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Physics/Air pressure and temperature

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Dear Steve,

Suppose there is very deep yet open mine. Not a pit mine like this one: http://cnx.org/content/m41470/latest/graphics3.jpg rather a well mine like this http://markanstice.files.wordpress.com/2014/01/mining2-010.jpg
only much, much deeper. Suppose there are no trouble factors involved (no water layers, no nearby volcanoes, no sands, just a vertical tunnel dug out in a solid rock. Suppose that neither the sunshine nor the rain ever reach the interior of this tunnel (there is a protective roof built above it) so that they do not influence the air inside. Suppose that the temperature outside has a very low yearly amplitude, like say on the Azores Islands, it never goes lower than +10 C, and never goes higher than +20 C so we may assume that it has a constant temperature of +15 C, which is told to be the average temperature of the Earth planet. Suppose that contrary to Azores, there are no strong winds which could influence the air in the mine as well.  

My terminology might not be 100% correct, but I hope you can understand me. I am not a native English speaker and I live in a coutry where English is rarely spoken.

Now my questions:
How deep has the mine to be so that the air pressure at the bottom be exactly 2 atm, while on the surface of the mine, on the sea level, there is exactly 1 atm pressure? What temperature would there be on the bottom of this mine if the temperature outside is +15 C?

This is not a homework question (I am 73 years old), I simply don't know how to calculate this. If, apart form the pure results of the calculation, you could provide me with some equations that could help me understand how this is calculated, I would be really grateful.

Thank you and all the best to you,

Sincerely yours,

Mathew

Answer
Hello.  Actually, your command of English is superior to many people I meet here.  In the atmosphere, this is a fairly well-understood calculation (matched with measurement) that results in what is known as the adiabatic lapse rate.  Your problem is somewhat more complex, becuase you're talking about going below the surface of the Earth.  In any such environment, temperature will be determined by the surrounding material in a practical (well, not very practical) situation like you describe.

If you want to make some further (mostly correct) simplifying assumption, you can simplify this greatly.  That assumption is that since you're only going to 2 atm, you're probably not going that deep relative to the Earth's radius.  This means you can ignore the slightly lower gravity.  Using either the adiabat I saw for dry air on wikipedia (an emagram that I used to go to 50% atmospheric, since doubling going down should be the same as going to 50% while going up) or this calculator: http://www.altitude.org/air_pressure.php I get the same answer.  The adiabatic lapse rate chart indicates that you will achieve such a pressure at approximately 6500m deep (almost exactly 1/1000th of the radius of the Earth).  The calculator says -6700m.   I used a couple quick tricks to extrapolate almost exactly the same result from engineeringtoolbox.com's data, which goes smoothly down to -1500m and seems to be consistent.  

Based on surface temperatures around 15 degrees C, and using another type of lapse rate calculator here: http://www.shodor.org/os411/courses/_master/tools/calculators/lapserate/ we result in a temperature of nearly 80 C...that's quite warm for humans to survive in.  It's perhaps possible in the deep ocean in a giant closed-ended tube where the water temperatures could cool the environment, but the resultant heat flow into the tube and out into the water would seriously mess with the adiabatic conditions.  That's when things get nearly impossibly complex, I might only think of being given such a problem in an advanced graduate thermodynamics course as pure conjecture.  But below what is known as the thermocline in the ocean, temperatures hover just a few degrees above 0 C.  It would be much hotter as an actual well, where temperatures (oil company data is abundant on this subject with a quick search) increase at about 29 C/km depth.

Hope that helps.

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Dr. Stephen O. Nelson

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I can answer most basic physics questions, physics questions about science fiction and everyday observations of physics, etc. I'm also usually good for science fair advice (I'm the regional science fair director). I do not answer homework problems. I will occasionally point out where a homework solution went wrong, though. I'm usually good at explaining odd observations that seem counterintuitive, energy science, nuclear physics, nuclear astrophysics, and alternative theories of physics are my specialties.

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I was a physics professor at the University of Texas of the Permian Basin, research in nuclear technology and nuclear astrophysics. My travelling science show saw over 20,000 students of all ages. I taught physics, nuclear chemistry, radiation safety, vacuum technology, and answer tons of questions as I tour schools encouraging students to consider careers in science. I moved on to a non-academic job with more research just recently.

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Ph. D. from Duke University in physics, research in nuclear astrophysics reactions, gamma-ray astronomy technology, and advanced nuclear reactors.

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