Understanding the Barometric Formula

The barometric formula expresses how atmospheric pressure decreases with altitude in a planet’s gravitational field. Starting from the condition of hydrostatic equilibrium, where the downward weight of a thin layer of air equals the upward pressure difference across it, we write $\mathrm{dP}=-\rho g\mathrm{dz}$. Here $P$ is pressure, $\rho$ is density, $g$ is gravitational acceleration, and $z$ is height. Combining this with the ideal gas law $\rho =\frac{P}{\mathrm{R\; T}}$ for a gas of mean molecular mass $M$ leads to a differential equation whose solution is an exponential profile.

Assuming temperature remains constant with altitude, integration yields $P=P{₀}_{}{e}^{-\frac{\mathrm{M\; g\; h}}{\mathrm{R\; T}}}$. The constants represent the universal gas constant $R$, gravitational acceleration $g$, average molar mass $M$, and geometric altitude $h$. Because the exponent is negative, pressure drops rapidly with elevation: an increase of about 5.5 kilometers approximately halves surface pressure on Earth.

While a perfectly isothermal atmosphere is an approximation, the expression provides a remarkably good first-order description of real conditions up to the lower stratosphere. In reality temperature decreases roughly linearly with altitude in the troposphere before stabilizing aloft, and the full International Standard Atmosphere (ISA) uses piecewise polytropic layers rather than a single constant temperature. Nevertheless, the simple exponential relation captures the essential physics for many educational and engineering calculations.

Using the Calculator

This tool evaluates the equation above for user-specified altitude, sea‑level pressure, and temperature. The inputs are flexible, permitting calculations for other planets by substituting an appropriate mean molecular mass or gravitational acceleration in the script. For Earth the defaults assume dry air with $M=0.0289644$ kg/mol, $g=9.80665$ m/s², and a standard temperature of $288.15$ K.

After pressing the calculate button, the script computes the exponential factor and multiplies by the provided sea‑level pressure. The result is displayed in pascals and in hectopascals (millibars) to allow easy comparison with weather reports. Because no external libraries are used, all arithmetic is performed locally in your browser, ensuring privacy and portability even when offline.

Example Pressure Table

The table below lists typical pressures predicted by the isothermal barometric equation for several common elevations when using the default constants. Values are rounded to the nearest whole pascal.

Altitude (m)	Pressure (Pa)	Pressure (hPa)
0	101325	1013
1000	89876	899
5000	54019	540
10000	26436	264

These figures illustrate the sharp decline in available oxygen with height. At the cruising altitude of commercial jetliners, pressure drops to roughly one quarter of its sea‑level value, necessitating cabin pressurization to protect passengers from hypoxia.

For mountaineers and pilots, knowing pressure at altitude helps predict weather trends, determine required oxygen supplementation, and calibrate altimeters. Meteorologists apply the same formula—albeit with temperature variations—to convert surface pressure readings to mean sea level, allowing comparison between stations at different elevations. Engineers designing pressurized vessels or environmental control systems also rely on barometric estimates when calculating leakage rates or structural loads.

Despite its simplicity, the barometric formula encapsulates fundamental principles of thermodynamics and mechanics. It embodies the interplay between gravitational force, molecular weight, and thermal agitation. As height increases, fewer molecules reside above a given layer to exert downward force, so pressure decreases. Warmer air expands, increasing scale height and slowing the drop‑off, while heavier gases or stronger gravity compress the atmosphere more tightly. Understanding these dependencies aids in planetary science, aeronomy, and even astrophysics, where similar equations describe gas distribution in stellar atmospheres.

Advanced models refine the calculation by incorporating temperature gradients, humidity, and variation in gravitational acceleration with altitude. For example, the full ISA uses a lapse rate of 6.5 K/km up to 11 km, leading to a power‑law rather than exponential pressure decrease. Nonetheless, the isothermal approximation remains a cornerstone of atmospheric physics education, offering a clear demonstration of how simple assumptions yield insightful quantitative predictions. By experimenting with the inputs above—perhaps adjusting temperature to simulate a hot day or changing the gas constant to represent Mars—you can explore how different worlds shape their atmospheres.