Barometric Formula
Atmospheric pressure as a function of altitude.
Formula
Variables
Example
Air pressure roughly halves for every 5.5 km you climb in altitude.
How it works
The barometric formula tells you how atmospheric pressure falls with height in an isothermal atmosphere, meaning the air temperature is assumed constant. It comes from combining hydrostatic balance, where pressure decreases upward because air has weight, with the ideal gas law, which links density to pressure and temperature. The result is an exponential decay: as you climb, there is less air above you, so the pressure drops more and more slowly in absolute terms but by a roughly constant fraction per unit height.
When to use it
Use this formula when you need a quick estimate of pressure at altitude, especially in physics, meteorology, aviation, and mountain-environment calculations. It is most appropriate when temperature changes with height are small enough that treating T as constant is a reasonable approximation.
Worked example, step by step
Find the pressure at h = 2500 m if sea-level pressure is P0 = 101325 Pa, with M = 0.02897 kg/mol, g = 9.81 m/s^2, R = 8.314 J/(mol·K), and T = 288 K. Step 1: Substitute into the exponent: Mgh/(RT) = (0.02897 × 9.81 × 2500) / (8.314 × 288) Step 2: Calculate the numerator: 0.02897 × 9.81 × 2500 = 710.7 Step 3: Calculate the denominator: 8.314 × 288 = 2394.6 Step 4: Divide: 710.7 / 2394.6 ≈ 0.297 Step 5: Apply the exponential law: P = 101325 × e^(-0.297) Step 6: Evaluate the exponential: e^(-0.297) ≈ 0.743 Step 7: Final pressure: P ≈ 101325 × 0.743 ≈ 75,300 Pa So the atmospheric pressure at 2500 m is about 7.53 × 10^4 Pa, or 75.3 kPa.
Common mistakes to avoid
- Using the altitude in kilometres instead of metres without converting units
- Forgetting that the exponent must be dimensionless, so all units must be consistent
- Assuming the formula is exact for all heights even though real atmospheric temperature changes with altitude
Did You Know?
Air pressure roughly halves every 5.5 km of altitude — which is why airliner cabins must be pressurized.
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