Standard Atmosphere Pressure from Altitude (11 to 20 km)

In the standard atmosphere the layer from 11 to 20 kilometres is isothermal: the air stays at 216.65 K, and the pressure falls by a constant fraction for each extra metre of height. That is why one exponential describes it. The formula is equation 33b of the U.S. Standard Atmosphere 1976 (COESA), $p = p_{11}\exp\left(-\frac{g_0\,(H - 11\,000)}{R\,T}\right)$, with $T = 216.65$ K, and it starts from the pressure $p_{11}$ of 22 632 Pa at 11 km, the value the troposphere formula of the same standard gives.

Each term has a plain meaning. The height climbed above the base of the layer is $H - 11\,000$, so at 11 000 m the exponent vanishes and the formula returns $p_{11}$. The gravity $g_0$ is 9.80665, and $R$ is the specific gas constant of air, 287.053 J/(kg K), which is 8.31432 divided by 0.0289644. The ratio $R\,T/g_0$ is a length: it appears in the inverse formula for $H$, and each time the height gains that length the pressure is multiplied by the same factor. The altitude $H$ is the geopotential altitude the standard tabulates, which at 15 km is 35 m lower than the geometric height.

The standard's own Table I, in geopotential metric units, gives 226.32 hPa at 11 km, 193.30 at 12 km, 141.01 at 14 km and 120.44 at 15 km, and the page reproduces those to the five digits the table prints. It also runs backwards: the altitude at which the standard pressure is 120.44 hPa is 15 000 m, within the table's rounding of 0.2 m of height. Divide the pressure by $R\,T$ and you have the density at the same altitude.

Use it only between 11 000 and 20 000 m. Below 11 km the standard has a temperature gradient of -6.5 K/km and a different formula applies; from 20 km the gradient is +1.0 K/km and the layer no longer has a constant temperature. Most airliners cruise between 10.7 and 12.5 km, so this is the pressure of the flight level above the point where the library's tropospheric standard stops. The real atmosphere is only the standard on average: pressure at a given height varies by several percent with the weather, and the flight level is set by the pressure, not by the height.