XOV.)
1960
VIRIAL COEFFICIEKTS OF HELIUM
dium and cesium iodides were found to be 1.33, 1.13, 1.60 and 1.58, respectively. I n order to test the validity of our treatment of alkali metal iodide-silver iodide mixtures, solutions containing varying concentrations of silver iodide were studied a t fixed potassium iodide concentration (0.5875 M ) and AE calculated from ~ I KMI.+I obtained from the values of K K I / K A and. the data in Table IV. As is seen in Table V, the agreement between observed and calculated values of AE is excellent even though the concentration of potassium iodide in these solutions is greater than in any solution used to evaluate the various constants. The P K A g C i - ~ K Mvalues I given in Tables IV and VI are corrected for the formation of MI.Ag1. The value of P K A g C l - p ~ increases m in going fr;m lithium to potassium and then decreases for rubidium and cesium. This trend indicates that some unknown effect in addition to polarization or intimate ion pair formation is operative. Comparing the effect of anion on PKMXwhen ;SI is fixed, it is seen from Table VI that the order of dissociation constants is K M C l > KMI> K M B r for all the alkali metal ions. An explanation of these
1607
results on t,he basis of only polarization effects does not seem possible. However, invoking the possibility of the existance of intimate and solvent separated ion pairs in EDA, it is apparent that the predicted lower stability of an alkali metal chloride intimate ion pair might lead to greater dissociation than for the alkali metal iodides and bromides. Hibbard and Schmidt3b calculated K A g N o s , K A g I , K N ~and I KB~,NI from their conductance data. The values of p K ~ x- ~ K Nare ~ I1.51, 0.11 and 0.03 for silver iodide, tetra-n-butylammonium iodide and silver nitrate, respectively. Our potentio1.11, $0.20 and -1.6 for metric data yield p K ~ x- p K N a l for silver iodide, tetra-n-butylammonium iodide and silver nitrate, respectively. The agreement between our results and Hibbard and Schmidt’s results is not very good. Our calculation of the absolute value of K A ~ N O ~ is in very good agreement with Hibbard and Schmidt’s value, but we cannot offer an explanation for the lack of agreement among the other values. Acknowledgment.-This work was sponsored by the Office of Ordnance Research.
+
THE VIRIAL COEFFICIENTS OF HELIUM FROM 20 TO 300’K.l BY DAVIDWHITE,THORRUBIN,PAUL CAMKY AND H. L. JOHNSTON Cryogenic Laboratory of the Chemistry Department, The Ohio State University, Columbus 10, Ohio Received January 27, 1960
The compressibility of gaseous helium from the boiling point of liquid hydrogen to 300°K. in the pressure range one to 33 atmospheres, has been determined. The 22 experimental PV isotherms are represented by an equation of state containing three virial coefficients. The second virial coefficientsare compared with those calculated from various intermolecular potential functions for helium suggested in the literature. The agreement is fair.
Introduction usually can be offset by a decrease in equilibrium The compressibility of gaseous helium, from the distance of approach without serious deviation boiling point of liquid hydrogen to room tempera- from the experimental results. Some of the uncerture, has been investigated with particular emphasis tainty in the relation of the depth of the potential on the low pressure region. The purpose of the well to the equilibrium distance can be removed if a research was twofold; first, to measure deviations self consistent set of B’s over a wide temperature from the ideal gas law which could be used for the range are available. Although a large amount of establishment of a temperature scale in this Labora- experimental work on the equation of state of tory from helium gas thermometry data; second, to helium has been done in the past, there, undetermine the second virial coefficients, B’s, over an fortunately, does not exist a set of data in the acextended temperature range a t sufficiently small cessible temperature range (say 1 to 1200’K.) intervals as to permit a reliable test of a number of which to date can be judged self consistent. I n the suggested intermolecular potential functions for temperature range 20 to 300°K., the work prior to the interaction of two helium atoms. Admittedly, that reported here does not yield B’s which join the use of second virial coefficients to establish in smoothly those of Schneider and Duffie2 from 300 any detail, the form of intermolecular potentials is to 1473’K. It is very difficult to ascertain not a satisfactory one, nevertheless, one can obtain whether the second virial coefficient in the liquid some indication of at least two of the parameters, helium range are consistent with the B’s at higher namely, the depth of the potential well and the temperature where measurements begin at approxidistance of separation a t the minimum. We say mately 14OK. Although the temperature gap is some indication since these parameters of the small, there is such a rapid change in the magnitude potential functions are not separable. In any cal- of B that it is difficult to make any reliable compariculation of the intermolecular potential function son. The present work does not resolve all of the from second virial coefficients, one finds that the above mentioned problems, however, it does preeffect of increasing the depth of the potential well sent a consistent set of B’s over a wide temperature (1) This work was supported in part by the Air Material Command, Wright Field.
