How likely do you drink Julius Caesar’s last water?

image: 

Picture of Julius Caesar holding a glass of water, based on Vincenzo Camuccini work
Julius Caesar holding a glass of water. Does your drink contain a part of it? The image is based on Vincenzo Camuccini work, with work by Goldfields Oasis combined. (©Masa Sakano, CC-BY-SA 4.0)

A little chemistry problem.

What is the likelihood that the next drink you are going to have contains some of the water molecules drunk by Julius Caesar on the day he was assassinated?

Here I give an answer in this extended and fairly thorough article, where all potentially significant factors are considered.


1 Question

What is the likelihood that the next drink you are going to have contains some of the water molecules drunk by Julius Caesar on the day he was assassinated?

For simplicity, you can assume Caesar drank a 300-millilitre glass of water on the day.


2 Answer

There are more than one way to answer this question and you can dig as deep as you want. You can derive a rough estimate for the answer only with mental calculations with some assumptions, or can evaluate answers with super-computers and complex modelling.

Here I describe two ways, an intuitive and more computational ways, on the basis of some simple assumptions and then assess how appropriate the assumptions are, before I conclude it.

2.1 Overview of how to estimate the likelihood

First, for simplicity, we assume that the water molecules in the water Caesar drank on the day are distributed across the present world (the earth crust and atmosphere), mixed uniformly with all the other water molecules, with no loss in quantity since Caesar’s death (such as, transformation into other forms of molecules or escaping to space). Let us hereafter refer to it as the “uniform-distribution assumption”. This assumption will be evaluated later. If the assumption turns out to be incorrect to a small but significant extent, we will modify the conclusion accordingly, unless the assumption turns out to be completely off the mark.

Second, in this type of problems, one of the following types of answer should be suffice, rather than a certain fairly precise number.

  1. Extremely unlikely
  2. Unlikely
  3. Neck and neck
  4. Likely
  5. Extremely likely

The first tactic to take should be an order estimate — so-called astronomers’ way.

Third, the numbers of the water molecules in your and Caesar’s glasses are critical. You can think of two extreme cases to prove the point intuitively. If a water molecule was as large as a glass of 300 ml, the probability the entire content of your 300-ml glass is identical to that drunk by Caesar would be hyper-extremely small, given that there is a huge amount of water on the Earth. Conversely, if a water molecule was infinitesimally small to the extent that a glass of 300 ml contains virtually an infinite number of water molecules, the probability that your 300-ml glass contains none of Caesar’s drink would be virtually nil; in other words, your water contains one or more of Caesar’s drink in (virtually) 100% confidence. In reality, the number is very large but finite, and therefore a quantitative, if rough, calculation is required to estimate the balance.

Fourth, I can think of two strategies to make an estimate.

Strategy A is to estimate the averaged number of Caesar’s water molecules contained in your glass, considering the volume of the water on the Earth. Then, you can estimate the probability in a statistical way.

Strategy B is to calculate or estimate the probability, using the standard theory of probability. Formulation is fairly straightforward, where you can apply some approximation, but the final computation may be problematic.

2.2 Solutions

2.2.1 Common grounds for strategies A and B

2.2.1.1 How many water molecules are there in a glass

The molecular formula of water is H₂O and so its molecular weight is 18. Using the Avogadro constant of NA = 6×1023 [/mol] (strictly, 6.02214076 × 1023), the number of the water molecules N300 contained in a 300-ml glass is calculated to be

N300=30018×NA(1)1×1025(2)

2.2.1.2 How many water molecules are there on the Earth (crust and atmosphere)

The total volume of the water on the Earth crust and atmosphere VE is (equivalent to) roughly 1.4 × 109 [km³] (or more strictly, 1.386 × 109 km³; see [1 2]).

