Suppose I - or more likely someone else - gave you access to the world's best instrumentation and then asked you to calculate the volume of water in earth's oceans. Think of the difficulties you would encounter. First off, the earth's oceans have an irregular bottom, and it swell know that below the ocean's surface there are huge trenches, mountain ranges, ridges and rifts. Any measurement would need to be empirical, involving the mapping of features that could be several kilometers below the surface. As for the surface itself, it is hardly constant on either a long or short time scale. Gravatational tides, waves, temperature gradients in the water itself, as well as pressure and temperature gradients in the surrounding fluid, the atmosphere, all effect the position of the surface, as do temporary fluctuations that may derive from higher or lower fluxes of fresh water into the sea and well as fluxes out of the sea owing to vapor pressure.
Even a moment's reflection would suggest that this fascinating question is a very difficult one to approach. (Calculating the mass of seawater, if it's any consolation, would be even more difficult, owing to compositional effects and hydraulic effects, as pressure, and thus density, varies with depth.)
Poking around for figures estimating the volume of earth's oceans, I've decided for the purpose of calculations in this diary to use the estimate published by the Yale Geologist Jun Korenaga, in the publication (linked in the link) Terra Nova, 20, 419–439, 2008, which is that the volume of the ocean is 1.51 billion cubic kilometers. We can estimate if this is a reasonable figure by noting that the mean radius of the earth is generally taken to be...
...6,371 km (exactly), implying from the formula for the volume of a sphere, that the volume of the earth (were it a perfect sphere) would be 1.08321 trillion cubic kilometers, the selected volume of the ocean would require a radius roughly 3 km higher than the given mean radius, meaning that a perfectly smooth ocean on a perfectly smooth earth would have a depth everywhere of 3 km.
That seems reasonable to me. (The Titanic, as an example, rests some 3.8 km below the surface on the decidedly not smooth seabed of the real earth.)
Seawater's composition is not everywhere the same, and these effects account for things like ocean currents and other phenomena. However it is well understood that a fair average may be obtained for its composition. Besides water, the majority of seawater is sodium chloride, but other elements are found in varying proportions.
Uranium is a constituent of water owing to the weathering of uranium containing rocks, in particular granite. Roughly the concentration of uranium in seawater is 33 mg per cubic meter (cf Ind. Eng. Chem. Res., 2009, 48 (14), pp 6789–6796) This suggest that the oceans contain about 5 billion tons of uranium, but I am going to ignore this large quantity of radioactive material and focus on the radioactive element potassium, because unlike uranium (and for that matter, rubidium a radioactive element that is a cogener of potassium) potassium is essential to life. I will say this: The ocean is saturated with respect to uranium, meaning that if uranium were removed from seawater - say to run power plants, something the Japanese have examined in some detail - it would continually be recharged by weathering of granite, assuming rivers actually flow to the sea, something that actually happens with less and less frequency in modern times.
As stated earlier, the ocean is not homogenous, and salinity gradients are well known. In fact these gradients drive ocean currents. The Gulf Stream, for instance, is driven by saline waters sinking in the arctic regions in relatively lighter polar waters. Some people have speculated that the melting of the polar icecaps might result in shutting the Gulf Stream down because of the dilution of these saline waters with waters from melting ice. However, a working figure for the concentration of potassium in seawater is 416 milligrams per liter.
The nuclear stability rules, which determine whether a particular isotope is radioactive or not, predict that elements have odd atomic numbers will have either zero (which is observed for elements 43 (technetium), and 61 (promethium)) one, or two stable nuclei represented by it. Examples of elements having only one stable isotope are fluorine, sodium, and cobalt. Examples of elements with odd atomic numbers having two stable isotopes are chlorine (35 and 37) and copper (63 and 65). Rubidium - the most common element in human flesh that has no known physiological purpose (except possibly has a potassium mimetic) - has two natural isotopes in all of its ores and in natural brines, 85 and 87, but only one of them (85) is stable, the other is radioactive, but has a half-life much longer than the age of the earth, roughly 47 billion years.
I have argued elsewhere - based on the nuclear stability rules and the mass "defect" - that it is almost certain that calcium-40 is radioactive, although the half-life must be so long that it essentially escapes detection. (Several years ago it was discovered that a similar case was obtained for bismuth-209, with bismuth being thought to be the element with the highest atomic number (83) to be stable - a surprise because 83 is an odd number. To my personal relief on this score, it was discovered that bismuth is radioactive, although the half life is 2.0 X 1019 years, way longer than the age of the universe. cf: Nature 422, 876-878 (24 April 2003), "Experimental detection of α-particles from the radioactive decay of natural bismuth.")
Potassium also has two stable nuclei, isotope 39, which represents 93.2581% of naturally occurring potassium, and isotope 41, which represents 6.7302% of naturally occurring potassium. These numbers do not sum to 100% because a third isotope, a radioactive isotope, potassium-40 is also always found with potassium. The percentage of potassium that exists today that is radioactive potassium-40 (K-40) is 0.0117%.
The K-40 isotope has a half-life of 1.277 billion years, which is sufficiently long to have allowed it to have existed since the formation of the earth 4.5 billion years ago. About 8.7% of the radioactive potassium-40 that was present when the earth formed is still here, although it must be said that potassium is less radioactive than it has been at any time in earth's history.
