Correction of Weighings for the Buoyancy of the Air

The object of weighing a body is to determine its mass, and the physical law upon which the measurement depends is that the weights of bodies are proportional to their masses, if they are sufficiently near together.

Now we have all along assumed that when an adjusted balance-beam was in equilibrium, the force of gravity upon the weights was equal to the force of gravity upon the body weighed, i.e. that their weights were equal, and this would have been so if we had only to deal with the force of gravity upon these bodies. But the bodies in question were surrounded by air, and there was accordingly a force upon each acting vertically upwards, due to the buoyancy of the air; and it is the resultant force upon the weights which is equal to the resultant force upon the body weighed. But the forces being vertical in each case, their resultant is equal to their difference; and the force due to the displacement of air by the body is equal to the weight of the air displaced, Le. it bears the same ratio to the weight of the body as the specific gravity of air does to the specific gravity of the body; while the same holds for the weights.

Thus, if w be the weight of the body, σ its specific gravity, and λ the specific gravity of air at the pressure and temperature of the balance-case, the volume of air displaced is w/σ and its weight wλ/σ (p. 105). Hence the resultant force on the body is w(1λ/σ); similarly, if ω be the weights, and ρ their density, the force on the weights is ω(1-λ/ρ). These two are equal, thus

since in general λ/σ is very small.

The magnitude of the correction for weighing in air depends therefore upon the specific gravities of the weights, the body weighed, and the density of the air at the time of weighing, denoted by ρ, σ, and λ respectively. The values of ρ and σ may be taken from the tables of specific gravities (tables, 17, 80) if the materials of which the bodies are composed are known. If they are not known, we must determine approximately the specific gravity. We may as a rule neglect the effect of the buoyancy of the air upon the platinum and aluminium weights, and write for ρ, 8.4, the specific gravity of brass, the larger weights being made of brass. The value of λ depends upon the pressure and temperature of the air, and upon the amount of moisture which it contains, but as the whole correction is small, we may take the specific gravity of air at 15° C. and 760 mm., when half-saturated with moisture, as a sufficiently accurate value of λ. This would give λ=0.0012.

Cases may, however, arise in which the variation of the density of the air cannot be neglected. We will give one instance. Suppose that we are determining the weight of a small quantity of mercury, say 3 grammes, in a glass vessel of considerable magnitude, weighing, say, 100 grammes. Suppose that we weigh the empty vessel when the air is at 10° C. and 760 mm., and that we weigh it with the mercury in at 15° C. and 720 mm. deducing the weight of the mercury by subtracting the former weight from the latter. We may neglect the effect of the air upon the weight of the mercury itself, but we can easily see that the correction for weighing the glass in air has changed in the interval between the weighings from 22 mgm. to 20.5 mgm. The difference between these, 1.5 mgm., will appear as an error in the calculated weight of the mercury, if we neglect the variation in density of the air, and this error is too considerable a fraction of the weight of the mercury to be thus neglected.

Experiment.

Determine the weight in vacuo of the given piece of platinum.

Enter results thus:

Weight in air at 15°C. and 760 mm. with brass weights 37.634 gm. Specific gravity of platinum 21.5. Weight in vacuo, 37.632.