign becomes Y 1} indicating that X t is less than X 2 . ^Work cited in footnote 3. For four equal weights, X (1 10 4) , six comparisons are made as follows: where X, and (1 to 4) (1 to 6) x x against X 2 = Y x x x against X 3 = Y 2 x x against X 4 = Y 3 x 2 against X 3 = Y 4 x 2 against X 4 = Y 5 x 3 against X 4 = Y 6 ) = weights to be compared I observed differences for the above comparisons. The correction values (B) for each weight (or combi- nation of weights) were calculated as follows: B 1 = [Y 1 K rxY ^ + Y 2 K /v vV + B, [Y^ _ Y ,+Y 2 K /V .^ + : (X 2 ,Y 2 )- + M (X) R]/D + M (X2) R]/D The deviations (Dev) are found in a similar manner to the B values, using the coefficients from the same designs. The deviations are calculated as follows: Dev(l) = [Y x L rz Y >> + Y 2 L^ WO Dev (2) = [Yt L r7 Y >> + Y 2 L,^ '(Z 2 ,Y 2 ) >]/D •]/D B ° = ^ K (X n ,Y 1 ) + ^ K (Xn,Y 2 ) + ' • ' +M (Xn) R]/D(l) where B n = the calculated correction value for weight X,,, mg, Y n = the observed difference in the (n)th comparison being made, mg, *VY Y1 = parameter value for the design being ^' J used, M (XJ = restraint value for the design being used, and R = the previously determined true mass correction value for the standard weight or combination of standard weights, mg, D = the divisor for the design being used. The true mass for each weight (X) is determined by adding or subtracting the correction value B to the nom- inal value of the standard weights being compared. Dev (n) [Y l L (7 Y ^ + Y 2 L(Z n ,Y 2 ) + Wi) -]/D(2) where Dev (n) = Y n = L (Z n ,YJ - and D = calculated deviation corresponding to observation Y n , mg, the observed difference in the (n)th comparison, mg, the parameter value (deviation) from the design being used, the divisor for the design being used. The standard deviation of the particular set of measure- ments is calculated as follows: The deviations are com- puted, squared, summed, and divided by the degrees of freedom for the particular design being used. The square root of this number is the standard deviation for the measurements. This should not be significantly different from the standard deviation of the balance of ± 2.91 mg. CALIBRATION PROCEDURE CALIBRATION OF 1-kg WEIGHTS NIST design A.1.2 (appendix A) was used in the cali- bration procedure for a 1-kg weight or a combination of weights equivalent to 1 kg. In the calibration procedure, a 1-kg weight (Xj) and a combination of weights equaling 1 kg (X 2 ) from the desired set were compared to an NIST- calibrated class M check standard (X 3 ) and standard (X 4 ). The check standard was a combination of weights equaling 1 kg, and the standard was a 1-kg weight. CALIBRATION WITHIN A SET OF WEIGHTS NIST design C.10.5.2.2.1.1.1 (appendix B) was used in the calibration procedure for weight sets with denomina- tions of 500 g, 200 g, 200 g, and 100 g; 50 g, 20 g, 20 g, and 10 g; and 5 g, 2 g, 2 g, and 1 g; or other combinations with the ratio of 5:2:2:1. In the calibration procedure, a 500-g, 50-g, or 5-g weight was used as X l5 a 200-g, 20-g, or 2-g weight was used as X 2 , a 200-g, 20-g, or 2-g weight was used as X 3 , and a 100-g, 10-g, or 1-g weight was used as X 4 . The desired set was compared to a check standard (X 5 ) comprised of a combination of 50-g, 20-g, 20g, and 10-g weights, 5-g, 2-g, 2-g, and 1-g weights, or 500-mg, 200-mg, 200-mg, and 100-mg weights. Weight X 6 was the standard obtained from the NIST- calibrated class M set and was a 100-g, 10-g, or 1-g weight. Eight different comparisons were made on the Voland balance as follows: x 1+ x 6 against X 2 +X 3 +X 4 +X5 = Y, x 1+ x 5 against X 2 +X 3 + X 4 +X 6 = Y 2 x 1+ x 4 against x 2 +x 3 +x 5 +x 6 = Y 3 x x against x 2 +x 4 +x 5 +x 6 = Y 4 X t against x 3 +x 4 +x 5 +x 6 = Y 5 x 2 +x 4 against x 3 +x 5 Y 6 X.