# Global warming/climate change: an easy calculation

-May 22, 2013

Energy comes in many flavors: nutritional calories stored in chemical bonds, electrical energy, the mass-energy of Einstein’s special relativity, etc, and heat. The defining quality of energy is its potential to do mechanical work. Work is defined as the sum of the force exerted on an object multiplied by the distance the object moves under that force.

When you exercise, your muscles use energy stored in chemical bonds (e.g., adenosine triphosphate) to do the work of exercise. Combustion engines convert energy stored in the chemical bonds of fuel to heat, that heat expands gas, which exerts force on a piston that turns a crank and ultimately moves things.

The sun radiates heat-energy at the atmosphere and heats air.

Heat capacity is the amount of heat-energy required to raise the temperature of one gram of a given substance by one degree centigrade, 1C. The scale is set by water, whose heat capacity is 1.0 cal/gmC at standard temperature and pressure (i.e., at 0C and 1 atmosphere pressure). One calorie of heat raises one gram of water one degree C.

Coffee mugs stay cool because they have high heat capacity (and low thermal conductivity), the silver spoon you stir it with, on the other hand, heats right up because it has low heat capacity (and high thermal conductivity).

The heat capacity for air is 0.24 cal/gmC, so every calorie of heat absorbed by air raises its temperature by 0.24C. Of course, night and the presence of big cool oceans keep air temperature in a rather narrow range (especially if you live in California).

The heat capacity of so-called greenhouse gases have lower heat capacity than air. For example, carbon dioxide’s heat capacity is 0.20 cal/gmC. Since it’s 20% lower than that of air, carbon dioxide heats up 20% faster than air when exposed to the same amount of sunlight.

Nice clean air is about 78% nitrogen (N2), 21% oxygen (O2), 0.9% argon (Ar), 0.04% carbon dioxide, plus trace quantities of other gases.

In the last 150 years or so, the concentration of carbon dioxide has increased by about 40%.

The controversial question is whether or not humanity is the cause of that increase. Calculating how much carbon dioxide people have released by burning fossil fuels is a 10th grade science homework problem. Here’s what you need to do the calculation yourself to a precision of about 20%:

First, you’ll need to know how much fossil fuel people have burned in the last several decades. The internet is rife with statistics on how much oil and coal is consumed each year around the world. (If you prefer industry rather than government statistics, here’s one from BP).

Second, to calculate the amount of carbon dioxide produced per molecule of fuel you’ll have to dig up some 10th grade chemistry: remember how to use Avogadro’s number to calculate the carbon dioxide produced by burning a barrel of oil, recall that one Mole of gas molecules occupies a volume of 22.4 liters—which I think is magic—and dust off the memory that hydrocarbon fuel is made up of carbon chains blanketed in hydrogen. Their molecular structure is CnH2n+2  with n mostly between 5 and 12. I assumed that an average hydrocarbon chain is about 8 carbon atoms long.

Third, you need the volume of the atmosphere. I made two approximations that seemed reasonable to me: I assumed that the earth is a sphere of radius 6250 km. The equatorial radius is 6378 km, but the planet is a bit squished so I took a little off to get a nice average. Then I assumed the atmosphere is a uniform shell of constant air pressure out to about 9000 m, just above Mt. Everest. Of course, air pressure is hardly constant at increasing altitudes, but as it gets thinner it extends way past Everest’s summit. I went for a give-and-take approximation. You’re welcome to use a more realistic model; knock yourself out.

Photo: Here's a picture of a glacier I took in Alaska.

Now you’re ready to calculate how many carbon dioxide molecules humans have introduced to the atmosphere as far back as you trust the data. I went back to 1930 because I have a book of physical constants from 1930 that quotes carbon dioxide concentration and I love to pull if off the shelf.

The answer I got was that people have burned enough fuel since 1930 to increase the atmospheric concentration of carbon dioxide by at least 100%. I figure my uncertainty is about 20% for no reason other than that I used to be a physics professor and got pretty good at making this stuff up.

Measurements from NOAA and Scripps Institution of Oceanography show that the actual concentration has increased by about 35%. The earth’s natural buffer system easily accounts for the discrepancy between the amount introduced and the increased concentration through absorption in oceans, forests, and plants. The absorption due to the ocean can be reverse-engineered from the increase in the ocean’s acidity over the same time period because carbon dioxide and water combine to form carbonic acid, CO2 + H2O yields  H2CO3.

Humanity’s contribution to global warming/climate change is controversial. But it’s science, so you or anyone who took 10th grade science can calculate the contribution on the back of an envelope. Whether you use data from the Department of Energy or Exxon-Mobile won’t alter your results.

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