(2) W . G. Schneider and J. A . H. Duffie, J . Chen. Phye., 17, 751 (1949).
1608
D. WHITE,T. RUBIN,P. CAMKY AND H. L. JOHNSTON
interval which joins without discontinuity those of Schneider and Duffie.
Apparatus and Procedure The experimental apparatue used in this research has been described briefly.* A further account of it can be found in the work of Friedman.' The importsnt difference in this a paratus from those in uee by earlier workers consisted of t f e pipet valve which allowed isolation of the noxious volume from the ipet volume. A similar device was subsequently used %y Keller6 for PV determinations of helium below 4°K. Pipet pressures above two atmospheres were measured by means of an M.I.T. type dead weight gauge, the lower pressurea by mercury manometers. The pipet temperatures were determined by two standardized6 copper-constantan thermocouples. The amount of gm in the pipet wm determined by expansion of the sample into Calibrated evacuated thermostated tanks connected to a manometer. I n a typical experiment, the pipet pressure and temperature were measured together over a time interval long enough to obtain thermal equilibrium. Heat leaks were minimized by evacuating the volume inside of the can and by maintaining the auxilliary block temperatures at the pipet temperature. When the apparatus waa operated in this manner the pipet temperature drift WBB less than 0.001 degree per half hour,' further, this drift waa always consistent with the ressure-time variation. At the end o f the pressure-temperature measurements, the pipet valve was closed and the gaa in the noxious volume was removed. The gas sample in the pipet waa then expanded into the calibrated tanks at 25". At leaat three successive expansions were employed in order to reduce the remaining gas in the pipet to a pressure of 50 mm. or less. Accurate measurement of the tank pressures were made with a calibrated cathetometer. The pipet tem eratures of the runs in a given isotherm were not identical u!t did not vary by more than a few thousandths of a degree. Small corrections were applied to the PV products to bring them to a common temperature. The dead weight gauge used in the experiments was accurate to one part er 10,000 and precise to one part in 50,000 or better. Afmanometer readings were accurate to 0.01 mm. These were corrected for capillarity* to the standard gravity and to the density of mercury at 0". The expansion tanks calibrated with water, were known to one part in 15,000 or better. The pipet volume of 120 cc. calibrated with mercury, a t room temperature was known to one part in 25,000 or better. The pipet volume at other temperatures was calibrated using thermal expansion data for nickel samples0 cut from the same billet aa the pipet. The amount of gas in the expansion tanks was computed by an iterative procedure using, at first, trial second virial coefficients at 25.00"' and finally the values derived from this work. The amount of gas left in the pipet after expansion was similarly corrected by the iterative procedure for the second virial coefficients finally derived from this work. The temperature of the gas left in the noxious volume between the pipet and tanks after the last expansion was taken as the mean of several thermocouple station measurements. Since this noxious volume was only of the order of 2 cc., the perfect gas law was used in the calculation of the number of moles in the space. Purification of the Helium.-The helium used in this research was obtained from the Air Reduction Company and was purified by first passing it over a high-pressure charcoal trap a t the boiling point of liquid nitrogen. The gas was then liquefied and triply distilled before storage in a cylinder. The purity of the gas was determined by means of a mass spectrometer and found to be 99.99%. The impurities consisted of traces of nitrogen and oxygen which were probably introduced while filling the evacuated storage cylinder. (3) H. L. Johnston and D. White, Trans. of A.S.M.E., 71, 785 (1950). (4) A. 5. Friedman, Thesis, The Ohio State University, 1950. (5) W. E.Xeller, Phys. Rea.,97, 1 (1955). (6) T . Rubin, H. L. Johnston and H. Altman, J . A m . Chem. Soc., 73, 3401 (1951). (7) D.White, A.5.Friedman and H. L. Johnston, J . Am. Chem. Soc.. 72, 3927 (1950). (8) Cawood and Patterson, Trans. Faraday Soc., 29, 514 (1933). (9) T.&Rubin,H. Altman and H. L. Johnston,.unpublished.
ml
Vol. 64
Experimental Results The experimental results are shown in Table I. The data consist of 22 isotherms between 20.58 and 299.99'K. These data were fitted to the three term equation of state PV = A ( T ) RT)P C(T)P* (1)
+
+
TABLE I P.
atm.