+X, against x 3 +x 4 Y 7 x 2 +x 5 against Xs + X, Y 8 The Y (i to 8) values are the observations obtained for the above comparisons. The correction values (B) and deviations were calculated using equations 1 and 2, respectively, and the parameters and restraint from NIST design C.10.5.2.2.1.1.1 shown in appendix B. The standard deviation of a particular set of measurements was calcu- lated using the method outlined previously. To calibrate weights larger than 1 kg, the same NIST design (C.10.5.2.2.1.1.1) was used, but with a different restraint, divisor, and parameter values as shown in appendix C. Correction values, deviations, and the stand- ard deviations were calculated using equations 1 and 2 and the same steps as outlined above. CALIBRATION RESULTS The calibration results of the class 2 and class 4 sets using the class M, NIST-calibrated set as standards are given in tables 2 and 3 respectively. The results of cali- brations for individual weights follow. Table 2.-lndividual comparisons of class 2 weights with the class M standard Weight Item Identification True mass of, Standard Standard mark correction, deviation of deviation for mg correction, mg a set, mg 1 5 kg None 93.052 7.06 3.62 2 2 kg X 34.935 3.69 3.62 3 2 kg None 18.507 3.69 3.62 4 1 kg None 14.789 2.30 2.04 5 500 g None 9.091 1.06 2.67 6 200 g X 3.065 1.25 2.67 7 200 g None .208 1.25 2.67 8 100 g None 3.675 1.41 2.67 9 50 g None 2.909 1.31 2.44 10 20 g X -.122 1.29 2.44 11 20 g None -1.551 1.29 2.44 12 10 g None .296 1.42 2.44 13 5g None 1.577 1.31 2.18 14 2g None 2.059 1.29 2.18 15 2g X -.798 1.29 2.18 16 ig None -.399 1.42 2.18 Table 3.-lndividual comparisons of class 4 weights with the class M standard Weight Item Identification True mass of, Standard Standard mark correction, deviation of deviation for mg correction, mg a set, mg 1 5 kg 66.624 7.06 5.87 2 2 kg •X 25.649 3.69 5.87 3 2 kg 15.649 3.69 5.87 4 1 kg 16.039 2.30 2.89 5 500 g 10.698 1.06 1.54 6 200 g •X .279 1.25 1.54 7 200 g .279 1.25 1.54 8 100 g 2.283 1.41 1.54 9 50 g 4.713 1.31 4.42 10 20 g •X .885 1.29 4.42 11 20 g -4.115 1.29 4.42 12 10 g .800 1.42 4.42 13 5g 2.543 1.31 2.44 14 2g •X .874 1.29 2.44 15 2g .160 1.29 2.44 16 ig -.634 1.42 2.44 CALIBRATION OF 5- TO 1-kg, CLASS 2 WEIGHTS Calibration of the 5- to 1-kg, class 2 set of weights was accomplished using the procedure outlined under "Calibration Within a Set of Weights" and NIST design C.10.5.2.2.1.1.1 (appendix C). Observations Y (1) Zw v (5) 10 mg 30 mg 30 mg 10 mg 25 mg 20 mg 10 mg 20 mg 11.039 mg These observation values and R x were then used in equation 1 to calculate the true masses, which are X 1 = 5 kg + 93.052 mg, X 2 = 2 kg + 34.935 mg, and X 3 = 2 kg + 18.507 mg, The deviations were Dev (1 Dev (2 Dev (3 Dev (4 Dev (5 Dev (6 Dev (7 Dev (8 -2.86 mg 2.86 mg 0.71 mg -0.71 mg 2.14 mg 2.14 mg -3.57 mg The sum of the deviations squared is 39.27, which divided by 3 degrees of freedom is 13.09. The square root of 13.09 is 3.62 mg, the standard deviation of this set of measure- ments. R^s found in the report of calibration for the class M set standard for 1-kg weight (table 1). CALIBRATION OF 1-kg, CLASS 2 WEIGHTS Calibration of the 1-kg, class 2 weight was