An Exercise in Global Warming
Global Warming is a highly-politicized issue. It seems like solid scientific pursuit has been set aside so that Really Pushy People can tell you how they will solve this problem through high taxes on SUVs while other Really Pushy People tell you it's impossible for human beings to affect their planet, so just mind your own business. The idea behind Global Warming (as far as I can tell) is that carbon dioxide (CO2) emissions from the burning of hydrocarbon-based fuels trap warmth from the sun much the way glass traps warmth in a greenhouse. Too much CO2 traps too much warmth. Rising temperatues cause global havoc. Hilarity fails to ensue.
Exactly how much CO2 are we talking about here? How much effect would it have on the atmosphere? How do you even figure that out?
Let's put an upper-bound on the damage by reasoning this way: If burning hydrocarbon-based fuels is the source of the carbon dioxide problem, then burning Earth's entire hydrocarbon reserve at one time should release all the available carbon into the atmosphere. We will include oil, coal, and natural gas reserves, which constitue new carbon sources. We won't include natural carbon sources such as corn ethanol, word-burning, or other biofuels, since these only recycle existing atmospheric carbon dioxide. Besides, I don't have any politicians pointing fingers at me for grilling burgers over an open fire.
Here is my attempt at getting some answers. The numbers and calculations are all here, plus resources for the values whenever I needed to look something up. If you disagree with the numbers or the methods, please let me know and I'll take your concerns into consideration.
How much oil is there? It's question with some staggeringly huge answers. The US Department of Energy estimates the total world petroleum reserves at 1.317 x 1012 barrels of crude. British Petroleum lists 1.208 x 1012 barrels of crude oil reserves. The folks at The Global Education Project show a reserve of 1.077 x 1012 barrels. There is most likely more undiscovered oil, but this estimate covers all the oil we think we can get to today, including the Alberta oil sands. We'll take the highest estimate and dig it all up.
Total World Oil Reserves: 1.317 x 1012 bbls.
I can't really do much math in barrels. Let's convert it into SI units. Standard Oil started shipping its crude in 42 US-gallon barrels in the 1860, buying them from the same coopers which provided these barrels for winemaking. Standard Oil is long gone, but the 42 gallon barrel is still around. Only now we measure them in liters.
Total World Oil Reserves: 1.317 x 1012 bbls x 159.0 L/bbl
= 209.4 TL (teraliters)
To do chemistry calculations with this, we will need to know the weight of all this oil. To do this conversion we will need the density of crude oil. However, this value varies from oil field to oil field, and even from year to year for the same field. The International Energy Agency formed a working group in 2004 to determine the average properties of oil and oil products; they determined that the average density for crude oil was 0.8484 kg/L.
Total World Oil Reserves: 209.4 TL x 0.8484 kg/L
= 177.7 Pg (petagrams)
We are interested in the carbon content of the oil, since the carbon atoms burn to form carbon dioxide. Crude oil is made up of long hydrocarbon chains, such as:
| Compound | Name |
|---|---|
| CH4 | methane |
| C2H6 | ethane |
| C3H8 | propane |
| C4H10 | butane |
| CnH2n+2 | n-carbon chain |
What we really want to know is how much of each of these compounds is made up of carbon, by weight. The atomic weight of carbon is 12.011, while hydrogen is a much ligther 1.008. This table attempts to calculate the molar weight of the hydrocarbon chains above, including the n-carbon chain as n gets very big.
| Compound | Molar Weight of Carbon | Molar Weight of Hydrogen | Total Molar Weight | Percent Carbon by Weight |
|---|---|---|---|---|
| CH4 | 12.011 | 4.032 | 16.043 | 74.868% |
| C2H6 | 24.022 | 6.048 | 30.070 | 79.887% |
| C3H8 | 36.033 | 8.064 | 44.097 | 81.713% |
| C4H10 | 48.044 | 10.080 | 58.124 | 82.658% |
| CnH2n+2 | n x 12.011 | (2n+2) x 1.008 | n x 12.011 + (2n+2) x 1.008 | (as n -> ∞) 85.628% |
I am certainly not suggesting that we have infinitely long carbon chains in crude oil! The point is that, no matter how long we make the carbon chain, we can't get any higher percentage than about 85.6% carbon. In reality, there are many impurities that could drive that percentage down, but there are also some other carbon-chained molecules that have double-bonds, which could increase the carbon ratio. This website claims the total carbon content of crude oil is around 84%. That's not far from our calculation!
