J. E. Lewis' and 1. A. Woolf Research schoolof Physical Sciences Australian National University Canberra, Australia
Air Buoyancy Corrections for Shgle-Pan Balances
I n a recent article Burg and Veith2 demonstrated the equivalence of the expressions used to make air buoyancy corrections to weighings made with either double- or single-pan balances. Care should, however, be exercised in using these equations when correcting weighings made with a one-pan balance which uses the method of substitution of weights. The most commonly used substitution balances are those of the Mettler Corporation, and these employ weights of stainless steel, actual density 7.76 g/cma (at O°C), which have been adjusted in mass to be equivalent to the same apparent mass as a brass weight, density 8.4 g/cma (at O°C), at a fixed air density of 0.0012 g/cm3. Thus a stainless steel weight of such a balance with a nominal mass of, say, 10 g will exactly counter-balance a hrass mass of exactly 10 g a t only that air density. At other air densities a correction factor must be included in the final equations of Burg and Veith to make them rigorously applicable for weighings made with balances of this type. An expression for this correction is derived below. The analysis of Burg and Veith may be used through their next to last equation which we write here as
where M. is the mass of the object being weighed; MI, the combined mass of the pan and removable weights; M,, the combined mass of the pan and the weights remaining after equilibrium has been attained with the object on the pan, do is the density of the object, d, that of the weights, and dAthat of air. Equation (I) may be rearranged to give
Expansion of the term (1 - dA/d,)-I in eqn. (2) and simplification of the resulting expression by neglecting terms in dAof higher order than the first yields
This is similar to the final equation of Burg and Veith who, however, make the additional assumption that the apparent mass of the object being weighed, Ma, is equal to MI - M2. Equation (3) is then rewritten as
(Equation (4) is identical with the last equation of Burg and Veith.) If M. (i.e., MI - M2) is taken as the reading indicated by the balance dials then eqn. (4) is correct only at one air density if the weights of the balance have been adjusted to have the correct nominal (or indicated) mass at that density. Consider a mass, MB,of brass, dB = 8.4 g/cm3 a t an air density of 0.0012 g/cmS. This will have an effective mass of 'MB where
A stainless steel mass of M , g, density d, = 7.76 g/cm,$ of the same effectivemass as the brass will have a different real mass, M,, from Ma hut since .M, = .Ms therefore
hence M. = 1.0000118M~
(6)
I n eqn. (3) the difference MI - Mz represents balance weights of stainless steel which have been adjusted for specific air density as described above. Therefore to be able to use the balance a t other air densities and to apply appropriate buoyancy corrections, eqn. (4) must he rewritten as
It is clear that the correction factor when weighing masses of the order of 100 g is of order 1 mg which is well in excess of the error in the balance weights. A further points are of interest in the context of buoyancy corrections. First, the above equations have involved the customary linear tmncation of the series expansion of the term [l - (dA/dJ-'. For objects of density of order 1 g/cma or less this approximation causes an error of close to 1 in lo6 and should it be unacceptable the exact equation should be used
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Present address: Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada. 'BURG, W. R.,and VEITA,D. A., J. CHEY.EDUC.,47, 192 (1970). I
Volume 48, Number 9, September 1971
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