Barometric formula - Citizendia

The barometric formula, sometimes called the exponential atmosphere or isothermal atmosphere, is a formula used to model how the pressure (or density) of the air changes with altitude. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) Temperature and layers The temperature of the Earth's atmosphere varies with altitude the mathematical relationship between temperature and altitude varies among five An isothermal process is a Thermodynamic process in which the Temperature of the System stays Constant: &Delta T = 0 An atmosphere (from Greek ατμός - atmos, " Vapor " + σφαίρα - sphaira, " Sphere " In Mathematics and in the Sciences a formula (plural formulae, formulæ or formulas) is a concise way of expressing information Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different Altitude is the Elevation of a point or object from a known level or datum (plural data

1 Pressure equations
2 Density equations
3 Derivation
4 References
5 See also

Pressure equations

See also: Atmospheric pressure

There are two different equations for computing pressure at various height regimes below 86 km (or 278,400 feet). 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 meter)

h

= Height above sea level (meters)

R *

= Universal gas constant for air: 8. The kelvin (symbol K) is a unit increment of Temperature and is one of the seven SI base units The Kelvin scale is a thermodynamic Relationship with the Boltzmann constant The Boltzmann constant kB (often abbreviated k) may be used in place of the gas constant by working 31432 N·m / (mol·K)

g 0

= Gravitational acceleration (9. 80665 m/s²)

M

= Molar mass of Earth's air (0. 0289644 kg/mol)

Or converted to Imperial units:^[1]

where

P

= Static pressure (inches of mercury)

T

= Standard temperature (kelvins)

L

= Standard temperature lapse rate (kelvins per foot)

h

= Height above sea level (feet)

R *

= Universal gas constant (using feet and kelvins and gram moles: 8. The kelvin (symbol K) is a unit increment of Temperature and is one of the seven SI base units The Kelvin scale is a thermodynamic Relationship with the Boltzmann constant The Boltzmann constant kB (often abbreviated k) may be used in place of the gas constant by working The mole (symbol mol) is a unit of Amount of substance: it is an SI base unit, and almost the only unit to be used to measure this 9494596×10⁴ kg·ft²·s^-2·K^-1·kmol^-1)

g 0

= Gravitational acceleration (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. 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 US Standard Atmosphere is a series of models that define values for atmospheric Temperature, Density, pressure and other properties over ^[2] The reference value for P_b for b = 0 is the defined sea level value, P₀ = 101325 pascals or 29. 92126 inHg. Inches of mercury, inHg or "Hg is a measuring unit for Pressure. 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 $h = h b + 1$ . :^[2]

Subscript b	Height Above Sea Level		Static Pressure		Standard Temperature (K)	Temperature Lapse Rate
Subscript b	(m)	(ft)	(pascals)	(inHg)	Standard Temperature (K)	(K/m)	(K/ft)
0	0	0	101325	29. 92126	288. 15	-0. 0065	-0. 0019812
1	11,000	36,089	22632. 1	6. 683245	216. 65	0. 0	0. 0
2	20,000	65,617	5474. 89	1. 616734	216. 65	0. 001	0. 0003048
3	32,000	104,987	868. 019	0. 2563258	228. 65	0. 0028	0. 00085344
4	47,000	154,199	110. 906	0. 0327506	270. 65	0. 0	0. 0
5	51,000	167,323	66. 9389	0. 01976704	270. 65	-0. 0028	-0. 00085344
6	71,000	232,940	3. 95642	0. 00116833	214. 65	-0. 002	-0. 0006096

Density equations

The expressions for calculating density are nearly identical to calculating pressure. The only difference is the exponent in Equation 1.

There are two different equations for computing density at various height regimes below 86 geometric km (84,852 geopotential meters or 278,385. 8 geopotential feet). 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:

${\rho}=\rho_b \cdot \left[\frac{T_b}{T_b + L_b\cdot(h-h_b)}\right]^{\left(\frac{g_0 \cdot M}{R^* \cdot L_b}\right)+1}$

Equation 2:

${\rho}=\rho_b \cdot \exp\left[\frac{-g_0 \cdot M \cdot (h-h_b)}{R^* \cdot T_b}\right]$

where

ρ

= Mass density (kg/m³)

T

= Standard temperature (kelvins)

L

= Standard temperature lapse rate (kelvins per meter)

h

= Height above sea level (geopotential meters)

R *

g 0

= Gravitational acceleration (9. 80665 m/s²)

M

= Molar mass of Earth's air (0. 0289644 kg/mol)