Total Crude Oil Carbon: 177.7 Pg crude x .8523 carbon/crude
= 152.2 Pg
Coal is basically fossilized peat. Because it is easy to collect, transport, and burn, it has been used as a fuel since antiquity. It is abundent and widely distributed around the world. The US Department of Energy estimates that the world's total recoverable coal reserve is about 998 billion short tons. Let's convert that to metric.
Total World Coal Reserves: 998 x 109 short tons x 2000 lbs/short ton / (2.2 lbs/kg)
= 907.3 Pg
The carbon content of coal is highly variable, depending on the kind of coal. The following table was generated from the Kentucky Educational Television, Australian Coal Association, and Environmental Literacy Council websites.
| Type of Coal | Carbon Content | Percent of World Resources | Total Carbon Mass |
|---|---|---|---|
| Lignite | 25% - 35% | 20% | 54.44 Pg |
| Sub-Bituminous | 35% - 45% | 28% | 101.6 Pg |
| Bituminous | 45% - 86% | 51% | 303.1 Pg |
| Anthracite | 86% - 98% | < 1% | < 8.35 Pg |
Total Coal Carbon: 467.5 Pg
Natural gas is the term for various kinds of gasseous hydrocarbons. Natural gas harvested from geological sources is largely made up of methane, with some ethane, propane and butate, as well as inert components such as nitrogen. The US Department of Energy lists world natural gas reserves at 6287 trillion cubic feet.
Total World Natural Gas Reserves: 6287 x 1012 ft3 x 28.32 L/ft3
= 178 TL
The following table looks at the natural gas and calculates the carbon content for each component. The percent makeup by mass is taken from the above DOE site.
| Component | Percent by Mass | Density (g/L) |
|---|---|---|
| Methane (CH4) | 70% - 90% | 0.717 |
| Ethane (C2H6) | 5% - 15% | 1.212 |
| Propane (C3H8) and Butane (C4H10) |
< 5% | 1.83 and 2.52 |
| Trace Non-Hydrocarbon Gases | 0% - 10% | - |
Let's consider the world's natural gas to be 80% methane and 10% ethane. For our purpose we will divide the remaining mass evenly -- 5% propane, 5% butane.
The math is reasoned as follows: We want the total mass of each of methane and ethane. We have the mass fractions for each, the density for each, and the total volume of natural gas. Mass (mi) can be calculated from mass fraction (fi) and total mass (mtot) as follows
mi = fi · mtot
Density is the ratio of the mass to the volume:
di = mi / Vi
Substitue for mi:
di Vi = fi mtot
Or in terms of Vi:
Vi = fi mtot / di
If we sum up the individual volumes we can get the total volume:
Vtot = V1 + V2 + ...
Write the individual volumes in terms of their individual mass fractions and densities, factoring out the total mass:
Vtot = mtot · ( f1/d1 + f2/d2 + ... )
Rewrite in terms of the total mass, for which we want to solve:
mtot = Vtot / ( f1/d1 + f2/d2 + ... )
If we substitute in the values we know, we can calculate our total mass of natural gas.
Total World Natural Gas Reserves: 178 TL / ( .8/0.717 + .1/1.212 + .05/1.83 + 0.05/2.52 )
= 142.9 Tg
Now we can go back and use the mass fractions combined with the carbon fractions to determine the total mass of carbon.
| Component | Mass | Carbon Fraction | Mass Carbon |
|---|---|---|---|
| Methane | 114.3 Tg | 74.868% | 85.59 Tg |
| Ethane | 14.29 Tg | 79.887% | 11.42 Tg |
| Propane | 7.145 Tg | 81.713% | 5.838 Tg |
| Butane | 7.145 Tg | 82.658% | 5.906 Tg |
Let's sum up that carbon!