Therefore, the total number of the water molecules on the Earth crust and atmosphere NE is

NEVE[km]×1015[g/km³](3)=1.4×1024[g]=1.4×1024×NA18[H₂O]5×1046[H₂O].(4)

Note that this can be estimated intuitively, even in mental calculations, using a set of more commonsense-type knowledge (see Appendix A — Intuitive estimates).

2.2.2 Strategy A

On the basis of the “uniform-distribution assumption”, the averaged number Nmean of the molecules originating from Caesar’s water that exist in your glass of water is,

Nmean=N300NEN300=N3002NE(5)2000[H₂O].(6)

The number of the the molecules originating from Caesar’s water that exist in your glass of water should follow the Poisson distribution with an average of Nmean (≈2000), hence a variance of √Nmean (≈45). Intuitively, an event whereby no molecules from Caesar’s water exist in your water should be extremely unlikely for Nmean ≫ 1. More quantitatively, such an event is 45σ.

Data scientists know off the top of their heads, for example, a 3σ event happens by chance with a probability of 0.3%, and 5σ with one in a million. A 45σ event is unfathomably highly unlikely.

This means that even if Caesar’s last drink was only 3 ml, or in other words even if only 1% of Caesar’s 300-ml last drink contributed to the uniformly available water molecules, part of which must sneak into your drink; the null hypothesis for such an event would still give 4.5σ (0.001% chance probability). If it was only 1 ml, it would be 2.6σ event (1%).

To conclude, your water should definitely one or more water molecules that were in Caesar’s last drink.

2.2.3 Strategy B

The probability that a molecule in the water in the 300-ml glass you are drinking did not exist in the water drunk by Caesar on the day is

NEN300NE=1N300NE.(7)

Hence the probability P- that none of the molecules in the water in your glass existed in Caesar’s water is, considering N300 ≪ NE,

P=k=0N3001(1N300NEk)(1N300NE)N300(8)(11×10255×1046)1×1025=(12×1022)1×1025.(9)

The probability P+ that the water in your glass contains one or more molecules that existed in Caesar’s water is, therefore,

P+=1(12×1022)1×1025.(10)

The actual analytical calculation of this P+ is difficult. The binomial approximation is often used for a formula in a form of (1 + Δx)n when Δx≪1. However, the approximation cannot be applied in this case because the other condition for the approximation to be applicable, |Δx|⋅n ≪ 1, does not hold:

|Δx|n=|2×1022|1×1025=2×1031 .(11)

Qualitatively, the fact that 2×103 is orders of magnitude higher than 1 implies that the resultant value for the former (P-) is very close to 0, and accordingly the latter (P+) is very close to 1.

Instead, numerical calculations, such as the website Wolfram Alpha, would be of essential help to get a quantitative value. In short, the probability of the former is P- ∼ 3×10-869 (nb, the given string is (1 - 2*10(-22.0))(10^(25.0))) and accordingly the latter P+ is almost ultimately close to 1 (by 3×10-869). Note that the index value of 869 in this solution has a great deal of uncertainty — remember only 1 significant digit has been used, and besides the assumption must not be so accurate to account for the precision in the first place.

Similarly, the case where Caesar’s last drink was only 1 ml would return the result ∼ 0.0013 (0.13%), which roughly agrees with the estimate in strategy A within an order.

To conclude, again, your water should definitely more than one water molecule that originate in Caesar’s last drink.


3 Evaluation of the assumption

I evaluate the “uniform-distribution assumption”, that is, the water molecules in the water Caesar drank on the day are distributed across the present world (the earth crust and atmosphere), mixed uniformly with all the other water molecules, with no loss in quantity since Caesar’s death (such as, transformation into other forms of molecules or escaping to space).

Did the (300-ml) water Caesar drank on the day of his death escape his body somehow and then has it spread out evenly across the globe by now?

3.1 Water molecules are not confined in Caesar’s body?

The water molecules a person drinks are discharged from her/his body in forms of bodily water discharges (sweat, urine, faeces, breathing out, shedding tears, spitting, bleeding etc), while s/he is alive.

