The isothermal atmosphere

As a first approximation, let us assume that the temperature of the atmosphere is uniform. In such an isothermal atmosphere, we can directly integrate the previous equation to give

$$p = p_0 \exp\left(-\frac{z}{z_0}\right).$$ (327)

Here, $p_0$ is the pressure at ground level ($z=0$), which is generally about 1 bar, or 1 atmosphere ($10^5$ N${\rm m}^{-2}$ in SI units). The quantity

$$z_0 = \frac{R\,T}{\mu \,g}$$ (328)

is called the isothermal scale-height of the atmosphere. At ground level, the temperature is on average about 15$^\circ$ centigrade, which is 288$^\circ$ kelvin on the absolute scale. The mean molecular weight of air at sea level is 29 (i.e., the molecular weight of a gas made up of 78% Nitrogen, 21% Oxygen, and 1% Argon). The acceleration due to gravity is $9.81\,{\rm m}\,{\rm s}^{-2}$ at ground level. Also, the ideal gas constant is $8.314$ joules/mole/degree. Putting all of this information together, the isothermal scale-height of the atmosphere comes out to be about $8.4$ kilometers.

We have discovered that in an isothermal atmosphere the pressure decreases exponentially with increasing height. Since the temperature is assumed to be constant, and $\rho\propto p/T$ [see Eq. (325)], it follows that the density also decreases exponentially with the same scale-height as the pressure. According to Eq. (327), the pressure, or density, decreases by a factor 10 every $\ln \!10\, z_0$, or 19.3 kilometers, we move vertically upwards. Clearly, the effective height of the atmosphere is pretty small compared to the Earth's radius, which is about $6,400$ kilometers. In other words, the atmosphere constitutes a very thin layer covering the surface of the Earth. Incidentally, this justifies our neglect of the decrease of $g$ with increasing altitude.

One of the highest points in the United States of America is the peak of Mount Elbert in Colorado. This peak lies $14,432$ feet, or about $4.4$ kilometers, above sea level. At this altitude, our formula says that the air pressure should be about $0.6$ atmospheres. Surprisingly enough, after a few days acclimatization, people can survive quite comfortably at this sort of pressure. In the highest inhabited regions of the Andes and Tibet, the air pressure falls to about $0.5$ atmospheres. Humans can just about survive at such pressures. However, people cannot survive for any extended period in air pressures below half an atmosphere. This sets an upper limit on the altitude of permanent human habitation, which is about $19,000$ feet, or $5.8$ kilometers, above sea level. Incidentally, this is also the maximum altitude at which a pilot can fly an unpressurized aircraft without requiring additional Oxygen.

The highest point in the world is, of course, the peak of Mount Everest in Nepal. This peak lies at an altitude of $29,028$ feet, or $8.85$ kilometers, above sea level, where we expect the air pressure to be a mere $0.35$ atmospheres. This explains why Mount Everest was only conquered after lightweight portable oxygen cylinders were invented. Admittedly, some climbers have subsequently ascended Mount Everest without the aid of additional oxygen, but this is a very foolhardy venture, because above $19,000$ feet the climbers are slowly dying.

Commercial airliners fly at a cruising altitude of $32,000$ feet. At this altitude, we expect the air pressure to be only $0.3$ atmospheres, which explains why airline cabins are pressurized. In fact, the cabins are only pressurized to $0.85$ atmospheres (which accounts for the ``popping'' of passangers ears during air travel). The reason for this partial pressurization is quite simple. At $32,000$ feet, the pressure difference between the air in the cabin and that outside is about half an atmosphere. Clearly, the walls of the cabin must be strong enough to support this pressure difference, which means that they must be of a certain thickness, and, hence, the aircraft must be of a certain weight. If the cabin were fully pressurized then the pressure difference at cruising altitude would increase by about 30%, which means that the cabin walls would have to be much thicker, and, hence, the aircraft would have to be substantially heavier. So, a fully pressurized aircraft would be more comfortable to fly in (because your ears would not ``pop''), but it would also be far less economical to operate.