Or converted to English units:^[1]

Where

ρ

= Mass density (slugs/ft³)

T

= Standard temperature (kelvins)

L

= Standard temperature lapse rate (degrees Celsius per foot)

h

= Height above sea level (geopotential feet)

R *

= Universal gas constant (8. 9494596×10⁴ ft²/(s·K)

g 0

= Gravitational acceleration (32. 17405 ft/s²)

M

= Molar mass of Earth's air (28. 9644 grams per mole)

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. The reference value for $ρ b$ for b = 0 is the defined sea level value, $ρ o$ = 1. 2250 kg/m³ or 0. 0023768908 slugs/ft³. Values of $ρ b$ 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 $h = h b + 1$ ^[2]

In these equations, g₀, M and R^* are each single-valued constants, while $ρ$ , L, T and h are multi-valued constants in accordance with the table below. 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 US Standard Atmosphere is a series of models that define values for atmospheric Temperature, Density, pressure and other properties over ^[2].

Subscript b	Height Above Sea Level (h)		Mass Density ( $ρ$ )		Standard Temperature (T') (K)	Temperature Lapse Rate (L)
Subscript b	(m)	(ft)	(kg/m³)	(slugs/ft³	Standard Temperature (T') (K)	(K/m)	(K/ft)
0	0	0	1. 2250	2. 3768908 x 10^-3	288. 15	-0. 0065	-0. 0019812
1	11,000	36,089. 24	0. 36391	7. 0611703 x 10^-4	216. 65	0. 0	0. 0
2	20,000	65,616. 79	0. 08803	1. 7081572 x 10^-4	216. 65	0. 001	0. 0003048
3	32,000	104,986. 87	0. 01322	2. 5660735 x 10^-5	228. 65	0. 0028	0. 00085344
4	47,000	154,199. 48	0. 00143	2. 7698702 x 10 $- 6$	270. 65	0. 0	0. 0
5	51,000	167,322. 83	0. 00086	1. 6717895 x 10^-6	270. 65	-0. 0028	-0. 00085344
6	71,000	232,939. 63	0. 000064	1. 2458989 x 10^-7	214. 65	-0. 002	-0. 0006096

Derivation

The barometric formula can be derived fairly easily using the ideal gas law:

$\rho = \frac{M \cdot P}{R^* \cdot T}$

When density is known:

$P = \frac{\rho \cdot {R^*} \cdot T}{M}$

And assuming that all pressure is hydrostatic:

$dP = - \rho g\,dz\,$

Substituting the first expression into the second we get:

$\frac{dP}{P} = - \frac{M g\,dz}{RT}$

Integrating this expression from the surface to the altitude z we get:

$P = P_0 e^{-\int_{0}^{z}{M g dz/RT}}\,$

Assuming constant temperature, molar mass, and gravitational acceleration, we get the barometric formula:

$P = P_0 e^{-M g z/RT}\,$

In this formulation, R is the gas constant, and the term $R T / M g$ gives the scale height (approximately equal to 7. The ideal gas law is the Equation of state of a hypothetical Ideal gas, first stated by Benoît Paul Émile Clapeyron in 1834 Fluid statics (also called hydrostatics) is the Science of Fluids at rest and is a sub-field within Fluid mechanics. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space Relationship with the Boltzmann constant The Boltzmann constant kB (often abbreviated k) may be used in place of the gas constant by working A scale height is a term often used in scientific contexts for a distance over which a quantity decreases by a factor of e. 4 km for the troposphere). The troposphere is the lowest portion of Earth's atmosphere. It contains approximately 75% of the atmosphere's mass and almost all of its Water vapor and

(For exact results, it should be remembered that atmospheres containing water do not behave as an ideal gas. See real gas or perfect gas or gas to further understanding)

References

^ ^a ^b Mechtly, E. Real gas effects refers to an assumption base where the following are taken into account Compressibility effects Variable heat capacity Van der Gas while Ideal gas and this article are constructed -->By definition a perfect gas is one in which intermolecular forces are neglected This page is about the physical properties of gas as a state of matter A. , 1973: The International System of Units, Physical Constants and Conversion Factors. NASA SP-7012, Second Revision, National Aeronautics and Space Administration, Washington, D. C.
^ ^a ^b ^c ^d U.S. Standard Atmosphere, 1976, U. S. Government Printing Office, Washington, D. C. , 1976. (Linked file is very large. )

Contents

Pressure equations

Density equations

Derivation

References

See also