Water a human takes can be retained in the body for a timescale of 10 days or so. This timescale is estimated as follows.

Roughly ∼60% in weight of a human body is water, which is translated into 36 kg for a person of 60 kg in body weight [3]. How much a person (needs to) takes a water greatly varies, depending on many factors, including the body weight, ambient temperature, exercise level, etc. The minimum amount for an adult is roughly 1.2 litres [4], which is a combined values of any forms of water intake, including breathing and that in foods. A couple to few litres of water per day is recommended to drink by health organisations, which is roughly 10% of the above-mentioned 36 kg. Therefore, though some water in a human body, such as that in bones, stays in the body for a much longer time, 10 days are a rough transit timescale of the water circulation in the human body.

Consequently, most of, say over 90% of, the 300-ml water Caesar took on the day of his death was retained in his body when he died.

If he had been buried or mummified, how much water escaped to the environment and then travelled across the globe would have been debatable. However, it is known that Julius Caesar was cremated (publicly in the Roman Forum [5]) shortly after his death in assassination on 15 March, 44 BC, roughly 2,100 years ago from the time of writing (2020).

Therefore, pretty much all the water molecules in Caesar’s body, including those Caesar took on the day of his death, were discharged into the environment — mostly into the air and possibly partly into sewage or on the ground in a form of urine and bleeding. Either way, any of these are easy ways for the water molecules to travel far since (see the following sections for the travel paths and timescales).

3.2 Bodily transformation of water molecules?

If a human body efficiently transformed water to different molecules (or single atoms), a significant portion of the water molecules Caesar took may no longer exist by the time of his death or cremation.

However, in general, an overwheming majority of the chemical reactions that happen in animals’ bodies do not transform water molecules to something else.

An exception in a human body I know is the chemical reaction to convert starch to glucose in the human digestion system, where water is actively used and is converted (see Fig.4 of [6]). However, given the facts that (1) the reaction uses only one-sixth of H₂O in amount contained in the glucose generated and (2) the reaction does not give energy to the (human) body, the total amount of the water used for the reaction should be many orders of magnitude smaller than the water generated by oxidation in a human body (calorie burning or broadly speaking, digestion). After all, if anything significant in a living organism’s body, it must generate energy on balance.

Thus, pretty much all the water molecules Caesar took on the day of his death were discharged without transforming into other forms of molecules.

3.3 Water molecule transformation, loss, and generation on the Earth

Water molecules experience cycles on the Earth. Do they possibly transform or disintegrate into different molecules or atoms during the cycles? Conversely, are water molecules newly generated from some other molecules and/or atoms?

3.3.1 Water molecules transforming into something: photosynthesis

The most major form of chemical reaction on the Earth is oxidation, because there is plenty of oxygen on the Earth, which is highly reactive substance. Water is basically oxidised (di-)hydrogen. Hence it is pretty stable. Besides, if there was anything active enough to react with water on the Earth, given water is everywhere on the Earth’s surface, they must have already reacted with water millions and millions years ago and almost none would have remained by now, 4.5 billion years after the Earth’s birth. For example, although sodium metal rapidly reacts with water and disintegrates water molecules, there is virtually no natural sodium metal remaining on the Earth’s surface, the fact of which keeps the remaining water safe.

Thus, there are not many natural modes to disintegrate water on the Earth. If anything, the possible exceptional cases should satisfy the condition that the reacting substance is on constant supply presumably due to some sort of cyclic chain of reactions and/or some sort of supply from the Sun. There is indeed an case on the Earth: photosynthesis in plants.

In photosynthesis, water and carbon dioxide are converted into sugar and oxygen molecules, powered by sunlight [7]. The yearly total amount of the oxygen molecules generated with photosynthesis on the earth is roughly 3×1014 [kg/yr] [8], where the amount of the water used is about a half of it, ∼1.5×1014 [kg/yr]. Accordingly, in 2,100 years, ∼3×1017 [kg] of water molecules break into different forms of molecules. This is lower by an order of 4 than the amount of the water that exist on the present Earth’s surface (crust and atmosphere). Therefore, only a negligible ratio of the water molecules on the Earth break in 2,100 years, and a vast majority of them stay as they were 2,100 years ago.

