By Bobby Neal Winters
Every glass of water you drink contains a few molecules from Caesar’s urine.
There, I have shared my burden with you. I first heard this myself more than 20 years ago. A mathematician was talking in the faculty lounge. He tossed it off, rocked my world, and then the conversation rolled along into other esoteric matters leaving me with this horrific bit of knowledge: I am drinking Caesar’s urine. Between the period on the end of the last sentence and the “B” at the beginning of this one, I took a sip of coffee; more water, more of Caesar’s urine.
For years I was able to blot this out of my mind, but then it came back at me in the form of a question: how much? How much of this rare liquid am I getting in every glass of water? I am a mathematician, so I can do the calculations. I did them. The answer is 2100 molecules in every eight-ounce glass of aqua not-so-pura. The point of this essay is to explain how I arrived at that number and explore some of the territory along the way.
Making this calculation requires an assumption, so I will begin by explaining my assumption with the promise that later in essay I will return to justify it. My assumption is this: Caesar’s urine is uniformly scattered throughout all of the water in the world. That is a big one, but it is absolutely necessary to the calculation. The calculation requires a simple formula—you knew it was coming—that I will give below:
Here C is the amount of Caesar’s urine in a glass of water, f is the fraction of the world’s water that is Caesar’s urine, and G is the amount of water in a glass.
The value for f is one that we will have to calculate. Here, I am going to limit the scope of my study. I am only interested in the urine that Caesar excreted upon his assassination. Calculating the amount of urine Caesar produced over his lifetime would introduce too many assumptions for my comfort-level which is going to be sorely tested in any case. Consequently, I will focus on that one last bladder-full. Here I am also having to make the assumption that Caesar had an average sized bladder and that bladder was full.
In any case, the average bladder has the capacity of 350 milliliters. This is almost exactly the same amount of fluid in a can of Mountain Dew. Coincidence? I think not!
I am not too worried about this figure as it is something that I can verify for myself given soft drinks, time, and a calibrated beaker that I am not using for anything else. Much more of concern is the next number I need, which is the amount of water available on the whole earth. This is a figure I had to look up: 1.36x10^9 km-cubed. That is to say about one billion cubic kilometers. This converts to 1.36x 10^21 liters.
Doing the math—dividing the amount of urine in Caesar’s bladder by the amount of water in the whole world—this gives us a figure of f=2.58x10^-22.
I have been using scientific notation here, which you might not deal with every day. The figure for f is incredibly small. Written out as usual, it would be
I meant to have 21 zeros between the point and the 2. You can count to see if I got it right. In any case, you will agree that this number is small, so one might wonder at this point whether this whole drinking Caesar’s urine thing is plausible at all. In order to justify this, I have to make a side trip to the world of the atom.
Avogad Row: The Place Where Chemists on the Skids Wind up
At this point, I get into an area where I don’t feel very comfortable: chemistry. I never had a chemistry course in college. I took one as a high school senior of which I only remember two things. I learned the metric system for the fourth or fifth time and we put some aluminum in some nitric acid.
Putting aluminum in nitric acid was really, really cool. I spilled some of the nitric acid on my fingers and it turned them yellow and when we put the aluminum in the acid it got all hot and it looked like it might explode. Learning the metric system for the fourth or fifth time was somewhat anti-climactic to this. And come on, we get all of this hype of how the metric system is so much simpler than the so-called British system, but this why does it have to be drilled-in over and over and over. When I went to college, I couldn’t stand that I might have to learn the metric system yet again, so I opted out of chemistry. I took physics instead where the first thing they taught me was…the metric system.
I paid for this deficit in my education a few years back when I was made Acting Chair of the Department of Chemistry. Among the things I learned there was that chemists are much more interested in blowing things up than they are in the metric system, but to blow things up in a predictable orderly way they must have an exhaustive knowledge of the metric system. That is neither here nor there.
