Thanks for taking the time to respond Kupa. Exhanges like these are always a good thing and I genuinely want to say thank you. You bring up good points and I have a feeing in the end that neither of us will budge on our view, because we are locked into seeing the ambiguities in the question in our own way. There is definitely ambiguity in this question, and I would ventrure to guess that if someone were to take the time to analyze each one, you'd find that it would all come down to the type of rubber the balloon was made out of.
And the truth of the matter is that in the last few years that I've tutored, no one has called out the flaws in this question so eloquently. I sincerely want to thank you and confess that after our exchange, I'll strongly suggest to my bosses that they add a few more terms into this question to make it work more clearly. To be perfectly honest, I had issue with the fact that water doesn't expand from 0 to 4 degreesC (it contracts), but I figured they'd shoot it down given that the purpose of their question was to emphasize POE and speed rather than physical science concepts.
I want to adjust for a few factors that were ignored in your Charles' law approach to the question, which will once again introduce ambiguity to the point where an ideal version of that law is not enough. I'm going to apply a middle value to my factors, which will hopefully give us a centered value to argue from.
For simplicity, I've chosen 27 degreesC and 57 degreesC, so the temperature change is 10%. This temperature change would result in a 1.8% change in the water density, and thus a rise in the water height of 1.8%. I'm going to consider a balloon exactly in the middle of the system, which will feel a 0.9% increase in the mass of water above it (assuming that if the top of the water increases by 1.8% and the bottom stays at the same point, then the middle goes up by 0.9%). I'm going to modify this number later to account for the rise in the water level caused by the balloon expansion.
Your assumption is that V
1/T
1 = V
2/T
2 applies for this system. In an ideal piston that exhibits no resistance to expansion or contraction, filled with an ideal gas, will follow that law. However, in a balloon that exhibits a restoring force when deformed, that law no longer applies. To what extend it deviates depends on the thickness of the balloon and the material of which it's made. As I'm sure you have personally experienced, it gets harder and harder to blow up a balloon as you continue to fill it. To look at the impact of the rubber and material thickness, consider blowing up a basketball or bike tire. They fill easily at first, but get much, much harder to expand as you pump more and more air inside, eventually reaching a point where it just won't expand anymore. In this question we are not told the degree to which the restoring force of the balloon increases, but at the surface of the water the restoring force will be its greatest (as the balloon is closest to maximum volume). The point is that your assumption of a 25% change in volume is for an ideal system and is over estimated in this case (to what extent would depend on the material and gas). Again, though, without those details in the question, we can't find an exact answer. If the balloon didn't expand, then it have a delta V of 0%. If it expands to the maximum (with a heating from 27 to 57 in degreesC), it would be a deltaV of 10%. It obviously expands because of the passage, so a 5% expansion for an ideal gas is a middle ground number we can use.
Second, the expansion of the submerged balloon you speak of will cause an additional increase in the water level (again we don't know to what extent), so the balloons end up more submerged not only because of the reduction in water density, but also because of the submerged balloons expanding and pushing the water up. There is no way to know the exact amount without knowing the dimensions of the tank and volumes of the balloon, but it's a factor to some extent. We'll make it a small one.
These two factors are significant enough to support a realistic scenario where the balloon becomes less buoyant. If we consider the 27 to 57 change I proposed, then the following are realistic numbers: 10% max expansion with no restoring force issue with the balloon will be reduced based on the material's natural restoring constant and the magnitude to which is extends. Let's use F = kx to determine our restoring term. If the volume were to increase by 10%, then x would increase by 2.1% in each direction. Now this again is ideal and we don't know k, but let's just say it reduces the expansion to 1.1% in all directions (about half of the maximum). 1.011^
3 = 1.033. So that gives us an expansion of the balloon of roughly 3.3%. If the gas behaves ideally, we'll get 3.3%, and if it doesn't it would be more in the area of 3.0%. Again, this is an estimate that can't be amde without knowing something about the gas. Deviations from ideality can be as large as 20% and as small as 0.000001%, so I went with a 10% decrease. This gives us a 3.0% increase in volume that could be larger or smaller, depending on the exact system.
So let's say the water height increases by the 1.8% due to the change in water density and for simplicity add 0.2% for the impact of the expanding balloons. This gives us a 2% increase in the water level, which will amount to a 1% increase in external pressure on the average balloon (in the middle), and thus a drop of 1% in the volume (using 1.0 for the density of water to make the gauge pressure calculation easier).
So now we can use Archimedes' Principle to see how the buoyancy changed. The volume of the balloon goes up by about 2.0% (again, this assumes an average restoring term, a 10% maximum increase in volume due to temp when we ignore other factors, a 1% decrease in volume due to pressure when we ignore other factors, and an average deviation from ideal gas behavior).
B = rhowater x Vdisplaced x g
rho goes down by 1.8% and Vdisplaced goes up by 2%.
I think I have been fair in taking the middle value for every assumpiton, and in the end the two factors are so close that no definite conclusion can be drawn. I am certainly not disputing that there is ambiguity in this question, and neither is BR if you read through their explanation. You have convinced me that they need to put a restoring term for the balloon and the exact temperature change in their passage or question.
Hopefully you are convinced that there is enough ambiguity that it could go either way with choice B.
So let's just say this was an exact MCAT question on your test. Having completely over-analyzed this question as much as we have, what do you think the best answer is? Not a correct answer, because I think we can all agree (including BR if you read their explanation) that there is no correct answer. But what is the best answer?