3.3.2 Water molecules generated: oxidation

A reaction in the reverse direction, namely, water molecules being generated, exists.

Basically, oxidisation of any substance that contains an hydrogen atom (can and does most likely) generate water molecules. Any fire or burning fuel would generate water, for example. Much more significantly in quantity and ubiquitously, internal combustion (often called digestion if it is by animals) in organisms generates water.

However, the fact the total amount of the water on the Earth is stable in a timescale of a million years implies that water generation is basically on balance with water molecule break-up. Note that although the Earth is believed to have lost a quarter of the water since its birth 4.5 billion years ago [9], the timescale of 2,100 years as in this context is less than 1 millionth of it, and accordingly, the amount of change is negligible.

3.3.3 Other modes of water reactions, generation, and loss on the Earth

The chemical reaction to convert starch to glucose in the human digestion system has been already discussed, and has been dismissed as a significant source to contribute to the water transformation. Basically, animals do not transform water to different forms for a significant amount — greatly less than their generation of water.

Since the Earth is an open system to space in the Universe, water loss constantly happens in the upper atmosphere as a result of photodissociation; a molecule is broken up by UV light from the Sun into hydrogen and oxygen and the former escapes to space. Also, water is brought to the Earth crust from the Earth internal. Their yearly amounts of water loss and generation are 4.8×10-4 km³/yr and 0.3 km³/yr, respectively [10]. The cumulated amounts for 2,100 years for both of them are totally negligible, compared with the current amount of the water on the Earth, 1.4×109 km³.

Finally, geologically, a reaction of methanogenesis exists in theory on the Earth , as in

2H₂O + CO₂ → CH4 + 2O₂,

where water molecules are broken into different molecules. However, the likelihood that this happens in the normal circumstances is extremely low, as one can infer from the fact that water and carbon-dioxide stably coexist in the atmosphere.

3.4 Spread of water molecules across the globe

We have already confirmed the water molecules were discharged from Caesar’s body to the ambient environment. Now, the question is whether these water molecules are distributed evenly across the globe since his death 2,100 years ago.

It is a reasonable assumption that the water molecules discharged from Caesar’s body while he was alive and before and at his cremation were evaporated quickly into the air in timescales of minutes to a day. Then, let us consider the fate of these water molecules since.

3.4.1 Water molecule transport in the atmosphere

According to an education article [11],

a drop of water spends an average of just nine days in the atmosphere before falling back to Earth.

As such, the life cycle of the water in the atmosphere and on the ground (on the land), where water should evaporate quickly unless the temperature stays below freezing, is very short, 5 orders of magnitude as short as the timescale of 2,100 years since Caesar’s death.

In the timescale of 9 days in the air, water molecules can travel a fair amount of distance as follows. The average wind speed on the surface of the Earth at the ground level is about 11 km/hour [12]. With this speed, a water molecule in the atmosphere travels over 2000 km in 9 days, which is more than 10% of a half of the Earth’s circumference (40,000 km in circumference). In addition, it is known wind speed in general is a positive function of the altitude; for example, the wind speed is on average twice as high at an altitude of 10 km above open water as the sea level or 10-m high [13].

Also, the wind on the Earth is not completely random but has a tendency of westerly, which is efficient to distribute water molecules across the globe, being in good contrast with the random work, which is less efficient for wide-spread distribution.

All in all, at least 10 cycles of water circulation in the atmosphere would be required to spread water across the globe, if the air circulation alone is considered. In reality, 10 cycles are not required because the water movement in the ocean should be playing an important role, as described below.

3.4.2 Life cycle of water in/on the oceans and ground

I now consider what happens with the water molecules when they come back to the Earth’s surface from the atmosphere.

