Atmospheric Pressure & the Barometric Formula

simulator beginner ~8 min
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540 hPa at 5 km — roughly half sea-level pressure

At 5 km altitude with standard sea-level conditions (1013 hPa, 15°C, 6.5°C/km lapse rate), atmospheric pressure drops to approximately 540 hPa — about 53% of the surface value.

Formula

Barometric formula: P(h) = P₀ × (1 - Lh/T₀)^(gM/(RL))
Isothermal approximation: P(h) = P₀ × exp(-Mgh/(RT))
Scale height: H = RT / (Mg) ≈ 8.5 km for standard atmosphere

Why Pressure Falls With Altitude

Atmospheric pressure at any point is simply the weight of the air column above it. As you ascend, there is less air above you, so the pressure drops. Near sea level, each cubic metre of air weighs about 1.2 kg, but at 10 km altitude, air density is only a third of that. The barometric formula quantifies this relationship, producing the characteristic exponential decay curve that governs everything from aircraft performance to mountain cooking.

The Barometric Formula

The barometric formula comes in two flavours. The isothermal version — P = P₀ exp(-Mgh/RT) — assumes constant temperature, which is a decent approximation for thin layers. The more realistic hypsometric version accounts for the temperature lapse rate (typically 6.5°C/km in the troposphere): P = P₀ × (1 - Lh/T₀)^(gM/RL). This simulation implements the hypsometric version and shows you the pressure, density, and boiling point at any altitude up to the stratosphere.

Practical Consequences

The pressure-altitude relationship has profound practical consequences. Aircraft altimeters work by measuring pressure and converting it to altitude via the barometric formula — which is why pilots must adjust for local pressure variations. Mountaineers above 8 km enter the 'death zone' where oxygen partial pressure is too low for sustained human survival. At 18 km (the Armstrong limit), pressure is so low that body fluids boil at 37°C without a pressure suit.

Pressure and Weather Systems

Variations in sea-level pressure drive the wind patterns we experience as weather. High-pressure systems (anticyclones, >1020 hPa) produce clear skies as air descends and warms. Low-pressure systems (cyclones, <1000 hPa) produce clouds and rain as air converges and rises. The tightest pressure gradients produce the strongest winds — a Category 5 hurricane can have a central pressure below 920 hPa, nearly 100 hPa below the surrounding environment.

FAQ

How does atmospheric pressure change with altitude?

Atmospheric pressure decreases approximately exponentially with altitude. Near sea level, pressure drops about 12 hPa per 100 metres of elevation gain. The rate of decrease slows with altitude because the air above becomes progressively thinner. The barometric formula captures this relationship: P = P₀ × (1 - Lh/T₀)^(gM/RL), where L is the lapse rate and h is altitude.

What is the barometric formula?

The barometric formula relates atmospheric pressure to altitude by assuming the atmosphere is in hydrostatic equilibrium (gravity balances the pressure gradient). In its simplest form, P = P₀ × exp(-Mgh/RT), where M is the molar mass of air, g is gravity, h is height, R is the gas constant, and T is temperature. The more accurate hypsometric version accounts for temperature changing with altitude.

Why does water boil at lower temperatures at high altitude?

Water boils when its vapor pressure equals the surrounding atmospheric pressure. At lower pressures, less energy is needed for water molecules to escape into the gas phase. At 5 km altitude (roughly 540 hPa), water boils at about 83°C instead of 100°C. At the summit of Everest (8,849 m), it boils near 70°C — making it impossible to cook food properly.

What is scale height in atmospheric physics?

Scale height (H) is the altitude over which pressure decreases by a factor of e (≈2.718). For Earth's atmosphere at 15°C, H ≈ 8.5 km. It provides a convenient single number to characterize how rapidly pressure falls: after one scale height, pressure is 37% of its surface value; after two, about 13.5%; after three, about 5%.

Sources

Embed

<iframe src="https://homo-deus.com/lab/meteorology/atmospheric-pressure/embed" width="100%" height="400" frameborder="0"></iframe>
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