Total Natural Gas Carbon: 98.75 Tg
To summarize, we have a lot of available carbon for burning:
| Total Crude Oil Carbon: | 152.2 Pg |
|---|---|
| Total Coal Carbon: | 467.5 Pg |
| Total Natural Gas Carbon: | 0.098 Pg |
| Total Available Carbon Reserves: | 619.8 Pg |
Now let's burn it! Each carbon atom gets combined with two oxygen atoms to form carbon dioxide. The atomic weight of carbon is still 12.011, with oxygen coming in at a hefty 15.999.
| Compound | Molar Weight of Carbon | Molar Weight of Oxygen | Total Molar Weight | Percent Carbon by Weight |
|---|---|---|---|---|
| CO2 | 12.011 | 31.998 | 44.009 | 27.292% |
So, if we take this and apply a little math, we will calculate exactly how much carbon dioxide can be released by burning the entire world's hydrocarbon reserves.
Total CO2 Released: 619.8 Pg carbon / .2729 (carbon/carbon dioxide)
= 2.271 Eg (exagrams)
= 2.271 million million metric tons
That's a lot of carbon dioxide!
Of course, Earth's atmosphere isn't small, either. According to NASA, the atmosphere consists of 5.1 x 1018 kg of matter. In terms of not-so-well-known SI units:
Total Mass of Earth's Atmosphere: 5.1 Zg (zettagrams)
The same site contains data for the atmospheric composition, which we can use to determine the fraction of the atmosphere which is carbon dioxide. CO2 is listed as 383 parts per million by volume.
Fraction of CO2 by Volume: 0.0383%
Since we have been working in terms of mass, we need to know how our per-volume fraction translates into a per-mass fraction. We can work out an approximation by noting that, for ideal gasses, the per-volume fraction is equivalent to the per-mole fraction. This means that 383 our of every million molecules in the atmosphere are CO2 molecules.
What is the average molecular weight of the components of air? NASA lists 28.97 g/mol. With that in mind we can calculate the fraction of CO2 by volume:
Atmospheric molecular weight: 28.97 g/mol
Carbon Dioxide molecular weight: 44.009 g/mol
Fraction of CO2 by mole: 0.000383 mol/mol
Fraction of CO2 by mass: 44.009 g/mol * 0.000383 mol/mol / (28.97 g/mol)
= 0.0005818 g/g
= 0.05818%
Now we can calculate the mass of atmospheric carbon dioxide.
Total Atmospheric CO2: 2.967 Eg
= 2.967 million million metric tons
So, according to our back-of-the-website calculations, if we burn all the world's available hydrocarbon reserves today, we will increase the atmospheric CO2 level by:
Increase in Atmospheric CO2: 2.271 Eg / 2.967 Eg
= 76.54%
What is the effect of an 77% increase in CO2? That is a much more difficult question to answer. Let's look at it in a couple different ways.
We can try to figure out the effect of this increase in atmospheric CO2 by looking at prehistoric levels of CO2. How do we do that? Scientists take samples of rock which was formed by compression of soil. Any air that was trapped in the soil when it was formed can be extracted, and the relative level of CO2 can be measured. I don't know how accurate the results are, but they make for a set of very pretty graphs at the Global Warming Art website.
An 77% increase in atmospheric CO2 would bring us to about 676ppmv. This is a level we might have expected to see during the Cretaceous period, when the world was warmer by about 4°C (7.2°F), according to The Geological Society of America.
The next set of calculations is almost certainly wrong. According to Beer's Law, the relationship between the percentage of radiation absorbed by a gas and the concentration of that gas is logorithmic, not linear. This implies that the actual increase in temperature will be less than the linear calculation. I'll work out the details when I have the time. ~JB
Another way we might figure out the effect of an increase in CO2 is to look at the effect of the current CO2 level and then calculate its increased effect. This relies on the assumption that an increase in CO2 concentration has a linear effect in the region of interest (from 383ppmv to 676ppmv). I don't see any reason why that assumption wouldn't hold, so I'll proceed... cautiously.
According to the US Environmental Protection Agency and Physical Geography, greenhouse gasses account for about 33°C of our current temperature. A couple of sites, including Real Climate state that CO2 causes about 25% of the effect.
Warming Effect of CO2: 33°C x 25%
= 8.25°C
Thus if we increase the CO2 levels by 77%, we should increase the effect of CO2 77%.
Temperature Increase Due to Increased CO2: 8.25°C * 76.54%
= 6.315°C (11.37°F)
This is reasonably in line with what we would expect from the geological record.
conclusions to follow...