Atmospheric pressure

Calculating variation with altitude

See also: Barometric formula

There are two different equations for computing the average pressure at various height regimes below 86 km (or 278,400 ft). Equation 1 is used when the value of standard temperature lapse rate is not equal to zero and equation 2 is used when standard temperature lapse rate equals zero.

Equation 1:

${P}=P_{b}\cdot \left[{\frac {T_{b}}{T_{b}+L_{b}\cdot (h-h_{b})}}\right]^{\frac {g_{0}\cdot M}{R^{*}\cdot L_{b}}}$

Equation 2:

${P}=P_{b}\cdot \exp \left[{\frac {-g_{0}\cdot M\cdot (h-h_{b})}{R^{*}\cdot T_{b}}}\right]$

where

$P$ = Static pressure (pascals)

$T$ = Standard temperature (kelvins)

$L$ = Standard temperature lapse rate (kelvins per m)

$h$ = Height above sea level (meters)

$R^{*}$ = Universal gas constant: 8.31432×10³ N·m / (kmol·K)

$g_{0}$ = Gravitational constant (9.80665 m/s²)

$M$ = Molar mass of Earth's air (28.9644 g/mol)

Or converted to English units:

where

$P$ = Static pressure (inches of mercury)

$T$ = Standard temperature (kelvins)

$L$ = Standard temperature lapse rate (kelvins per ft)

$h$ = Height above sea level (feet)

$R^{*}$ = Universal gas constant (using feet and kelvins and gram moles: 8.9494596×10⁴ kg·sq ft·s^-2·K^-1·kmol^-1)

$g_{0}$ = Gravitational constant (32.17405 ft/s²)

$M$ = Molar mass of Earth's air (28.9644 g/mol)

The value of subscript b ranges from 0 to 6 in accordance with each of seven successive layers of the atmosphere shown in the table below. In these equations, g₀, M and R^* are each single-valued constants, while P, L, T, and h are multivalued constants in accordance with the table below. (Note that according to the convention in this equation, L₀, the tropospheric lapse rate, is negative.) It should be noted that the values used for M, g₀, and $R^{*}$ are in accordance with the U.S. Standard Atmosphere, 1976, and that the value for $R^{*}$ in particular does not agree with standard values for this constant. The reference value for P_b for b = 0 is the defined sea level value, P₀ = 101325 pascals or 29.92126 inHg. Values of P_b of b = 1 through b = 6 are obtained from the application of the appropriate member of the pair equations 1 and 2 for the case when $h=h_{b+1}$.:

Subscript b	Height Above Sea Level		Static Pressure		Standard Temperature (K)	Temperature Lapse Rate
	(m)	(ft)	(pascals)	(inHg)		(K/m)	(K/ft)
0	0	0	101325	29.92126	288.15	-0.0065	-0.0019812
1	11,000	36,089	22632	6.683245	216.65	0.0	0.0
2	20,000	65,617	5474	1.616734	216.65	0.001	0.0003048
3	32,000	104,987	868	0.2563258	228.65	0.0028	0.00085344
4	47,000	154,199	110	0.0327506	270.65	0.0	0.0
5	51,000	167,323	66	0.01976704	270.65	-0.0028	-0.00085344
6	71,000	232,940	4	0.00116833	214.65	-0.002	-0.0006097

Sample Calculation

Find the average pressure at 30,000 meters.

First note that 30,000 meters is above 20,000 but below 32,000 so it therefore falls in the range of subscript b=2 in the chart above. Also note that the temperature lapse rate for that region is not equal to zero; therefore, equation 1 is appropriate.

${P}=P_{2}\cdot \left[{\frac {T_{2}}{T_{2}+L_{2}\cdot (h-h_{2})}}\right]^{\frac {g_{0}\cdot M}{R^{*}\cdot L_{2}}}$

Or

${P}=5474.89\cdot \left[{\frac {216.65}{216.65+0.001\cdot (30,000-20,000)}}\right]^{\frac {9.80665\cdot 28.9644}{8314.32\cdot 0.001}}$

${P}=5474.89\cdot \left[{\frac {216.65}{226.65}}\right]^{34.163195}$

${P}=5474.89\cdot 0.214044$

${P}\ =1171.867$ Pascals at 30,000 meters

Local atmospheric pressure variation

Hurricane Wilma on 19 October 2005 – 88.2 kPa in eye

Atmospheric pressure varies widely on Earth, and these changes are important in studying weather and climate. See pressure system for the effects of air pressure variations on weather.

Atmospheric pressure shows a diurnal (twice-daily) cycle caused by global atmospheric tides. This effect is strongest in tropical zones, with amplitude of a few millibars, and almost zero in polar areas. These variations have two superimposed cycles, a circadian (24 h) cycle and semi-circadian (12 h) cycle.

Atmospheric pressure based on height of water

Atmospheric pressure is often measured with a mercury barometer, and a height of approximately 760 mm (30 inches) of mercury is often used to teach, make visible, and illustrate (and measure) atmospheric pressure. However, since mercury is not a substance that humans commonly come in contact with, water often provides a more intuitive way to conceptualize the amount of pressure in one atmosphere.

One atmosphere (101.325 kPa or 14.7 lbf/sq in) is the amount of pressure that can lift water approximately 10.3 m (33.9 ft). Thus, a diver at a depth 10.3 meters under water in a fresh-water lake experiences a pressure of about 2 atmospheres (1 atm for the air and 1 atm for the water). This is also the maximum height to which a column of water can be drawn up by suction.

Low pressures such as natural gas lines are sometimes specified in inches of water, typically written as w.c. (water column). A typical gas using residential appliance is rated for a maximum of 14 w.c. which is approximately 0.5 PSI.

Non-professional barometers are generally aneroid barometer (Figure 3) or strain gauge based. See Pressure measurement for a description of barometers.

Atmospheric pressure's relation to water's boiling point

Although water is generally considered to boil at 100° C (212° F), water actually boils when the vapor pressure is equal to the atmospheric pressure around the water. Because of this, the boiling point of water is decreased in lower pressure and raised at higher pressure. This is why baking cookies at altitudes beyond 3,500 feet above sea level requires special baking directions.

See also

• Plenum
• NRLMSISE-00
• Barometric formula
• International Standard Atmosphere - a tabulation of typical variation of principal thermodynamic variables of the atmosphere (pressure, density, temperature etc.) with altitude, at mid latitudes.

References

• US Department of Defense Military Standard 810E
• Burt, Christopher C., (2004). Extreme Weather, A Guide & Record Book. W. W. Norton & Company ISBN 0-393-32658-6
• U.S. Standard Atmosphere, 1962, U.S. Government Printing Office, Washington, D.C., 1962.

External links

• NASA page on the 1976 Standard Atmosphere
• Source code and equations for the 1976 Standard Atmosphere
• A mathematical model of the 1976 U.S. Standard Atmosphere
• Calculator using multiple units and properties for the 1976 Standard Atmosphere

Experiments

• An exercise in air pressure
• Movies on atmospheric pressure experiments from Georgia State University's HyperPhysics website - requires QuickTime