Hydrological Cycle; adopted from Fig.1 of “Water Cycles and Climate Change”, Kevin Trenberth, 2011, Global Environmental Change
Hydrological cycle on the earth. Estimates of the main water reservoirs, given in plain font in thousand (10³) km³, and the flow of moisture through the system, given in slant font in thousand (10³) km³/yr, equivalent to Exagrams (1018 g) per year. Text and figure adopted from [14].
3.4.2.1 Life cycle of water in the oceans

A majority of precipitation occurs over the ocean, or specifically, roughly 77% [14], the fact of which is unsurprising, considering that ∼71% of the surface of the Earth is the ocean (70.9%, more precisely [15]).

Water molecules in the ocean can stay for a long time with an average of ∼3,000 years [11, 16], before evaporating into the air. Given that (1) the average residence time is 3,000 years, that (2) the water close to the deep ocean bed has a considerably longer lifetime, whereas that close to the surface should have a shorter lifetime, and that (3) the water that originated from Caesar’s drink should initially arrive at the ocean by one of rainfall, rivers, and ground water flows and hence should start at the surface of an ocean, the timescale of Caesar’s drink-origin water should be somewhat shorter than 3,000 years, though a quantitative discussion is difficult.

The water in the ocean, being liquid, moves around. There are oceanic currents near the surface (typically up to 400 metres in depth) with a characteristic surface speed ranging 0.05—0.5 metres per second [17]. These speeds translate into 1,600—16,000 km/yr, and thus technically, water takes up to 25 years to circle around the Earth (40,000 km for the equatorial circumference). In reality, most of the ocean currents bridge neither multiple oceans nor multiple hemispheres. Therefore, the spread of water molecules with this move is limited mostly in a hemisphere in the single ocean, where it started the travel, and accordingly the required travelling timescale is more like 10 years from one end to the other. In reality, not all the surface water in the ocean are in the current, and so the averaged travel time is probably longer, perhaps by a factor of a few, such as 30 years. In any case, the timescale for the water to spread is 2 orders of magnitude shorter than 2,100 years, though the spread is limited to a part of an ocean.

By contrast to the horizontal movement, vertical movement of the water in the ocean (often referred to as upwelling and downwelling) exhibits much lower speeds of a few metres per month or ∼1×10-6 m/s [17]. With these speeds, a water molecule on the surface takes 300 years to reach the averaged depth of the ocean, ∼4,000 metres, or 30 years to reach 400 metres, which is the depth of the surface ocean currents. The latter timescale is comparable with that of the ocean-wide horizontal movement of the surface ocean currents. In this sense, the water that starts its journey from the ocean surface will be distributed in a considerable part across the surface of the same hemisphere of the ocean in a timescale of a few decades, though some of them can sink deep in the ocean and do not travel along the surface ocean currents any more.

In addition to surface oceanic currents, there is a global oceanic circulation called the Global Conveyor Belt [18] or the thermohaline circulation [19]. The circulation completes the entire global cycle in 1,000 years, travelling over all the major oceans. Thus, in 2,100 years, the water molecules in an ocean should complete 2 cycles in average, with which water molecules are spread in the oceans world-wide more or less evenly.

More specific history for this case would be like this. Given that Caesar’s drink-origin water started its journey from the surface of an ocean when it first reached it, and considering the fact that vertical movement of the water molecules in oceans is slow, a considerable portion of it must have remained in the ocean surface layer till the molecules got somewhat evenly distributed across the same ocean in an timescale of the initial few decades. Then, eventually (a timescale of a few hundred years), a significant part of the water molecules reached deeper in the oceans and eventually get on the Global Conveyor Belt, which spread the molecules world wide in a timescale of 1,000 years.

Consequently, the oceanic water move alone should achieve a fairly uniform world-wide distribution of Caesar’s water molecules across all the oceans. Given that the transit time of water molecules in the ocean is 3,000 years, which is longer than the timescale of interest of 2,100 years, this fact is critical.