accomplished using the procedure outlined under "Calibration of 1-kg Weights" and NIST design A.1.2 (appendix A). Observations Y (1) Zw v (5) 3P 5mg 5mg 5mg 11.039 mg These observation values and R t were then used in equation 1 to calculate the true masses, which are and X : = kg + 14.789 mg X 2 - kg + 16.039 mg. R 2 = 16.039 mg, which will be the restraint used in the next weighing set. The deviations were Dev (1) Dev (2) Dev (3) Dev (4) Dev (5) Dev (6) 1.25 mg -2.50 mg 1.25 mg 1.25 mg -1.25 mg The sum of the deviations squared is 12.50, which divided by 3 degrees of freedom is 4.17. The square root of 4.17 is 2.04 mg, the standard deviation of this set of measure- ments. 7 Rj is found in the report of calibration for the class M set (table 1) and is the true mass correction value for the 1-kg weight. CALIBRATION OF 500- TO 100-g, CLASS 2 WEIGHTS Calibration of the 500- to 100-g, class 2 set of weights was accomplished using the procedure outlined under "Calibration Within a Set of Weights" and NIST design C.10.5.2.2.1.1.1 (appendix B). Observations lo-) = 5 n mg Y( 2 ) = Omg Y (3) = 5mg Y (4) = Omg Y(5) = Omg Y (7) = Omg Y (8) = 5mg 8 R 2 - 16.039 mg These observation values and R 2 are then used in equation 1 to calculate the true masses, which are X x = 500 g + 9.091 mg, X 2 = 200 g + 3.065 mg, X 3 = 200 g + 0.208 mg, X 4 = 100 g + 3.675 mg, and Xs = 100 g + 1.533 mg. R 3 = 1.533 mg, which will be the restraint used in the next weighing set. The deviations were Dev (1) = = 2.14 mg Dev (2) = = -1.43 mg Dev (3) = = -0.71 mg Dev (4) = 1.43 mg Dev (5) = = -1.43 mg Dev (6) = -- Dev (7) = = -1.43 mg Dev (8) = = 2.86 mg The sum of the deviations squared is 21.44, which divided by 3 degrees of freedom is 7.15. The square root of 7.15 is 2.67 mg, the standard deviation of this set of measure- ments. CALIBRATION OF 50- TO 10-g, CLASS 2 WEIGHTS Calibration of the 50- to 10-g, class 2 set of weights was accomplished using the procedure outlined under "Calibration Within a Set of Weights" and NIST design C.10.5.2.2.1.1.1 (appendix B). Observations Y(i) = 5 mg Y(2) = 5 mg Y 3 = Y 4 = Y 5 = Y 6 = Y (7) = ° I( 8 ) = ?S5 *r; 1.533 mg These observation values and R 3 were then used in equation 1 to calculate the true masses, which are Xj = 50 g + 2.909 mg, X 2 = 20 g + (-0.122) mg, X 3 = 20 g + (-1.551) mg, X 4 = 10 g + 0.296 mg, and Xs = 10 g + 2.439 mg. R 4 = 2.439 mg, which will be the restraint used in the next weighing set. The deviations were Dev(l Dev (2 Dev (3 Dev (4 Dev (5 Dev (6 Dev (7 Dev (8 2.14 mg -0.71 mg -1.43 mg 0.71 mg -0.71 mg 0.71 mg -2.14 mg 2.14 mg The sum of the deviations squared is 17.80, which divided by 3 degrees of freedom is 5.93. The square root of 5.93 is 2.44 mg, the standard deviation of this set of measure- ments. Rj is found in the calibration of the 1-kg, class 2 weight and is the 'Rj is found in the calibration data of the 500- to 100-g, class 2 set CALIBRATION OF 5- TO 1-g, CLASS 2 WEIGHTS CALIBRATION OF 5- TO 1-kg, CLASS 4 WEIGHTS Calibration of the 5- to 1-g, class 2 set of weights was accomplished using the procedure outlined under "Calibration Within a Set of Weights" and NIST design C.10.5.2.2.1.1.1 (appendix B). Observations Y (1) v (5) Y (6) Y(7) - 5mg Tp : 5mg = = = = = = Calibration of the 5- to 1-kg, class 4 set of weights was accomplished using the procedure outlined under "Calibra- tion of 1-kg Weights." Comparisons and calculation of correction values, deviations, and the standard devia- tion were made using NIST