Using differential equations and derivatives of functions, maybe, I think. I don't want to blow off your question, but it is something I too am very new to and I'm not really sure where to even start, but I'll give it a crack. I should defer to Watsisname
, who is probably a Mensa member and as glib as any bard I've known and could probably answer this poetically in his sleep. So lets dive in!
For a crash course in Hyper-physics concerning gases, humans, and the environment I've found these possibly helpful links.
- Differential Equations: http://hyperphysics.phy-astr.gsu.edu/hbase/diff.html#c4
- Atmospheric Pressure: http://hyperphysics.phy-astr.gsu.edu/hb ... n.html#atm
- The Barometric Formula: http://hyperphysics.phy-astr.gsu.edu/hb ... or.html#c1
- Respiration (Inspiration & Expiration): http://hyperphysics.phy-astr.gsu.edu/hb ... ir.html#c3
- Oxygen vs CO2 diffusion rates and solubility: http://hyperphysics.phy-astr.gsu.edu/hb ... ry.html#c5
- Gas Exchange in the Lungs & membrane transport by diffusion: http://hyperphysics.phy-astr.gsu.edu/hb ... ir.html#c3
So, where to begin? Let us first look at ground level atmospheric pressure.
You proposed a starting atmospheric composition of: 0.21 atm pressure, consisting of the following
O2 94.2945314% = 0.1980185 partial atm
CO2 5.4840000% = 0.1151640 partial atm
Ar 0.1102857% = 0.0002316 partial atm
N2 0.1080000% = 0.0002268 partial atm
CO 0.0031829% = 0.0000067 partial atm
I hope I am starting off on the right foot here.
The ground level atmospheric pressure on a planet is not fixed by any of the standard planetary data such as gravity and temperature. It depends on just how much atmospheric gas is there and available and has not yet escaped into space. For example, Venus is a little smaller than Earth and has slightly lower gravity, yet its atmospheric pressure is about 90 times that of Earth. Once the ground level pressure is decided on, the atmospheric pressure (P) at other altitudes is governed by the following formula:
P = Po exp((-g*h) / H)
Po = the ground level pressure
g = the surface gravity in units of Earth gravity
h = the altitude
H = the atmospheric scale height
On Earth the scale height is 7400 meters. On other worlds it will be:
H = (Ho*Mo) / M
Ho = the scale height on Earth (7400 meters)
Mo = the mean molecular weight of Earth's atmosphere (29 atomic mass units) †
M = the mean molecular weight of the planet's atmosphere
† For breathable nitrogen-oxygen atmospheres, M will be close to 29. (I didn't do the math for that, I just cheated and looked it up.) Now we can figure pressures at the top of Mount Olympus or at the bottom of Hellas Crater. Plus, for later on, don't forget temperature changes with elevation.
The surface of the earth is at the bottom of an atmospheric sea. The standard atmospheric pressure is measured in various units: 1 atmosphere = 760 mmHg = 29.92 inHg = 14.7 lb/in2 = 101.3 KPa
The fundamental SI unit of pressure is the Pascal (Pa), but it is a small unit so kPa is the most common direct pressure unit for atmospheric pressure. Since the static fluid pressure is dependent only upon density and depth, choosing a liquid of standard density like mercury or water allows you to express the pressure in units of height or depth, e.g., mmHg or inches of water. The mercury barometer is the standard instrument for atmospheric pressure measurement in weather reporting. The decrease in atmospheric pressure with height can be predicted from the barometric formula. (That comes next, below)
The unit mmHg is often called torr, particularly in vacuum applications: 760 mmHg = 760 torr. For weather applications, the standard atmospheric pressure is often called 1 bar or 1000 millibars. This has been found to be convenient for recording the relatively small deviations from standard atmospheric pressure with normal weather patterns.
The constituents of dry air (Water Vapor will come soon, below) can be expressed as volume percentages, as you have done in your starter atmosphere, which will translate to the partial pressures out of the total atmospheric pressure. What we are trying to figure out though is how much volume a certain gas takes up at some standard temperature and pressure and how much of it is actually there. That means we need to know it's mass or how many molecules of it are in a volume of space. That leads us to the unit of measure called the mole. This is also where I fell asleep in 6th period class.
So, lets ask, how many molecules of a gas, at zero degrees Celsius and at exactly 1 atmosphere of pressure will take up a standard volume? Let's also make the standard of volume equal to say... something easy to remember, like... 22.4 liters! There, now we can measure the number of molecules of the gasses, that you suggested at the beginning, that are in the air. This is important so we know how much gets into your blood, because now we know the air's molecular density. Baricity refers to the density of a substance compared to the density of human cerebrospinal fluid. I think this is how you can determine how much Oxygen enters the blood stream through the Alveolar membranes in the lungs too.
Probably the simplest model of an atmosphere is something like this:
The density ρ, temperature T, and pressure p depend only on altitude y.
The weight g of a unit mass does not depend on y.
(Because atmospheres are not thick compared to the planet's radius).
A balance of vertical forces on a horizontal slab of area A between y and y+Δy gives
which says that the pressure just balances the weight.
Divide by Δy and take the limit as Δy tends to 0 to get the basic formula p′(y)=−ρg
Where you go from here depends on what else you assume about the atmosphere.
- Example #1) constant density: p(y)=p(0)−ρgy
- Example #2) ideal gas with constant temperature: Rρ′(y)T=−ρg
and you get density decreasing exponentially with altitude, and find p(y)=p(0)e−gy/RT
- Example #3) ideal gas with temperature decreasing linearly with y (as in our troposphere): T(y)=T0−ky
Then you have a harder differential equation to solve, and find p(y)=p(0)[1−(ky/T0)]^[2−g/(kR)]
...and it's time for that cigarette!
I'm headed to the underground, fortified, radiation shielded, pressurized, warmed up, concrete bunker for a bit, back at PlutonianEmpire
I will revisit this around 3a.m. again when I have some time. I'll be looking at Water Vapor and Respiration then.
PS If you get bored until then, I think you will enjoy this solar system generator with random breathable atmospheres! http://fast-times.eldacur.com/StarGen/RunStarGen.html
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