P-V-T DATAFOR GASEOUSHELIUM PV(ca1cd.) PV(obsd.) P v, cc. atm. mole-' ac. atm. mole-'
4.5149 9.9769 14.9050 20.9709 26.5339 33.2163
Temp. = 299.99"K. 24,670.6 24,738.4 24,805.3 24,879.8 24,946.5 25,016.2
2.4 -0.2 -8.1 -9.5 -10.5 0.5
4.6677 9.8100 14.8556 21.1120 26.8544 33.1539
Temp. = 273.16'K. 22,480.1 22,541.8 22,597.0 22,681.2 22,744.2 22,820.8
-1.9 -1.5 4.3 -4.4 2.0 1.5
4.6702 10.5414 15.6522 20.7861 26.5900 33 .OH2
Temp. = 249.99"K. 20,572.1 20,635.3 20,713 .O 20,774.8 20,843.3 20,921.3
2.2 9.7 -6.4 -5.3 -4.9 -5.4
4.6920 5.3862 10.3046 11.0990 14.7136 20.7363 26.7631 32.8367
Temp. = 200.11"K. 16,482.4 16,493.9 16,551.3 16,559.3 16,610.8 16,678.8 16,755 .O 16,826.2
2.2 -8.4 1.9 3.6 -3.6 2.0 -0.5 2.6
4.6452 10.0888 14.6037 20.8028 26.4976 32.2868
Temp. = 175.02"K. 14,425.8 14,492 .O 14,548.0 14,623.3 14,693.9 14,762.9
4.5926 9.7597 14.5789 20.8465 26.4396 32.8310
Temp. = 150.04"K. 12,368.8 12,434.7 12,491.4 12,570 .O 12,634.5 12,711 .O
1.2 -1.9 0.0 -2.5 8.0 2.1
4.5670 9.7922 14.7720 20.2545 26.5302 33.1357
Temp. = 125.03"K. 10,313 .O 10,379.6 10,440.2 10,510.9 10,582.2 10,668.4
3.6 0.7 0.7 -3.2 1.9 -3.8
-0.1 0.3 -0.4 0.1 -0.8 1.1
Nov., 1960 TABLEI (Continued)
-
PV(ca1cd.) PV(ob6d.) cc. atm. mole-1
p, atm.
p v, cc. atm. mole-'
4.7092 10.2396 15.9445 20.7658 25.7881 32.6609
Temp. = 100.02'K. 8,264.55 8,329.81 8,393.52 8,451.51 8,515.43 8,596.63
4.5869 9.6796 14.9762 20.1910 25.8985 31.8234
Temp. = 90.04'K. 7,439.44 7,497.27 7,559.25 7,618.79 7,687.04 7,755.21
1.50 -1.07 -0.84
4.7353 9.5004 14.8323 19,4208 26.2664 29.2778
Temp. = 80.02"K. 6,618.37 6,665.01 6,730.12 6,779.63 6,855.14 6,898.14
-0.92 4.98 -0.67 0.72 1.88 -7.25
4.5877 9.7942 14.6939 19.8699 24.9626 30 I5448
Temp. = 75.01OK. 6,202.37 6,258.02 6,314.94 6,374.62 6,427.57 6,493.84
1.43 2.37 -0.71 -2.89 -1.34 -1.54
4,5815 10.2984 14.5849 19,2922 24.3462 29.7154
Temp. = 69.00"K. 5,708.03 5,766.18 5,812.16 5,861.82 5,921.12 5,981.82
-1.38 0.52 0.82 2.39 -1.06 -1.26
-1.39 -1.62 -1.83 0.75 -4.01 -3.13 1.54 1.15 1.oo
Tamp. = 60.03°K.