design C.10.5.2.2.1.1.1 (appendix C). Observations 2.439 mg These observation values and R 4 were then used in equation 1 to calculate the true masses, which are Y (1) Zw v (5) Zf® 3g> 20 mg 15 mg 10 mg 15 mg 15 mg 10 mg 10 mg 15 mg 11.039 mg These observation values and R : were then used in and The deviations were Xj - 5 g + 1.577 mg, equation 1 to calculate the true masses, which ; X 2 = 2 g + 2.059 mg, X, = 5 kg + 66.624 mg, X 3 = 2 g + (-0.798) mg, x 2 = 2 kg + 25.649 mg, X 4 = 1 were g + (-0.399) mg. and X 3 = The deviations were 2 kg + 15.649 mg. Dev (1) = Omg Dev (1) = 3.57 mg Dev (2) = -1.43 mg Dev (2) = -1.43 mg Dev (3) = 1.43 mg Dev (3) = -2.14 mg Dev (4) = 1.43 mg Dev (4) = 5.00 mg Dev (5) = -1.43 mg Dev (5) = -5.00 mg Dev (6) = -1.43 mg Dev (6) = 2.14 mg Dev (7) = 1.43 mg Dev (7) = -2.14 mg Dev (8) = 1.43 mg Dev (8) = 5.00 mg The sum of the deviations squared is 14.31, which divided by 3 degrees of freedom is 4.77. The square root of 4.77 is 2.18 mg, the standard deviation of this set of measure- ments. The sum of the deviations squared is 103.53, which divided by 3 degrees of freedom is 34.51. The square root of 34.51 is 5.87 mg, the standard deviation of this set of measure- ments. 10 R 4 is found in the calibration data of the 50- to 10-g, class 2 set and is the true mass correction value for mass X 5 . Rj is found in the report of calibration for the class M set (table 1) and is the true mass correction value for the 1-kg weight. CALIBRATION OF 1-kg, CLASS 4 WEIGHTS CALIBRATION OF 500- TO 100-g, CLASS 4 WEIGHTS For calibration of the 1-kg weight, the procedure outlined in the "Calibration of 1-kg Weights" section was used. NIST design A.1.2 (appendix A) was employed for calculation of the correction values, deviation values, and standard deviation. Observations Y (1) Y (5) Y(6) 12 R, = 5mg 5 mg 5mg = 11.039 mg These observation values and R x were then used in equation 1 to calculate the true masses, which are and X, = 1 kg + 16.039 mg, X^ = 1 kg + 13.539 mg. The deviations were Dev (1) Dev (2) Dev (3) Dev (4) Dev (5) Dev (6) 2.50 mg -2.50 mg 2.50 mg -2.50 mg The sum of the deviations squared is 25.0, which divided by 3 degrees of freedom is 8.33. The square root of 8.33 is 2.89 mg, the standard deviation of this set of measure- ments. Calibration of the 500- to 100-g, class 4 weights was accomplished by following the steps outlined under "Cali- bration Within a Set of Weights." Comparisons and calcu- lation of the correction values, deviations values, and standard deviation were made using NIST design C.10.5.2.2.1.1.1 (appendix B). Observations Y (1) I® v (5) *8 : 5 mg 10 mg 10 mg 5 mg 5mg Omg Omg Omg 13.539 mg These observation values and R 5 were then used in equation 1 to calculate the true masses, which are Xj = 500 g + 10.698 mg, R 5 = 13.539 mg, which will be the restraint used in the j next weighing set. X 2 = X 3 = x 4 = X, = 200 g + 0.279 mg, 200 g + 0.279 mg, 100 g + 2.283 mg, 100 g + 2.283 mg. R 6 = 2.283 mg, which will be the restraint used in the next weighing set. The deviations were Dev (1) = Dev (2) = Dev (3) = Dev (4) = - -1.43 mg = 0.71 mg = 0.71 mg = Dev (5) = = Dev (6) -- = Dev (7) = Dev (8) = = 1.43 mg = -1.43 mg The sum of the deviations squared is 7.14, which divided by 3 degrees of freedom is 2.38. The square root of 2.38 is 1.54 mg, the standard deviation of this set of measure- ments. 12 Rj is found in the report of calibration for class M set (table 1) ^Rj is found in the report of calibration of the 1-kg, class 4 weight and is the true mass correction value for mass X 2 . 