Using the figures above for the volume of the seas, and the concentration of potassium in that volume, as well as the percentage of potassium that is the radioactive isotope, as well as the fact that the isotopic atomic mass of K-40 is 39.9639987 grams per mole we can calculate directly the amount of radioactive potassium-40 in the ocean. It is about 75 billion metric tons, outstripping uranium by a factor of 13 and being, gram for gram, more radioactive than the uranium, if one ignores (as maybe one shouldn't), the uranium daughters like radium, radon, etc, etc.
The total activity of the ocean owing to radioactivity associated with potassium-40 is approximately 2 X 1022 Bequerel, or roughly 530 billion curies.
This is an enormous amount of radioactivity, but it is very diffuse, spread throughout the ocean.
One may ask how much energy is released by this nuclear decay, and the answer is actually a rather large number given the branch ratio adjusted nuclear decay energy of K-40, which is about 1.33 million electron volts. The decay of potassium-40 in the Earth's oceans represents a power output of around 1.3 X 1013 watts, or 13 million megawatts, which is roughly the power output of 4,400 nuclear power plants of average size. The power output is roughly equal to 82% of the average continuous power consumed by humanity as a whole.
For the record, at the risk of encouraging simpletons of the Amory Lovins type who believe that it would be wise to cover the entire planet and all of its ecosystems with toxic semi-conductors and the like, the solar flux experienced by the earth is much larger than the power output of potassium 40, 13 thousand times larger, meaning that it is trivial for the earth to radiate the energy produced by potassium-40.
Nevertheless, this is significant energy.
Lord Kelvin - for whom our temperature scale is named - and one of the major founders of the science of thermodynamics - made a thermodynamic argument, a good one given that he, like many Kossacks, knew nothing at all about nuclear science, that since the earth was clearly hot, as evidenced by the existance of volcanos, that the earth could not possibly be more than 400 million years old.
It is now understood that the internal heat of the earth is provided by, among other radioactive elements, potassium-40. By the way, this should establish something that is often not recognized, that geothermal energy is nuclear energy.
By the way - just to be a little off topic - there is a paper published recently that indicates that the reason that the Gulf of Mexico is more radioactive than other bodies of water has to do with the release of "natural" radioactivity associated with oil and gas drilling, if, unlike me, you believe that oil drilling is "natural." (cf. Journal of Environmental Radioactivity 89 (2006) 1-17). If we explained to people - and I have done this - that oil and gas drilling produces significant releases of radioactivity, they suddenly might begin to give a shit, or, um, maybe not.
One might ask the ridiculous question, "What is the risk associated with potassium radioactivity?" but although I could put together an answer to this question, I'd rather not dignify it with an answer, but rather would ask "What is the risk of avoiding potassium?" which is of course a risk of 100%, since you would die without potassium.
About 3,000 curies of radioactive potassium is added to U.S. soils each year as fertilizer, without which the soils would most likely die.
Anyway, every year, as of this writing (the number will continually decrease for all recorded time) about 4,100 metric tons of potassium decays in earth's oceans.
The molar specific activity (the activity for 136.907 grams) of cesium-137, a common fission product, is 440 trillion Beq, or about 1,189 curies per mole.
This particular isotope, cesium-137, has been fetishized by the Fukushima crowd, who regard the failure of the reactors at Fukushima in an earthquake and a tsunami as the worst energy disaster of all time, meaning they never heard of the renewable energy disaster at Banqiao in 1975, which killed about 200,000 people in a matter of days. (It also means that most of them have never heard of climate change, which is very easily the worst energy disaster of all time.)
It follows that in order to make the Earth's oceans as radioactive as they were 1.227 billion years ago from potassium - a period in which life on earth was rapidly evolving - one would need to directly dump roughly 6,100 tons of pure cesium-137 directly into the ocean.
Because cesium has a different absorption profile than potassium, it is doubtful that this amount would remain solubilized in its entirety however. In particular cesium has a stronger tendency to be adsorbed onto the surfaces of certain minerals and has several insoluble aluminates, which accounts for the existance of the mineral pollucite, an aluminosilicate.
In any case, in a few hundred years all of this radioactivity would essentially be gone, and the ocean would be about as radioactive as it is now.
If one makes certain simplifying assumptions about the accumulation of cesium-137 in nuclear fuel, first that the concentration of cesium-136 is very small owing to its short half-life (about 13 days) and that nuclear fuel spends most of its life outside the reactor and not inside it, one can discern that a solution of the Bateman Equation takes the form of P(σ,E) (1-e-kt) where P(σ,E) is a function of the total power output of one, or even all, of the nuclear reactors in operation in the entire planet and the fission yield of a particular isotope, in this case, cesium-137 and the neutron energy spectrum of said reactors, which is overwhelmingly a thermal spectrum on the planet right now, t is the time that a particular power level has been operating, a k is the decay constant of the isotope, the natural logarithm of 2 divided by the half life of the isotope..
It is very clear that this is an asymptotic function, and by appeal to it, it is relatively easy to show that 6,100 tons of cesium-137 is considerably more than has accumulated in the entire history of nuclear power, nuclear war, and nuclear weapons testing and manufacture.
Have a nice day tomorrow.