3.4.2.2 Life cycle of water on/in the ground

Globally, roughly 23% of the world precipitation occurs over the land [14].

The fate of the water that falls on the land highly depends where it falls on; the numbers in the following sentences in this paragraph are taken from the article [11] unless otherwise noted. More than a half of the water molecules that fall on the land evaporate back to the air fairly quickly with timescales ranging from minutes to 2 months, the timescale heavily depending on exactly where they fall. Water that falls on tropical land will evaporate quickly like a matter of minutes, unless it permeates the soil, whereas that in freezing environments would take much longer than 2 months before it finally melts and then evaporates. For example, it takes 6 months with the Arctic snow, unless it permeates glaciers or large ice sheets, in which case it can stay even greatly longer like a century for glaciers or even million years in extreme cases for ice sheets. Water that stays in soil takes 1 to 2 months before it is evaporated (or transpired). The residence times of the water in rivers and reservoirs/lakes are typically 2 weeks and 10 years, respectively [20].

However, in reality, evaporation (plus transpiration) from the land accounts for only ∼65% of the precipitation on the land [14]. In other words, ∼35% of the water falling on the land does not evaporate back to the atmosphere (in the above-mentioned relatively short timescales). It either flows into a river/lake or sinks into groundwater, both of which eventually end up in an ocean/sea unless the molecules of interest evaporate back to the air during the process (n.b., the evaporation process accounts for a part of the ∼65%). The residence time (also called transit time) of the water in groundwater varies greatly from 2 weeks to over a million years [21].

To summarise, wherever water ends up on the land, except for the rare ice-sheet case and some deep groundwater, the timescale for it to evaporate to the atmosphere or to move to the ocean is by an order of 4 (or 2 for reservoirs) shorter than 2,100 years we are considering. In other words, we can interpret that the water that falls onto a land move to either back to the atmosphere or to the ocean in no time, compared with the timescale we are considering, 2,100 years.

3.5 Summary of the evaluation of the assumption

First, I have confirmed that the water molecules in the water drunk by Caesar 2,100 years ago remain basically on the Earth without transforming into other molecules or escaping to space. Also, there is no significant amount of water generated on the Earth during the period.

Second, the water molecules in the water drunk by Caesar were almost completely discharged into the environment, mostly into the air, shortly after his death at latest (stage 1).

Third, on and immediately after the day of Caesar’s death and cremation, the discharged water molecules travel in the air for typically 2,000 km in 9 days before they fall back down to the Earth surface, mostly on an ocean but some on land, in a form of rainfall (stage 2).

Fourth, those that have landed on land move either back to the atmosphere (hence repeating stage 2) or to an ocean in no time.

Fifth, the water that landed on an ocean tend to stay in the ocean for a long time of typically 3,000 years. It will initially in large part flow in the surface current and spread across the same hemisphere of the ocean which it has landed on in a couple of decades, while some evaporate to the air (back to stage 2) and some sink deeper in the ocean. In the first iteration since the discharge from Caesar’s body, few of their water molecules reached Pacific Ocean for this reason, and spread to this area must wait for either further iterations or the process explained in the next item.

Sixth, in the longer timescale, the water that landed on an ocean joins the Global Conveyor Belt and travels across all the oceans typically twice in 2,100 years. Combined with the facts that some are transported by surface ocean currents and that some evaporate to repeat stage 2, this process distributes the water molecules across the globe fairly uniformly. In particular, given the iteration starts from the ocean surface, there is much greater chance of evaporation for the water molecules to repeat stage 2, with which the process to establish uniform distribution must be enhanced.

