Thermodynamics & Continuum Mechanics: Atmospheric Pressure & Barometric Hypsometry
In fluid mechanics, meteorology, and thermodynamics, pressure ($P$) is defined as the perpendicular normal force ($F_\perp$) exerted per unit surface area ($A$) of an enclosed boundary:
$$P = \lim_{\Delta A \to 0} \frac{\Delta F_\perp}{\Delta A} = \frac{dF_\perp}{dA}$$
Atmospheric pressure (barometric pressure) represents the hydrostatic weight per unit area of the column of terrestrial air extending from a reference elevation to the top of the atmosphere. At sea level, Earth's standard atmospheric pressure is universally calibrated to $101,325\text{ Pascals}$ ($101.325\text{ kPa}$ or $1.01325\text{ bar}$).
1. Hydrostatic Fundamental Equation
For an incompressible fluid of uniform density $\rho$ subject to gravitational acceleration $g$, pressure increases linearly with depth $h$:
$$P = P_0 + \rho g h$$
This equation forms the mathematical operating basis of mercury and liquid manometers. In 1643, Evangelista Torricelli demonstrated that atmospheric pressure supports a column of liquid mercury ($\rho_{\text{Hg}} \approx 13,595.1\text{ kg/m}^3$) to a height of exactly $760\text{ mm}$ at $0^\circ\text{C}$, establishing the conventional unit Torr ($1\text{ Torr} \equiv 1\text{ mmHg} \approx 133.322\text{ Pa}$).
2. The Barometric Formula & Hypsometric Elevation Modeling
Unlike liquids, atmospheric air is a compressible ideal gas governed by the ideal gas law $\rho = \frac{P \cdot M}{R \cdot T}$, where $M$ is molar mass of dry air ($M \approx 0.0289644\text{ kg/mol}$), $R$ is the universal gas constant ($8.31446\text{ J/(mol}\cdot\text{K)}$), and $T$ is absolute thermodynamic temperature in Kelvin.
Combining the differential hydrostatic condition $dP = -\rho g \, dh$ with the ideal gas density equation yields:
$$\frac{dP}{P} = -\frac{M g}{R T} \, dh$$
Assuming an isothermal atmosphere of constant temperature $T$:
$$P(h) = P_0 \exp\left( -\frac{M g h}{R T} \right) = P_0 \exp\left( -\frac{h}{H_s} \right)$$
where $H_s = \frac{R T}{M g} \approx 8,400\text{ meters}$ represents the atmospheric scale height—the vertical distance over which atmospheric pressure drops by a factor of Euler's constant $e \approx 2.71828$.
3. Comprehensive Pressure Unit Definitions & Conversion Matrix
The SI base derived unit of pressure is the Pascal ($1\text{ Pa} = 1\text{ N/m}^2 = 1\text{ kg}/(\text{m}\cdot\text{s}^2)$). Standard engineering conversions include:
- Standard Atmosphere ($atm$): $$1\text{ atm} \equiv 101,325\text{ Pa} = 1.01325\text{ bar}$$
- Bar ($bar$): Metric industrial standard: $$1\text{ bar} \equiv 100,000\text{ Pa} = 100\text{ kPa} = 0.1\text{ MPa}$$
- Hectopascal / Millibar ($hPa$ / $mbar$): Meteorological standard: $$1\text{ hPa} \equiv 100\text{ Pa} = 1\text{ mbar}$$
- Pounds per Square Inch ($psi$): US Imperial standard ($1\text{ lbf/in}^2$): $$1\text{ psi} \approx 6,894.757\text{ Pa} \implies 1\text{ atm} \approx 14.69595\text{ psi}$$
- Inches of Mercury ($inHg$): Aviation altimeter setting standard: $$1\text{ inHg} \approx 3,386.389\text{ Pa} \implies 1\text{ atm} = 29.9213\text{ inHg}$$
- Technical Atmosphere ($at$): $$1\text{ at} \equiv 1\text{ kgf/cm}^2 \approx 98,066.5\text{ Pa}$$
4. Clausius-Clapeyron Relation & Water Boiling Point Variation
As atmospheric pressure drops with altitude, the boiling point of liquids decreases according to the Clausius-Clapeyron equation:
$$\ln\left(\frac{P_2}{P_1}\right) = -\frac{\Delta H_{\text{vap}}}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right)$$
At sea level ($101.325\text{ kPa}$), water boils at $100^\circ\text{C}$ ($212^\circ\text{F}$). At the summit of Mount Everest ($8,848\text{ m}$, $P \approx 33.7\text{ kPa}$), the boiling point drops to approximately $71^\circ\text{C}$ ($160^\circ\text{F}$), preventing conventional culinary cooking.