3.7087 4.5097 9.0293 12.5493 15.0393 20.2008
4,960.46 4,963.67 5,014.43 5,047.32 5,074. I:% 5 127.85
0.74 .68 .03 1.45 -0.07 -0.52
-
Temp. = 55.09"K. 0,98985 1.94156 3.6123 3.8748 5.7183 8,2170
4 , 5:30.97
1 545.12 4 ,556.07 4,557.2G 4,576,28 ~
12.1679
4 , :i95.2Y 1' , 6U 1 .0:3 4 , ti36. 77
1'2,320O9
4,636.14
8 9897
1.05823 3.6228 5.1015 5.8284 9.0043 11.0025
1609
VIRIALCOEFFICIENTS OF HELIUM
Temp. = 50.09'K. 4,122.46 4,142.44 4,154.56 4,161.26 4,188.89 4,206.57
0.80 -4.73 0.R1 0.70 -2.87 2.81 4.34
-
0.96827 1.93306 3.7090 4.9968 5.7705 5.8514 8.8762 12.0745 1.03425 1.99666 1.99968 3.7347 5.6241 6.8053 8.8665 11.6206 1.01995 1.97597 1.99698 3.8634 5.8795 6.0746 8.5398 12,3545 0.99736 1 ,87998 1 ,97717 3.7467 5.98'26 6.1212 9.1316 12.1595 0.95165
1 923'31 2.09017 3 . 7585 5 . 7035 6.34s:~ 8.25155 11.6O812 0.9G5.29 1 ,94976 1.99145 3.7522 5.0155 5.3707 6.7835 8.7033 9.7900
-1.31
0.78 -0.88 .29 .61 .09 .06
-
.04
1.07801 1.95987 3,7017 3.8445 5.0244 6.2025 6.6123 8.9292
Temp. = 45.10'K. 3,712.61 3,711.93 3,728.46 3,740.00 3,743.65 3,743 .OO 3,771.11 3,80 1.32 Temp. = 40.09"K. 3,298.42 3,304.99 3,306.42 3,315.15 3,330.52 3,340.15 3,355.73 3,377.62 Temp. = 35.10"K. 2,887.73 2,891 .75 2,891.87 2,902.91 2,913.55 2,914.94 2,930.24 2,953.25 Temp. = 33.00'K. 2,710.99 2,714.85 2,714.87 2,722.53 2,734.83 2,738.30 2,753.58 2,773.52 Temp. = 28.82'K. 2.366.95 2,368.04 2,370.28 2,374.32 2,382.06 2,387.95 2,396.72 2 , 4 21.62 Temp. = 24.65"K. 2,025.52 2,024.75 2,024.36 2.026.81 2,030.85 2,028.97 2,035.35 2,034.55 2,042.10 Temp. = 20.58"K. 1,685.47 1,684.53 1,682.60 1,678.34 1,682.05 1,675.96 1,683.53 1,680.71
-5.53 2.56 0.02 -1.11
1.61 2.93 0.57 -1.03 -0.52 -0.52 -1.93 1.60 0.16 -0.45 0.29 1.35 -0.78 0.31 0.30 -0.48 0.38 0.13
-0.46 0.50 0.03 0.01 0.43 1.09 0.41 -2.30 0.11 0.21 -0.45 1.38 -0.40 1.57 2.08 -0.78 0.37 -3.74 -2.99 -0.94 -0.49 0.20 -0.94 1.86 -0.45 6.97 3.72 -0.49 -1.24 -1.61 2.53 -1.87 4.18 -3.26 1.77
Vol. 64
I>. ~\;HITE, T. RUBIN,P. CAMKY AND H. L. JOHNSTOX
1610
12.0
8.C
0
This Research
V
Wiebe, Gaddy 81 Heins
x
Holborn & Schultre Holborn 8 Otto recomputed by Otto
B
Leiden Work
6.(
0.1
I
-4.1 0
60
I 120
I
I
I
180
240
300
T (OK.). Fig. 1.
