ACKNOWLEDGMENT
(4) B. L. Karger and L. V. Berry, Anal. Chem., 44, 93-99 (1972). (5) D. H. Freeman and W. L. Zielinski, Natl. Bur. Stand. (U.S.), Tech. Note, 589, 3-18 (1971). (6) R . A. Mowery, Jr., and R. S. Juvet, Jr., J. Chromatogr. Sci., 12. 687-695 (1974).
Conversations with G. R. Beecher and R. F. Doherty, the skillful1 technical assistance of D. J. Higgs, and the art work of Paul Padavano are gratefully acknowledged. LITERATURE CITED (1) D. H. Spackman, W. H. Stein, and S.Moore, Anal. Chem., 30, 1190-1206 ( 1958). (2) L. T. Skeggs, Am. J. Clin. Parhol., 2 8 , 311-322 (1957). (3) L. Berry and B. L. Karger, Anal. Chem., 45, 819A-827A (1973).
RECEIVED for review April 18, 1977. Accepted July 6, 1977. Mention of a trademark of proprietary product does not constitute a guarantee or warranty of the product by the U.S. Department of Agriculture, and does not imply its approval to the exclusion of other products that may also be suitable.
Air Buoyancy Equations for Single-Pan Balances Michael R. Winward,' Earl M. Woolley," and Eliot A. Butler" Department of Chemistty, Brigham Young University, Provo, Utah 84602
Most derivations of the correction for air buoyancy are made considering a two-pan, equal-arm balance that is restored to its zero point of rest. The exact equation for such a balance is
5)(1 F) -1
Mo = Mwts (1 -
-
where Mo is the true mass of the object that is compared to standard weights of mass MMs. The density of air is represented by clair,and the densities of the object and weights by doand d, respectively. Substituting the series expansion of the term [l - (dair/do)]-linto Equation 1 yields
M, = Mwts(1-
")
dWtS /?I \ 2
[1+ daix + d0
1
apply to both types of balances. However, their derivation is based on the implicit assumption that the balance is restored to the null position when a weighing is made. We present a more rigorous derivation to show that Equation 1 is not exact for a single-pan balance except when the beam is a t the null position and we determine the errors that are involved in using that equation. DERIVATION OF EQUATIONS Figure 1 shows a single-pan balance that has had the pan, weights, and counterpoise removed so that the beam is at equilibrium at the null point. In this figure, a and b represent the centers of mass of the two sides of the beam, cy' is the angle to the center of mass of the left side of the beam from the horizontal plane that passes through the central knife edge, and p' is the corresponding angle to the center of mass of the right side. L, and Lb are the horizontal distances between the central knife edge and the respective centers of mass. Since the sum of the torques about the central knife edge must be zero, we may write
Neglecting all terms that are higher than first order in d h gives
(3) The neglecting of the higher order terms involves an error of approximately 0.0001% if do = 1. For do > 1, the error is smaller. Either Equation 1 or, more commonly, Equation 3 is given in quantitative analysis texts. If in a weighing operation a balance is not restored to its zero point of rest, part of the object's weight is calculated from the deflection of the beam and the sensitivity of the balance. That part of the mass is not subject to the unequal buoyant forces as is the part obtained from the weights on the pan plus the rider. On the equal-arm analytical balance, a maximum of about 3 mg can be determined from the deflection of the beam. Hence, only 3 mg or less of the mass of the object is not subject to unequal buoyant forces. However, on the single-pan balance, the deflection of the beam is used for determining as much as 1 g of the object's weight. Thus, on the single-pan balance a much more significant portion of the object's mass may not be subject to the inequality of buoyant forces., Burg and Veith (1) have pointed out that it is not obvious that the same equation for buoyancy corrections applies to both types of balances. They derived an equation for the single-pan balance and concluded that the same equations did Present address, Union Oil Research Center, Brea, Calif. 2126
ANALYTICAL CHEMISTRY, VOL. 49, NO. 13, NOVEMBER 1977
where M is the mass of, and F, the buoyant force on each respective side of the beam. The acceleration of gravity is represented by g. The term (Mg - F) represents the net downward force acting on the respective sides. (Table I gives a listing of the subscripts used to identify the various masses and forces.) Figure 2 shows the balance with the pan of mass Mp,the weights of mass M I , and the counterpoise, whose mass is M,, replaced on the beam such that the beam is still a t the null position. Let cy be the angle from the horizontal plane to the terminal knife edge and L1 the horizontal distance between the two knife edges. Let @ be the angle from the horizontal plane to the center of mass of the counterpoise and L, be the horizontal distance from this point to the central knife edge. The right hand and left hand torques are equated to give
Figure 3 shows the balance with an object of mass M o placed on the pan and an approximately equivalent mass of weights removed so that the beam is at equilibrium at a position other
--.-
Figure 1. Balance beam at null position, Point a Is the center of mass of left (forward) side of beam; Point b is the center of mass of right (rear) side of beam
Table I. Subscripts Used to Identify Masses and Forces left (forward) side of beam a right (rear) side of beam b 1 weights (all removable weights on pan side of balance) P pan c counterpoise o object 2 removable weights that remain on pan side of balance when object is approximately counterbalanced
