Altitude Pressure Calculator

In the lowest layer of the atmosphere, from sea level to 11 km, the 1976 standard makes temperature fall in a straight line and pressure follows from it:

T = 288.15 K - 0.0065 K/m x H P = 101,325 Pa x (T / 288.15 K)^5.2559

The exponent is g0 x M0 / (R* x L): standard gravity 9.80665 m/s2, the molar mass of sea-level air 0.0289644 kg/mol, the standard's gas constant 8.31432 J/(mol K), and the lapse rate L of 0.0065 K per meter. H is geopotential altitude, which is the one subtlety in the formula. Gravity weakens as you go up, so lifting air a meter at 20 km takes less work than at sea level, and the standard measures height in "geopotential meters" of equal work. It converts from ordinary height Z with H = r0 x Z / (r0 + Z), where r0 = 6,356,766 m. The difference is 0.16 m at 1 km and about 63 m at 20 km: small, but enough to miss the document's printed tables if it is ignored.

A worked example at 5,000 m above sea level. H = 6,356,766 x 5,000 / 6,361,766 = 4,996.07 m'. T = 288.15 - 0.0065 x 4,996.07 = 255.68 K, which is -17.47 °C. P = 101,325 x (255.68 / 288.15)^5.2559 = 54,048 Pa, or 540.48 hPa, 53.3% of sea-level pressure. Density is P x M0 / (R* x T) = 0.7364 kg/m3, and water boils there at 83.3 °C. The 1976 document's own Table I prints 540.48 mb, 255.676 K and 0.73643 kg/m3 for 5,000 m, and this calculator reproduces it to the last printed figure.

Above 11 km the temperature stops falling. From 11 to 20 km' it holds at 216.65 K (-56.5 °C), and with no temperature change the pressure equation becomes an exponential: P = Pb x exp(-g0 x M0 x (H - Hb) / (R* x Tb)), where Pb and Tb are the values at the bottom of the layer. Above 20 km the air warms again through the stratosphere, then cools through the mesosphere; the standard describes the whole column up to 86 km as seven layers, each with a fixed temperature gradient, and every layer's base pressure follows from the one below. This calculator uses all seven up to 80 km.

The boiling point is a second calculation on the first. Water boils when its vapor pressure equals the pressure of the air above it, so lower pressure means a lower boiling point. The page uses the saturation-temperature equation from IAPWS-IF97, the international formulation for water and steam, which is valid from water's triple point (611.657 Pa) upward. Near the ground the boiling point drops about 1 °C for every 300 m of height: 99.97 °C at standard sea-level pressure (the round 100 belongs to an older definition of the Celsius scale), 94.6 °C at one mile, 90.0 °C at 3,000 m. Food cooked in boiling water cooks at that lower temperature however hard the pot boils, which is why simmering times lengthen at altitude.

Two common mistakes. The first is reading a weather app at altitude and taking its pressure as the pressure around you: weather services report pressure reduced to sea level so maps are comparable, which is why a forecast in a mountain town still says something near 1013 hPa. The standard figure in this calculator is the actual air pressure at the height. The second is assuming the air "has less oxygen" up high in the sense of a smaller percentage. The composition is the same 20.9% oxygen; it is the pressure that falls, so each breath of the same volume holds fewer molecules of everything.

What this does not model. The standard is a yearly average for mid-latitudes. A real day's pressure at a given height moves a few percent with the weather (entering today's sea-level reading scales the column to match it), and the real temperature can sit well away from the standard's, which changes density and, slightly, the pressure higher up. Humidity, latitude and season are not in the model. It is the reference that altimeters and aircraft performance tables are built on, not a forecast.