I need chemistry because I want to know how many molecules of water there are in a liter. Chemists are interested in numbers of molecules because when they mix stuff together the proportions they mix are important. The reactions go on at an atom-to-atom level.
Chemists use a special unit to measure substances used in chemical reactions. That unit is called the mole. The mole is an odd unit because it is not a unit of volume nor a unit of mass, though in controlled circumstances it can be related to either of those. Instead, the mole contains a fixed number of molecules. That number is called Avogadro’s Number.
Avogadro’s number is the number of molecules in 12 grams of pure carbon 12. Carbon 12 is an isotope of carbon whose nuclei contain 6 protons and 6 neutrons giving it an atomic weight of 12, hence the name Carbon 12. Does one get kind of a funny feeling about chemists here? Do you get the impression they are strange sorts of ducks? On one hand they have this mind-numbing obsession with the metric system and a total lack of imagination in naming their isotopes, but on the other hand they like to blow things up.
I’ve got all sorts of questions here. Why carbon? Why Carbon 12 in particular? Why 12 grams of Carbon 12? Were those two 12s chosen at random or is there more going on? Hmmm. Were the Knights of the Templar involved?
I must put those questions aside to focus on the task at hand. What is Avogadro’s Number? Here it is:
Again this is in scientific notation and that 10^23 in there means that if I were to write in out normally there would be 23 places between the 6 and the decimal. In other worlds this is a big number.
As chemistry arose out of alchemy, I might doubt the veracity of number figured out by people who in their heart of hearts are still looking for the Philosopher’s Stone, but as they love blowing things up so much and this number is so important to that goal, I will accept it.
A trip to www.wolframalpha.com will assure us that a liter of water contains 55.5 moles of water, so a quarter liter will contain 13.875 moles and, therefore, 8.33x10^24 molecules. The fraction of those which are Caesar’s urine is f=2.58x10^-22. Multiplying these two numbers together rounds off to 2100, which is the number I gave at the beginning of the article.
At the beginning of this article, we had to make the assumption that Caesar’s urine was scattered uniformly throughout all of the water in the world. I promised that this would be justified later, and now is the time when the chickens come home to roost.
I will start with things you already know. Rain falls from the sky onto the land; it trickles down hill into ditches, creeks, and streams; it gathers into lakes and rivers; it flows into the ocean; and at various points along the way it evaporates back into the air where it turns again into rain. It is part of a cycle that has been going on for billions of years.
Of course I’ve left out a few details such as at various places along the way folks—like Julius Caesar for instance—take a drink and then pee, putting it back into the cycle once again.
As a part of this system, many things happen to in and this is where mathematical modeling comes in. In the part of mathematics called Ergodic Theory, they model the behavior of such systems using a function called The Baker’s Transformation. While this can be described in mathematically excruciating detail—and nobody knows about excruciating detail like a mathematician—you can think of it simply as follows.
A baker takes a ball of dough, stretches it to twice its width, gives it a quarter turn, and then folds it back over onto itself. If you put a drop of red food coloring on this, it will eventually be spread uniformly throughout the loaf.
The idea is that similar things happen to Caesar’s urine. It flowed onto the street where he lay dying; it evaporated into the air; wind sheared it and spread it around; and, after a suitable interval of time, it is spread out everywhere.
Mathematicians study strong mixing, weak mixing, and topological mixing depending upon the type of system they are interested in. The Baker’s Transformation and Caesar’s urine are examples of strong mixing.
It occurs to me that some folks might find the prospect of drinking Caesar’s urine to be disturbing. In that case, you probably shouldn’t dwell to long upon the fact that you are also drinking the urine of Caesar’s horse, his dog, and of the beggar who was lying on the street in front of Caesar’s house. We live in an interconnected world. I’m touched by people who lived thousands of years ago and thousands of miles away. Those people (or should I say pee-ple) are literally a part of me.
As the author Norman Maclean said, “Eventually all things merge into one and a river runs through it.”
And that river will be carrying some of Caesar’s urine. I just thought you should know.