Substituting into Equation 8 the following identity II
+ p ‘ )- C O S @ - a ’ ) = - sin e cos a’ cos a’
I
COS@
i
(tan a’
+
Flgure 2. Balance at null position with pan, weights, and counterpoise restored. Point c is the center of mass of the counterDoise
Since the buoyant force F may be expressed in the following manner
Mk? F = da, -
d where d represents the density and M represents the true mass, Equation 10 may be rewritten to give
2)+
[
M1 (l -
M,,(I. -
COS CY COS COS
p
COS
(e - p ) (e + a )
Flgure 3. Balance with object on pan and appropriate welghts removed: beam at equilibrium but not at the null position
than the null position. Let M 2 equal the mass of the weights that are left on the balance and 6’ be the angle of deflection. The equation describing this situation is
(tan a‘
+ tanp’)
(12)
Substituting into Equation 12 the identity C O S & COS
COS p C O S
(e - p ) tan 01 + tan 0 ( e + 0 ) = 1 + cot e - tan CY
and letting dl = dz = d,
(13)
results in the following equation:
(7) Substituting Equations 4 and 6 into Equation 7 gives ANALYTICAL CHEMISTRY, VOL. 49, NO. 13, NOVEMBER 1977
2127
Table 11. Comparison of Buoyancy Correction Equations An a1y tical MOPt
Mdial
M, Equation 19
Mo Equation 1 M,, Equation 3
(tan a
=
Micro
1.0000 0.0000
1.0000 159.0000
0.10000
0.00000
0.10000 159.90000
0.0 10000 0.000000
1.0012 1.0010 1.0010
160.1674 160.1673 160.1671
0.10012 0.10010 0.10010
160.16728 160.16727 160.16706
0.010012 0.010010
g/mL, and d&
=
Data calculated for do = 1.0 g/mL, dah.= 1.2 x
A
Semi-micro
-
+ tan p ) (MI + MP)- (tan a‘ +
La tan p ’ ) Ma -
0.010010
0.010000
19.990000 20.020910 20.020908 20.020883
7.7 g/mL.
1, one commonly uses the observed weight for the value of
Ma as in the equation (15)
LI
Substituting this expression into Equation 1 and subtracting Equation 19 from the result yields
and
‘air error = -Mop, -
aa‘ (tan a’ + tan p ’ ) M daL1 Since we are considering a constant-load balance, the bending of the beam should be invariable and A and B are both constant. The reading on the dial of the balance is equal to the total mass of the weights that have been lifted from the beam.
The optical portion of the balance reading, Mopt,is determined by the deflection of the beam from the null point. The second term of the right-hand side of Equation 14 is a function of the angle of deflection 0. Therefore, let
That the optical reading is also a function of the density of air indicates that the sensitivity of the balance will vary with changes in air density. Also the fact that A and B are functions of C Y ,CY’,0,and 0‘reflects the well-known fact that the sensitivity of the balance depends upon the location of the center of mass of the system. Equation 14 may now be rewritten using the above definitions in the following form
Equation 19 represents a more precise equation for applying air buoyancy corrections to weighings made on a single-pan, constant-load substitution balance than has been previously reported. DISCUSSION The error involved in using Equation 1 for weighings that are made on a single-pan balance is equal to the difference between Equation 1 and Equation 19. In applying Equation
This means that the M , calculated using Equation 1 will be less than the true M,, calculated from Equation 19 by an amount equal to approximately 0.00015 M,,,,. An indication of the magnitude of the error that is introduced by using Equation 1 or Equation 3 instead of Equation 19 is given in Table 11. Here the results that are obtained from these three equations for the same apparent masses are shown. Data are given for apparent masses that are equal to the full range of the optical scale and also for the maximum load of an analytical balance, a semi-micro balance, and a microbalance. These data indicate the maximum error that can be introduced by using Equation 1since, in all cases shown, Moptis assigned to be the maximum value that is possible for each respective balance. As shown by Equation 21, this error is directly proportional to the optical scale reading. The results for Equation 3 approach those for Equation I for objects of small mass or for objects having densities near that of the weights. The data in Table I1 are calculated for do = 1. Whether or not it is acceptable to use Equation 1 instead of Equation 19 will depend upon the allowable experimental error and the value of the optical reading of the weighing in question. In addition, one must consider the density of the object and magnitude of its mass in deciding whether the convenience of using Equation 3 is justifiable. Finally, it should be noted that many of the newer balances employ stainless steel weights that have been adjusted to have the same apparent mass as a brass weight at a particular air density. When using these balances a t other air densities, appropriate corrections, which are on the order of 1 part in IO5, must be made (2). LITERATURE CITED (1) W. R. Burg and D.A . Veith, J . Cbern. Educ., 47, 192 (1970). (2) J. E. Lewis and L. A. Woolf, J . Cbern. Educ., 48. 639 (1971).
RECEIVED for review July 7, 1977. Accepted August 9, 1977.
CORRECTION Factors Affecting Quantitative Determinations by X-ray Photoelectron Spectroscopy In this article by C . D. Wagner, Anal. Chem., 49, 1282 (1977), the captions for Figures 1 and 2 are reversed. 2128
(21)
dwts
ANALYTICAL CHEMISTRY, VOL. 49, NO. 13, NOVEMBER 1977
