Barometric formula
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 pressure drops approximately by 11.3 pascals per meter in first 1000 meters above sea level. Pressure equations. There are two different equations for computing pressure at various height regimes below 86 km (or 278,400 feet). The first equation is used when the value of standard temperature lapse rate is not equal to zero: formula_1 The second equation is used when standard temperature lapse rate equals zero: formula_2 where: Or  converted to imperial units: where 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, "g", "M" and "R" are each single-valued constants, while "P," "L," "T," and "h" are multivalued constants in accordance with the table below. The values used for "M," "g", and "R" are in accordance with the U.S. Standard Atmosphere, 1976, and the value for "R" in particular does not agree with standard values for this constant. The reference value for "P" for "b" = 0  is the defined sea level value, "P" = 101 325 Pa or 29.92126 inHg. Values of "P" 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". 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). The first equation is used  when the value of standard temperature lapse rate is not equal to zero; the second equation is used when standard temperature lapse rate equals zero. Equation 1: formula_19 Equation 2: formula_20 where or, converted to U.S. gravitational foot-pound-second units (no longer used in U.K.): 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" for "b" = 0 is the defined sea level value, "ρ" = 1.2250 kg/m or 0.0023768908 slug/ft. Values of "ρ" 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". In these equations, "g", "M" and "R" are each single-valued constants, while "ρ", "L", "T" and "h" are multi-valued constants in accordance with the table below. The values used for "M", "g" 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. Derivation. The barometric formula can be derived using  the ideal gas law: formula_35 Assuming that all pressure is hydrostatic: formula_36 and dividing the formula_37 by the formula_38 expression we get: formula_39 Integrating this expression from the surface to the altitude "z" we get: formula_40 Assuming linear temperature change formula_41 and constant molar mass and gravitational acceleration, we get the first barometric formula: formula_42 Instead, assuming constant temperature, integrating gives the second barometric formula: formula_43 In this formulation, "R" is the gas constant, and the term "RT"/"Mg" gives the scale height (approximately equal to 8.4 km for the troposphere).