Physics of marriage and hypoxia

Abstract:

Got married, took a hike,

got high, got dehydrated and almost suffocated.

oh and a gerbil.

I lied.
I said the next post would deal with how being fat can save your life, and there are actually several ways that can happen. but that will have to wait for another instance.

A couple of years ago I got married.
Yes. I, like loads of better men than I, donned the world's smallest handcuffs, and walked the "green mile" as it were of single men, to happy cheers of "dead man marrying".

his: +5 ring vs. reason&accountability.
hers: one ring to rule them all and in the darkness bind them.

Well, actually I count myself lucky. most men get fucked on the deal, but as for myself, I got myself a ninja for a wife and a Jedi knight to boot. You should see her performing Jedi mind tricks on me or using The Voice on people - damn she's good!

Anyway after about a year of marriage we decided it's time to go for our much overdue honeymoon.
so somehow we came up with the obscene amount of money it now costs to fly to the USA, and we went to visit grandma, the family and take a hike. no really. we traveled California's national parks and hiked some...
at some point we even had oxygen shortage due to height sickness.

What had happened was, we trekked from Camp White Wolf to Ten lakes in Yosemite national park, and back, in one day.
Now Camp White Wolf is at \(\approx 2,400 m\), and the mountain overlooking Ten lakes is at \(\approx 3000 m \). so all in all it doesn't seem like a distinctive height difference right?
BUT (and it's a big butt for sure!) being overzealous and not so experienced hikers we spent the night near Briceburg which is at \(\approx 350 m\).

Now for a bit of physics:
Humans usually breath air which is comprised among other things of roughly 21% \(O_2\).
Assuming air is an ideal gas we get :
\[PV=NRT\]
Which means in a cubicle meter at sea level pressure, at \(298^\circ K\), we get \(\approx 40.9 {} moles \) of air which translates roughly to \(8.6\text{ moles of }O_2\).

Now we will consider a thin spherical layer \(\Delta r\) of gas:
\[ A(r)\cdot P(r)-P(r+\Delta r) = V\cdot\rho (r) \frac{GM_{\oplus}}{r^2} \]
\[\Rightarrow -4\pi r^2 \Delta r \frac{\partial P}{\partial r}=4\pi r^2 \Delta r \rho (r) \frac{GM_{\oplus}}{r^2} \Rightarrow -\frac{\partial P}{\partial r}= \rho (r) \frac{GM_{\oplus}}{r^2}\]

For all intents and purposes the gravitational acceleration on the earth's surface and at 3 km differs by a factor of \(1 \pm 10^{-4}\) so really there's no need to take into account the gravitational change in that small a difference in distance.
Thus we will treat the gravitational potential as \(\Phi_N=gr\), and thus the force per kg is directed downwards and is given by \(F_g=g\hat{r}\).
\[\rho = \frac{N\cdot m}{V} \Rightarrow \rho=\frac{Pm}{RT}\]
and so we quickly get:
\[\frac{dP}{P}= -\frac{gm}{RT}dr\]
Which assuming a constant temperature (which is a WRONG assumption as we will see immediately) we get:
\[ P=P_0\,e^{-\frac{gmr}{RT}} \]
Where \(m\) denotes the molar mass of air, \(R\) the ideal gas constant, and \(T\) the temperature.

Now using this formula we get that the air pressure at 3000 m above sea level is \(\approx 52595\) pa. which means the amount of moles in a cubic meter in 3000 meters of height is 21.2 moles. keeping in mind that \(O_2\) is only slightly heavier the \(N_2\) we allow ourselves the assumption that the \(N_2\sim O_2\) ratio in 3000 meters is similar to that of about ground level we get that the air up there has only ~11% oxygen in the same volume.
Just for the sake of good form - at 15-19% you get decreased ability to work strenuously.  impaired coordination may occur and there is a chance of induced early symptoms with individuals that have coronary, pulmonary, or circulatory problems. at 12-15% respiration and pulse increases; impaired coordination, perception, and judgment occurs. , at 10-12% respiration further increases in rate and depth; poor judgment and bluish lips occur. At 8-10% symptoms include mental failure, fainting, unconsciousness, an ash-colored-face, blue lips, nausea, and vomiting.

you get the picture right?

So 11% is quite close to being knee-deep in shit.

Let's have a more realistic estimate though, because we know the temperature decreases with height (at least in the atmosphere).

So basically the temperature is a physical function and we like to think of those as analytical so the first approximation of \(T\) around \(T_0\) is given generally by:
\[T(r)\approx T_0 - ar\]
The minus sign owing to our understanding of temperature decreasing with height.
and so the above differential equation becomes:
\[\frac{dP}{P}= -\frac{a\,gm\,dr}{a\,RT_0\left(1-\frac{ar}{T_0}\right)} \Rightarrow P=P_0\left(1-\frac{ar}{T_0}\right)^{\frac{gm}{aR}}\]
And so using the corrected formula with \(a\) denoting the linear-approximate rate of temperature decline in the atmosphere which is given by \(a=6.5 \cdot 10^{-3}\) we get the pressure up there was about 70970 pa which is much likelier, whereas the effective air percentage there is almost 15%.

Meaning that we we're only in an ankle-high depository.

Just as an additional reference, in normal circumstances the oxygen percentage in our exhaled breath is about 16%, and in some circumstances could go as low as 14%.
So imagine yourselves in a hot, non-ventilated room, stuffed with cheap foreign workforce, and imagine the wicked headache you'll get. Now multiply it, as we were climbing up at a quick pace thus taking even shallower and quicker breaths.

At some point we actually sat there at the top of the mountain, breaking for lunch, not realizing we are only making matters worse...

The payoff was worth it though - it's such a beautiful view up there - it's AMAZING!

Ten Lakes view, headache and dehydration included.

Anyway, on our way back we also came to the undeniable conclusion we were idiots, as we ran out of water, we had a pounding headache, and we had to virtually run all the way back to Camp White Wolf before dark, since we were really afraid to walk in the woods at night, with no light, no fire, no reception, and loads of wolves howling in the distance.

We got there in the nick of time,  the sun was already setting as we hit the marked trail back to camp, about 20 minutes later, in full dark, we got there.
Sore-foot, dead-tired and otherwise completely wretched, we decided that we are NOT making camp to sleep on cold bare ground. I drove us to a nearby motel outside the national park, and we took a shower, and slept the deepest sleep achievable by mankind...


oh, and a gerbil:

Gerbil

Next time: Lenses and idiots.