I make an effort to say nice things about pop-science books that I read, whether for book research or blog reviews. Every now and then, though, I hit a book that has enough problems that I have a hard time taking anything positive from it.
I got David Bodanis’s E=mc2: A Biography of the World’s Most Famous Equation from Union’s library because I like the subtitle, and plan to reference it in the relevant chapter of the book-in-progress. I figured that, if I’m going to swipe his subtitle, I should at least be able to say something substantive about the book.
Bodanis takes pains to say that this isn’t a book about the science of Einstein’s equation, but rather a “biography” of it, basically a collection of interesting historical anecdotes about the equivalence of mass and energy. The problem is, on the occasions when he does talk about the science, it’s always a bit off, by enough that I’m not sure I trust his historical anecdotes, either.
In an effort to find something good to say, though, I’ll say that he at least managed to stimulate some physics-y thought with one particular paragraph. Toward the end of the section about the sinking of the Lake Tinnsjo ferry to prevent a shipment of heavy water from reaching the Nazi atomic bomb program, he writes:
A few of the barrels that had been only slightly purified bobbed on the top of the lake, and the passengers who’d managed to get off but hadn’t made it onto lifeboats… grabbed on till a rescue boat came. But the barrels that contained the concentrated heavy water demonstrated, in slow-motion free fall, what they contained. Since the H2O molecules are composed of a nucleus heavier than ordinary water, the barrels sank as if weighted, swirling around the ferry and its innocent trapped passengers down to the bottom.
You can see what I mean by the science being a little off– he uses “H2O” as if it was something different than “ordinary water” (usually, heavy water is written “D2O,” from the name “deuterium” given to heavy hydrogen), he talks about a water molecule as if it were a thing with a single nucleus (which might arguably make sense at a coarse enough approximation, but just sounds wrong), and most of all, the way he talks about the sinking of the barrels makes it sound like heavy water is double the mass or ordinary water, when in fact it’s only about 10% heavier (most of the mass of a water molecule comes from the single oxygen, with a mass of 16 atomic units; hydrogen adds 2 for a total mass of 18, deuterium adds 4 for a total mass of 20, so an increase of 1/9 the original mass).
But is the picture described plausible? That is, would it be reasonable to think that a barrel full of heavy water would sink, where a barrel full of ordinary water would float?
My first reaction was “That’s ridiculous. They sank because the water was in barrels. It’s the mass of the container, not the mass of the contents that makes the difference.”
The key factor in determining whether an object sinks or floats is whether the mass of that object is greater than the mass of an equal volume of water (for the details, look up buoyancy) as has been known since the time of Archimedes. This is why ships float when all goes well– the air inside the ship has basically no mass compared to water, so while the hull materials are all heavier than water, the total mass of the ship is less than the mass of an equivalent volume of water– but sink when they hit an iceberg and fill up with water– once there’s enough water in the hull to make up for the mass difference, the whole thing goes straight to the bottom. You can demonstrate this for yourself with a cake pan and a sink full of water– an empty pan will float, but if you start pouring water in from a glass, the pan will sink below the surface, long before you have filled it from the glass.
So, if you have a barrel that is entirely full of water, it will sink, whether it’s heavy water or ordinary water. The mass of the water inside the barrel is obviously equal to the mass of the same volume of water outside the barrel, while the mass of the metal making up the barrel itself is far greater than the mass of the same volume of water– around 8 times greater, for most useful metals.
But is it possible that you could have a situation where a barrel containing ordinary water would float, while a barrel of heavy water would sink? That is, what would you need to do for that 10% difference in mass to make the difference between sinking and floating?
Well, if you want a barrel of water to float, you need to not fill it up all the way. How much space do you need to leave? To know that, you need to know the volume, and the weight of the barrel. If we assume the barrel to be the canonical 55 gallon drum, the volume is around 200 liters, and the mass of an equivalent volume of water is about 200 kg. So, what’s the mass of a 55 gallon drum? A little Googling turns up this hilarious page which boldly states a value for the weight, and then gives a bunch of examples, none of which are anywhere near as large as the first number. People selling steel drums claim a mass of about 50lbs, while the lightest mass claimed at WikiAnswers is around 20 lbs, so let’s say between 10-20 kg total mass.
If you want a steel barrel of water to float, then, you need to leave out at least enough water to add up to the mass of the metal. At the high end of barrel masses, that’s about 10% of the mass of the full volume of water, so you would only fill the barrel to 90% of its total capacity. The same volume of heavy water would be about 10% heavier, so if you got the percentages exactly right, there probably is a point where a barrel of ordinary water would float while a barrel of pure heavy water would sink (these are very rough numbers, but I’m only after a back-of-the-envelope estimate of the plausibility).
Is that a reasonable fill level? My first inclination would be to say no, at least not for a resource that was sufficiently valuable to the Nazis for the allies to sink a ferry with innocent passengers on board to stop it from reaching Germany. You would think that they would fill those barrels right to the top.
However, if you look at that link all the way back up at the top, you’ll see that the ferry sinking took place in February. In Norway. Given that water expands when it freezes, you would need to leave a little extra space inside the barrels so that the water would have room to expand, should it freeze along the way. How much space? Well, an iceberg is famously 9/10th below water, which tells you that the mass of a given volume of ice is about 90% of the mass of the same volume of water. So, the expansion during freezing must be about 10% of the volume, which means you would want your barrels to be at most 90% full.
So is it plausible that barrels of heavy water might’ve sunk, while barrels that contained ordinary water floated? Yes, barely. It would depend on the details of the barrels– their exact dimensions and mass– and how they were filled and shipped. With slightly lighter barrels and a 90% fill, the 10% mass difference between heavy water and ordianry water might make the difference between sinking and floating. You’d need to do the math a little more carefully than I did above, but there’s a reasonable chance that it might work.
I suspect, though, that the fill fraction is still a bigger factor than the mass difference. That is, the barrels that floated on the surface were probably not filled up to the same level as the ones that sank. (We’ll also ignore the fact that most of them were probably tied onto something so they didn’t roll around loose, and thus they went down as one unit with the ferry…) While it’s plausible that the mass might’ve made the difference, the numbers are close enough that it would seem kind of a big coincidence for that to be the determining factor. Particularly since they were supposedly incompletely purified, not pure light water, which would imply some increase in mass, of not the full 10%.
But, hey, on the bright side, I passed a few amusing hours thinking about the physics of buoyancy. And time spent thinking about physics is never completely wasted…