Seventh, in the timescale of 2,100 years, the distribution of Caesar’s water molecules across the existing water in the Earth crust and atmosphere should not yet be entirely uniform in some localised places as follows. Given the fact that the residence times in polar ice sheets, deep-underground groundwater, and deepest sea bed exceed several thousand years to even a million years in some cases, Caesar’s water molecules are unlikely to permeate them to the extent to achieve the number equilibrium with the existing molecules. In other words, the distribution is more patchy than complete uniformity, biased towards the accessible water sources for humans. This biased distribution of Caesar’s water molecules works as the tendency that the glass of water you drink is likely to have more of Caesar’s water molecules than in the case with the original ideal assumption.

Eighth, the fact that the number of the cycles with the Global Conveyor Belt, which is the most major driving force towards uniform distribution of Caesar’s water molecules across the globe, is limited to only 2 suggests that there may remain some deviation from the uniform distribution, namely patchiness, in the distribution of Caesar’s water molecules across the oceans of the Earth. As a conservative estimate, let us assume only 50% of Caesar’s water molecules have been distributed uniformly across the water on the Earth surface.

Finally, the drinking (fresh) water humans use is usually collected from one of reservoirs or lakes inland, surface flows (rivers), and shallow groundwater (wells). Their residence times are up to 10 years only. Therefore, the drinking water basically originates from recent evaporation, a majority (∼85%, [14]) of which comes from an ocean. Consequently, the uniformity of the water molecules in your drinking water basically reflects that of the ocean waters, regardless of the averaged residence time of the water in the ocean.

To summarise, the “uniform-distribution assumption” is mostly justified. In a conservative estimate, one can assume only 50% of Caesar’s original water molecules are distributed uniformly across the globe.


4 Conclusion

I accept the “uniform-distribution assumption”, i.e., the water molecules in the water Caesar drank on the day are distributed across the present world (the Earth crust and atmosphere), mixed uniformly with all the other water molecules, with virtually no loss in quantity since Caesar’s death. This assumption is mostly justified. As a conservative estimate, I assume that 50% of Caesar’s original water molecules are distributed uniformly across the globe and that the other 50% have no chance to get in your water to drink.

Assuming that the probability distribution for Caesar’s water molecules to exist in your 300-ml drink follows a Poisson distribution, which is the mathematically correct assumption, the event where the 300-ml water contains none of Caesar’s water molecules is a (over) 30-σ event, which is impossibly unlikely, say, by many orders of magnitude lower than 10-10%.

Therefore, the 300-ml water you are going to drink in the 21st century must most definitely contain one or more water molecules from the glass of water drunk by Julius Caesar on his last day.


5 Acknowledgement

I thank Jonathan Miles, an expert of chemistry, for bringing up this little quiz to me. It has been a great problem, which requires an extensive knowledge and study across many fields of science from statistics, earth science, to even history. I also thank Ed Nind, an expert of chemistry and school teacher, especially for pointing me towards the starch-to-glucose reaction, which I did not know at all. He impressed me by solving this quiz thoroughly through mental calculations during our approach to a crag for rock climbing, if his calculation slightly missed the target by an order or two as it often happens in complicated calculations.

This quiz was first shared in my Facebook wall, taken from Jonathan’s wall. A series of comments given by Simon Perry and Ian Stewart were intellectually exciting and stimulating, and I am most grateful for them. Without their comments, I would have not thought of compiling this piece. In addition, some of the items and solutions in this article were hinted by or taken from their comments.

The first image partially utilises a clip art made by Goldfields Oasis.


Appendix A — Intuitive estimates

A.1 Total amount of the water on the Earth surface

It is possible to make a rough (but precise enough) estimate of the total amount of the water on the Earth from some basic knowledge of science.