10 CALIBRATION OF 50-to 10-g, CLASS 4 WEIGHTS Calibration of the 50- to 10-g, class 4 set of weights was accomplished using the procedure outlined under "Calibration Within a Set of Weights." The comparison and calculation of correction values, deviations, and the standard deviation were made using NIST design C.10.5.2.2.1.1.1 (appendix B). Observations Y m = 5mg Y (2) = 10 mg *0) = 5mg Y (4) = 5 mg Y (5) = *w = 5 mg Y (7) = 5 mg T(8) = 10 mg X = 2.283 mg These observation values and R 6 were then used in equation 1 to calculate the true masses, which are Xj = 50 g + 4.713 mg, X 2 = 20 g + 0.885 mg, X 3 = 20 g + (-4.115) mg, X 4 = 10 g + 0.800 mg, and X 5 - 10 g + 2.943 mg. R 7 = 2.943 mg, which will be the restraint used in the next weighing set. The deviations were Dev (1) = Dev (2) = Dev (3) = = 0.71 mg = -1.43 mg = -0.71 mg Dev (4) = Dev (5) = = 5.00 mg = -5.00 mg Dev (6) = = 0.71 mg Dev (7) = Dev (8) = - 0.71 mg = 2.14 mg The sum of the deviations squared is 58.67, which divided by 3 degrees of freedom is 19.56. The square root of 19.56 is 4.42 mg, the standard deviation of this set of measurements. 14 Rg is found in the calibration data of the 500- to 100-g, class 4 CALIBRATION OF 5- TO 1-g, CLASS 4 WEIGHTS Calibration of the 5- to 1-g, class 4 set of weights employed the procedure outlined under "Calibration Within a Set of Weights." The comparisons and calcula- tion of the correction values, deviations, and standard deviation were made using NIST design C.10.5.2.2.1.1.1 (appendix B). Observations Y [(.2) :( 3 ) (4) :(8) = 5mg - = = = 5mg = = = 15 R 7 - 2.943 mg These observation values and R 7 were then used in equation 1 to calculate the true masses, which are X x - 5 g + 2.543 mg, X 2 - 2 g + 0.874 mg, X 3 = 2 g + 0.160 mg, and X 4 = 1 g + (-0.634) mg. The deviations were Dev(l Dev (2 Dev (3 Dev (4 Dev (5 Dev (6 Dev (7 Dev (8 1.43 mg -0.71 mg -0.71 mg -2.14 mg 2.14 mg -0.71 mg -2.14 mg 0.71 mg The sum of the deviations squared is 17.80, which divided by 3 degrees of freedom is 5.93. The square root of 5.93 is 2.44 mg, the standard deviation of this set of measurements. 15 Rj is found in the calibration data of the 50- to 10-g, class 4 set and is the true mass correction value for mass Xj. 11 DISCUSSION OF RESULTS The data reported in tables 2 and 3 show the precision that can be achieved using the Voland model HCE 100G analytical balance as a comparator, when calibrating class 2 and class 4 sets of weights. The standard deviations of each set of weights were generally within ± 1 mg of the ±2.91-mg balance sensitivity. CONCLUSIONS This method of calibration of standards of mass is adequate to obtain the true masses of weights that are used as counterbalances in determining the true masses of individual gases in primary gas standard preparation. A unique feature of this method is that weighing designs are employed that provide a self- consistency check of the calibration process. The standard deviation for the class 2 weights, when compared to the NIST class M calibrated 1-kg weight, was determined for each set. The standard deviation was ±3.62 mg for the 5-kg weight set, ±2.67 mg for the 500-g weight set, ±2.44 mg for the 50-g weight set, and ±2.18 mg for the 1-g weight set. The standard deviation for the class 4 weights, when compared to the NIST class M calibrated 1-kg weight, was determined for each set. The standard deviation was ± 5.87 mg for the 5-kg weight set, ± 1.54 mg for the 500-g weight set, ±4.42 mg for the 50-g weight set, and ±2.44 mg for the 1-g weight set. 12 APPENDIX A.