density of interior liquid - density of bath liquid
--------------------------------------------------- = volume fraction of air
density of interior liquid
This formula is an extrapolation of the idea that the density of an object is a combination, by volume proportions, of the densities of its components. The mathematical derivation goes as follows (next page):
Approximations and assumptions made in deriving this formula:
1. The density of the entire antibubble is the simple relationship indicated in equation 1, with negligible interaction between air and liquid.
2. The density of the air is much smaller than those of the liquids, such that it is essentially zero for the derivation of the final two equations.
3. The pressure of the surrounding liquid acting on the volume of the antibubble is negligible near the surface of the bath liquid.
In theory, equilibrium would be considered achieved when the antibubble neither rose or sank. However, in practice, antibubbles rarely stay stationary in the bath. For every antibubble that formed, it was recorded on which drop it was formed and whether it rose, sank, or sank and then rose. Most of the time, antibubbles did not form on the first drop. Additional drops were added to each test tube until an antibubble formed. This was repeated f or each solution of liquid B. Concentrations of salt in liquid B ranged from 0.15M to 0.50M NaCl, by 0.05M increments. The highest and the lowest B densities that would form an antibubble that sank and/or rose were used to calculate the range of plausib le volume fractions of air in the antibubble using Equation 3.
The results of this procedure were
tested against those from an alternate method, proposed by Connett
(1974). This method required the formation of an antibubble (by
dropping liquid A into liquid A) in an overflowing flat-bottomed
container. Once the an tibubble formed, the container would be capped
without trapping any other air and inverted so that the antibubble rises
to the flat surface so that the diameter could be measured. This proved
very difficult. Thus, instead of capping and inverting, the vessel
remained upright and the diameters of both the antibubble and the
resulting regular air bubbles (when the antibubble popped) were measured
simply and roughly by holding a ruler over the open surface. The volume
fraction of air in the antibubble wa s then found by dividing the volume
of the air bubble by that of the entire antibubble.
Figure 4: The purple bar in this graph shows the range of volume fractions of air. This was made by including the B solutions in the purple zone of the graph above.
Connetts method was then tested to see if these results could be replicated. Connetts method proved very difficult, however, and the results were very inaccurate. The diameter of the antibubble could only be determined to be about 4mm, and t he resulting air bubble was less than 1mm, greater than .7cm. Despite these imprecisions, the results generated by further calculations fall mostly within the predicted range. Generally, the antibubbles formed were brief and they all hung at the top. F or this reason, only the minimum side of the spectrum can be shown with these results.
Connetts method, though imprecise, fall mostly into the r
ange predicted (Figure 5).
Figure 5 shows the air volume fraction results from the first method as blue dots and from Connetts method as the connected red dots. The results from Connetts method mostly fall within the range predicted by the first method (sh own with a purple bar)
The results of the experimental method agreed with those obtained from the alternate (Connetts) method. The range of air volume fraction values for antibubbles containing a 0.02-0.03 mL drop was found to be from 0.009 to about 0.017, or 0.9 - 1.7% of the entire antibubble volume.
First, I would like to thank my father for his incredible moral support. I would also like to acknowledge Chris Nadovich, the creator of the Antibubble website which inspired this project.
|B solution||test tube #||Description|
|O.15||8||hung at top, long-lasting|
|20||hung at top, brief|
|21||hung at top, brief|
|0.2O||9||hung 1cm from top, brief|
|17||slowly up to hang on top, long-lasting|
|O.25||9||hung on top, brief|
|15||too brief to tell|
|0.3O||5||sank, popped on touching bottom|
|O.35||5||too brief to tell|
|19||slowly sank 1.5cm|
|21||very slowly sank 2.5cm, pause, rose 1.3cm|
|0.4O||1||too brief to tell|
|8||sank, very brief|
|12||hung at top, brief|
|O.45||17||settled on bottom briefly|
|B solution||B density||Volume Fraction|
|Results (Estimates) from Connett's Method|
|Antibubble||bubble||volume fraction of air|