where P is the pressure in atmospheres, V is the volume in cc./mole, T is the absolute temperature] and A ( T ) , B ( T ) , C ( T ) are, respectively, the first,
second and third virial coeEcients. The number of significant figures shown in this table are consistent with the precision previously
Nov., 1960
VIRI.4L C O E F F I C I E X T S O F
HELIUM
1611
given. The differences, P V c a l c d - PVobsd are the deviations obtained for each data point of the experimental values from the values calculated from relation 1 using the virial coefficient values given in Table 11. The second and third virial
A(T)/RT (Table 11) where R is the gas constant, 82.0567 cc. atm./mole deg., T is the absolute temperature of the isotherm measured by the pipet thermocouple in terms of the temperature scale previously mentioned. It is obvious that these data are in accord wit'h this scale to four parts in 10,000 TABLE I1 except at 20.58 and 50'K. where the agreement is VIRIALCOEFFICIENTS FOR GASEOUS HELIUM about 7 parts in 10,000. The second virial coefficient calculated in terms of relation 1 appears to A(T) B(T), C(T), T . OK. RT cc. cc. atm. -1 have random errors of about =k0.2-0.3 cc./mole at -0.000 1.oooog 11.99 299.99 high temperature and about *0.5 cc./mole at t'he -0.000 1,00032 12.08 273.16 lowest temperature. -0.000 1.00023 12.15 249.99 A comparison of the individual results for the -0.000 1.00041 12.23 200.11 second virial coefficients given by other authors +o. 000 1.00050 12.24 175.02 with the present work are shown in Fig. l.I1-l4 12.15 $0.000 150.04 1.00020 It is evident that the results of this research does +o.ooo 1.00015 12.18 125.03 not reconcile any differences among the earlier 11.85 0.ooog 100.02 i.00006 investigations. It is even questionable whether 11.60 0.0015 90.04 0.99963 these differences in B are real since these researches, 0.0040 .99982 11.01 80.02 in the main, were carried out in pressure ranges 10.70 0.0118 75.01 ,99990 where the effect of the higher virial terms could 10.30 0.0172 69.00 ,99951 influence t,he choice of the B. This is not the case 0.0219 .99991 9.55 60.03 in the present work where the pressure range was 8.96 0.0240 55.00 1.00053 so chosen as to minimize this effect. It is interest8.06 0.0400 50.09 1.00068 ing that the B's of this research compare favorably 7.48 0.0700 45.10 0.99972 with t,he earlier values only at temperatures where 0.0860 1.00040 6.57 40.09 the third virial coefficients are similar. The values 0,0540 1,00049 5.18 35.10 for B ( T ) ,of Wiebe, Gaddy and Heins agrees with 0.1230 0.99963 4.00 33.00 our results to within 0.3 cc./mole between 200 and ,99963 2.46 5.1880 28.82 273'K., where those authors found a value for 0.80 0.1710 24.65 ,99946 C ( T ) of about -0.009 compared to -0.000 from -2.62 0.2300 20.58 .99932 our dat'a. The data of Holborn and Otto and Holborn and Schultze yield B(T)'s0.3 cc./mole smaller coefficients were determined using a method first in value t'han those of this research, with fair agreedeveloped by Cragoe.l0 It is as follows: from a ment among the C ( T ) values above 170'K. Below set of points constituting an isotherm, one point is this temperature the values of both B ( T ) and C ( T ) chosen as a reference, designated by Po and Vo. of t,hese set's of data diverge. The Leiden data of Now since Nijhoff and Keesom near 70'K., agrees the best PoVo = A ( T ) B(T)Po C(T)Po2 with our results for both B ( T ) and C ( T ) . Comparison of Second Virial Coefficients Comon subtraction from (l),one obtains puted from Intermolecular Potentials with Experimental Values.-The determination of t,he parameters in the intermolecular potential function of helium has been attempted several times by correlations mdh experimental second virial coefficient It is obvious that the determination of B ( T ) and and or with transport properties of the gas. An C ( T ) do not depend upon the temperature scale, exhaustive review of this work will not be made here chosen, providing the set constitutes a true since that has already been done by Hirschfelder, isotherm. Because of the nature of equation 2 in et UZ.,'~ as well as by the authors to whom reference which differences are being used, it is only appli- is made below. cable to rather precise data. The most detailed calculations are tthose of KilThe ranges of experimental pressure were so patrick, et ~1.,'~-~8 who accounted for the quantum chosen, that in nearly all of the cases a straight line effects on the second virial coefficients using a numwas obtained on plotting the left-hand side of ber of different potential functions. Their first equation 2 us. (P/Po 1). Within the experi(11) W. H. Keesom, "Helium" Elsevier Publication, 1942. mental error, no higher virial than the third can be L. Holborn and H. Schultre, Ann. Physik, 141 47, 1089 (1915) justified whether relation 2 or a least squares and(12) L. Holborn and J. Otto, 2. Physik, 10, 367 (1922): 23, 77 (1924). method is employed.6 After evaluation of B ( T ) (13) C. W. Gibby. C. C. Tanner and I. Mason, Proe. R o y SOC.(Lonand C ( T ) , A ( T ) was calculated from relation 1. don), A122, 283 (1929). (14) R.Wiebe, 1'. L. Gaddy and C. Heins, Jr., J . am^. Chem. Soc., 13, All the virials calculated appear in Table 11, the 1721 (1931). graph of the second virial coefficients with tempera(15) J. Hirschfelder, C. F. Curtiss and R. B. Bird, "Molecular ture is Fig. 1. Theory of Gases and Liquids," John Wiley and Sone, New York, N. Y., The consistency of these coefficients with the Chapman and Hall (1954). J. E. Kilpatrick, W. E. Keller, E. F. Hammel and N. Metropotemperature scale is shown by the value of the ratio lis,(16) Phus. Rev., 94, 1103 (1954).