  1. The radius of the Earth is 6,400 km, or the circumference of the earth is 40,000 km. This could be derived alternatively from one of the following facts, depending on what knowledge one has:
    • Light can travel around the Earth 7.5 times in a second, and the speed of light is 300,000 km/s.
    • The speed of the spin of the Earth is 1700 km/h or 500m/s, and the Earth rotates every 24 hours.
    • A trans-continental aeroplane flies at a speed of roughly 850 km/h (as you see in the monitor on a seat onboard) or a Mach number of 0.8, and it takes roughly two 12-hours flights (namely 24 hours) to get to the opposite side of the Earth.
  2. The averaged depth of the ocean is 4 km (or more precisely 3.68 km [22]). This could be guessed from the following facts combined.
    • Each continent has a geological structure called a continental shelf at the boundary with the oceans that surround the continent. Beyond the continental shelf, the seabed steeply drops to deep sea.
    • The continent shelves cover only a tiny portion in terms of the area of the oceans.
    • The depth of a continental shelf is about 200 metres (as referenced in the Convention on the Continental Shelf (1964, Geneva), though the scientific definition is different.
    • From these facts, one can surmise the average depth of the ocean is an order of magnitude larger than 200 metres, that is, a few km.
    • The fact that the depths of the deepest oceanic trenches are more than 10 km are consistent with it.
  3. The oceans and seas cover roughly 70% (70.9%, more precisely [15]) of the Earth’s surface.
  4. Almost all the water (96.5%, precisely) on the Earth’s surface (crust and atmosphere) in all forms (solid, liquid, and gas) combined exists in the oceans and seas.

Appendix B — Balance sheet between the precipitation and river flow

Here I estimate the balance sheet between the precipitation on the land and flow of the rivers in the area, taking China as a typical case.

In China, the total amount of the water discharges of the 9 major rivers is of the order of 1011 m³/year; so I conjecture that the total discharges of all the rivers in China is 1012 — 1013 m³/year. The annual rainfall in China’s land is 6×1015 m³/year. Therefore, only 1 per cent or less of the water that falls on to China’s land ends up in the sea/ocean via rivers.

It is known that the amount of the water held as soil moisture is of the same order (two-thirds) of that stored in the rivers and lakes combined and that held in groundwater is 2 orders as much as the latter. The groundwater plays a much more significant role in terms of the quantity in the hydrological cycle on the land.


References

  1. “Water distribution on Earth” in Wikipedia↩︎

  2. “Where is Earth's Water?” in USGS↩︎

  3. “The Water in You: Water and the Human Body” in US Geological Survey (USGS)↩︎

  4. “Survival Hydration: Minimum Water Requirements”, WaterCures.org↩︎

  5. “Famous Funerals: Julius Caesar”, Consolidated Funeral Services, Inc↩︎

  6. “Bio-fuel agro-industrial production system in sweet sorghum”, Ray & Behera 2011↩︎

  7. “Photosynthesis”, Wikipedia↩︎

  8. “Photosynthesis: Earth’s life support system”, Richard Cogdell, 2013, New Scientist↩︎

  9. “The Earth has lost a quarter of its water”, Sybille Hildebrandt, 2012, Science Nordic↩︎

  10. “Origin and evolution of the hydrosphere”, Britannica↩︎

  11. “The Water Cycle”, in the University Corporation for Atmospheric Research (UCAR)↩︎

  12. “The World’s Winds Are Speeding Up”, Chelsea Harvey, 2019, Scientific American↩︎

  13. “Evaluation of the variability of wind speed at different heights and its impact on the receiver efficiency of central receiver systems”, Delgado et al., 2016, AIP Conf. Proc., 1734, 030011↩︎

  14. “Water Cycles and Climate Change”, Kevin Trenberth, 2011, Global Environmental Change↩︎

  15. ETOPO1 model, NOAA↩︎

  16. “The water cycle”, Britannica↩︎

  17. “Ocean current”, Britannica↩︎

  18. “The Global Conveyor Belt”, National Ocean Service↩︎

  19. “Thermohaline circulation”, Wikipedia↩︎

  20. “Estimated Residence time of water resources”, Philippe Rekacewicz, February 2006, UN Environmental Programme GRID↩︎

  21. “How Old is our Groundwater?”, Alberta WaterPortal↩︎

  22. “How deep is the ocean?”, NOAA Ocean Exploration and Research↩︎


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