— MULTIPLIERS OF THE OBSERVATIONS FOR DETERMINING THE CALCULATED CORRECTION VALUES AND DEVIATION VALUES FOR FOUR EQUAL WEIGHTS X 1 TO X 4 , NIST DESIGN A.1.2 The restraint for this design is that weight x 4 has previously been calibrated and is used for the known standard. Parameter values [K(x y )]> * divisor = 4 Observations X 1 Xj X 3 X 4 Yj 1 A (T Y 2 1 -1 Y 3 2 1 1 Y 4 1 -1 Y 5 1 2 1 Y 6 1 1 2 M 4 4 4 4~ where degrees of freedom and * Defined in equation 1. K-(X y ) = parameter values, = 3, M^x ) = restraint values for the design. Deviation parameter values [L/£ y )]>** divisor = 4 Observations Z : Z 2 Zj Z 4 Z 5 Z 6 Yj 2 ^ ^1 1 1 0~ Y 2 -1 2 -1 -1 1 Y 3 -1 -1 2 -1 -1 Y 4 1 -1 2 -1 1 Y 5 1 -1 -1 2 -1 Y 6 1 -1 1 -1 2 where ** Defined in equation 2. L(Z ,Y ) = deviation values for the design. 13 APPENDIX B.— MULTIPLIERS OF THE OBSERVATIONS FOR DETERMINING THE CALCULATED CORRECTION VALUES AND DEVIATION VALUES FOR WEIGHTS X, TO X 6 , LESS THAN 1 kg, NIST DESIGN C.1 0.5.2.2.1. 1.1 The restraint for this design is that the sum of weights X 1} X 2 , X 3 , and X 4 has previously been calibrated and is used for the known standard. Parameter values [Kyy Y )]> * divisor = 70 Observations X x Xj X 3 X 4 X 5 Xg Y 1 15 ^8 ^8 1 1 2T Y 2 15 -8 -8 1 21 1 Y 3 5-12-12 19 -1 -1 Y 4 2 12-14-14 -14 Y 5 12 2-14-14 -14 Y 6 -5 8-12 9-11 -1 Y 7 5 12 -8 -9 1 11 Y 8 10-10 10-10 M 35 14 14 7 7 T where degrees of freedom and * Defined in equation 1. Kvv Y ) = parameter values, = 3, M/v -\ = restraint values for the design. Deviation parameter values [L/£ y )]>** divisor = 7 Observations Z x Z 2 Zj Z 4 Z 5 Z 6 Zy YJ 2 ^1 ~\ C) 3 Y 2 -1 2 -1 2 Y 3 -1 -1 2 -2 2 Y 4 3 -3 1 1 Y 5 -3 3 -1 -1 Y 6 2 -2 1 -1 3 -1 Y 7 -2 2 1 -1 -1 3 Y fi 2 -2 1 -1 -1 -1 where ** Defined in equation 2. L(Z Y ) = deviation values for the design. 2 -2 1 -1 -1 -1 3 14 APPENDIX C.— MULTIPLIERS OF THE OBSERVATIONS FOR DETERMINING THE CALCULATED CORRECTION VALUES AND DEVIATIONS FOR THE CALIBRATION OF WEIGHTS X, TO X 6 , LARGER THAN 1 kg, NIST DESIGN C.1 0.5.2.2.1. 1.1 The restraint for this design is that weight ^ has previously been calibrated and is used for the known standard. Parameter values [L/^ Y )]» * divisor = 7 Observations X 1 X 2 X 3 X 4 Xj X$ YJ ^9 ^5 ^5 ^2 ^2 0~ Y 2 1 -1 -1 2 Y 3 1 -1 -1 2 Y 4 7 3 4 Y 5 7 4 3 Y 6 1 -1 1 -1 Y 7 -5 -1 -3 -2 -1 Y 8 5 3 1 1 2 M 35 14 14 7 7 7~ where degrees of freedom and * Defined in equation 1. K/y Y ) = parameter values for the design, = 3, M/vj = the corresponding restraint values for the design. where Deviation parameter values [L/^ Y )]»** divisor = 7 Observations Zj Z 2 Zj Z 4 Z 5 Z 6 Z, Y : 2 ^ ^1 3 Y 2 -1 2 -1 2 Y 3 -1 -1 2 -2 2 Y 4 3 -3 1 1 Y 5 -3 3 -1 -1 Y 6 2 -2 1 -1 3 -1 Y 7 -2 2 1 -1 -1 3 Y 8 2 -2 1 -1 -1 -1 ** Defined in equation 2. L(Z Y ) = deviation values for the design. 2 -2 1 -1 -1 -1 3 U.S. GOVERNMENT PRINTING OFFICE: 611-012/00,117 INT.BU.OF MINES,PGH.,PA 28993 ■TJ m z > ■n O X u 33 H m c u> m I v» w 8 o > r- □3 C CO z m c/> CO ^ w roc 52. -* it 3 £ °~3 £ a S2 ■* cn -a 00 o o 3" 3 (D > z m O c > i - o -o "0 O -< m "D I - o -< m 33 45 90 %/^V ~V' : ^V" V^V V*^> °v^V V x"i- V %. • •» .0- A <^ *°* • ' & %> '•^tT 9 ' a. 0,3 % " * • "*« • ' < • " ° " A v A «o . .0° .0 ^ *.,!- « • • • « v> % t* <>&' ^ • 4^ ^ • c ♦ HECKMAN BINDERY INC. ^ NOV 90 N. MANCHESTER, IMniAMA /COCO : %^ -*Jte-* \/ .*^K^ V.^ .'Jfev \/ :&ttk. +^* :\