+
+
+
(10) C. S. Cragoe, "Temperature, Its Measurement and Control in Science and Industry," Reinhold Publ. Corp., New York, N. Y.,1941, P. 89.
(17) J. DeBoer and A. Michels, Physica, [VI] 1, 409 (1939). (18) J. E. Kilpatrick, W. E. Keller and E. F. Hammel, Phys. Rev., 97,9 (1955).
D. WHITE,T. RUBIN,P. CAMKY AND H. L. JOHNSTON 14 13
T'ol. G i
-
-
paper,lB contains calculations based on a 6 to 12 Lennard-Jones function using constants proposed by DeBoer and Michels." This work is summarized by curve 1 of Fig. 2. For temperatures above 60°K., the calculated results are lower than the experimental data of the present work. This is expected since the constants were chosen to fit the data of the Leiden workers and the results of Holborn and Otto.'' This potential function, although empirical! permits a qualitative estimation of the second virial coefficient values over an unusually wide temperature range. Kilpatrick, Keller and Hammel18 also computed the second virial coefficients using an exp-Six potential function. This is a form previously suggested by Slater and KirkBuckinghamZ0and by Mason and Rice.21 Using this form together with constants suggested by Mason and RicelZ1good agreement with the higher temperature results of Schneider and Duffie is achieved (curve 3 of Fig. 2) as well as the B's presented here. The agreement with the lowest temperature virial data however is very poor.6 To improve agreement with the low temperature data, Kilpatrick, et aZ.,'* selected a new set of constants (hilR5). The results are shown in Fig. 2 as curve 2. The agreement of experimental B's with the calculations from the (MR5) potential function is worse than the Lennard-Jones potential function at temperatures above 20". It is obvious from the discussion and the com(19) J. C. Slater and J. G. Kirkwood, rbtd., 32, 349 (1928); 37, 682 (1 93 1).
(20) H. S. W. Massey and R. A. Buckingham, Proc. Roy. SOC. (London),Af68, 378 (1936). (21) E. A. Mason and W. E. Rice, J . Chem. P h y s , 2 2 , 522 (1956).
parisons given in Fig. 2 that neither the form nor the choice of constants for the potential function of helium made so far, have given satisfactory quantitative agreement with all of the experimental virial coefficients. It is questionable whether the form or choice of constants in the potential function can ever be resolved from virial data alone. There is considerable merit, however, in providing an empirical function so that a compact form may be used to express indirectly the second virial coefficients. We suggest the use of Yntema and Schneider's function for this purpose, it is E(Y) 6(Y)
= = 0
< 1 A. > 12 A.
Y
QJ
Y
(4)
ergs, 1 A.
< < 12 A.
where Y is the interatomic distance between two helium atoms in angstroms. This set of expressions substituted in the classical formula where v has the meaning of relation 4, N is Avogadro's number, k is Boltzmann's constant (the other terms have their usual meaning), is simple to use. The agreement with the present results, shown in Fig. 2 as curve 4,is good. Acknowledgments.-The authors wish to express their thanks to Mr. H. Altman for assistance in some of the measurements, to Mr. L. E. Cox for some of the modifications of the apparatus and to Dr. A. Friedman for cdibration of the pipet volume,
