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included in table 1 (the basis for our formula). Note, in particular, the values for P = 2.5 percent, a level that

was totally absent from table 1.

The sum of squares of residuals for these 80 values is 0.01812. Thus, the root mean square deviation

per value is 0.048. Interpolation would of course be better if a larger table of F values had been used
for the fitting process.

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The total number of parameters is the sum of 10 (one for each or 7), and 4 x 17 (four for each of the three eigenvectors v, and four for each of the 14 vectors occuring in eqs (5), (6), and (7)); i.e., 78. As mentioned in Part II, this number can be somewhat reduced through algebraic manipulation, but this is unnecessary for a fitting process carried out on a programmable calculator or on a computer.

We finally repeat our previous assertion (see also Part II) that these 78 parameters fit not only a table of 100 observations (Table 1), (which would be a waste of time) but actually any F value, for P between 1 and 25 percent, and for v, and vz between 4 and co.

7. Conclusion for Part III

Through repeated application of the procedure given in Part II, it is possible to fit functions of more than two arguments, provided the data appear as a complete factorial. This is accomplished by first combining all combinations of two or more factors into one factor until a two-way table is obtained. The parameter vectors of the SVD of this table are then expressed as two-way tables themselves and further SVD's are carried out. The procedure is simple in principle but can become quite cumbersome in practice. It is not recommended for functions of more than three arguments, unless no other appropriate fitting procedure is available.

8. References

[1] Pearson, E. S., and H. O. Hartley, Editors, Biometrika Tables for Statisticians, (Cambridge University Press, London, 1970).

JOURNAL OF RESEARCH of the National Bureau of Standards

Vol. 86, No. 1, January February 1981

The Refractivity of Air

Frank E. Jones*

National Bureau of Standards, Washington, DC 20234

July 23, 1980

The air density equation of Jones, Edlén's dispersion formula for standard air, and Edlén's empirically. derived expressions for the effects of Co2 abundance and water vapor partial pressure on refractivity have been combined into a simplified equation for the refractivity of air, and estimates have been made of uncertainties in calculated refractivity. Under ambient conditions typical of metrology laboratories, the agreement between the simplified equation and Edlén's formulation is well within the uncertainty in each. The simplified equation is valid in the visible region.

Key words: Air density; index of refraction of air; refractivity of air; wavelength of light in air.

1. Introduction

101325 Pa and a CO2 abundance of 0.0003 by volume. Ed

lén [1] expressed the refractivity, (n-1)ip of dry air at temIn metrological applications of wavelengths of light in air, perature t (in °C) and pressure p(in torr) as it is necessary to calculate the wavelength at ambient conditions of temperature (T), pressure (P), effective water vapor

(n-1).p = K, D.,

(2) partial pressure (e), and CO2 abundance (xco), using the refractive index of air under these conditions. The relation where K. (3] is a dispersion factor which is independent of t between drac, the vacuum wavelength, dain, the wavelength and p, and the density factor, Dip, is in air, and n, the refractive index of air, is Ivac = n hair Edlén (1) 'has derived a dispersion formula for standard air Don = p (1 + €.p)/{(1 + ary[1 - (n-1). 1}, (3)

6 (T 288.15K, P = 101325 Pa, e' 0, xco2 = 0.0003 by volume) and a formulation for the refractivity of ambient where a = 1/273.15 and e, is a factor which multiplies p in air, (n 1).pf. Edlén's formulation is in general use in an expression for the nonideality of the gas. By substituting metrology. Jones (2] has recently published a reformula- suitable values, (3) becomes tion of the equation for the density of air and applied it to the transfer of the mass unit. It is the purpose of the present Dep=p[1 +p (0.817 – 0.0133 t)x10-)/(1+0.0036610 t). (4) paper to combine the air density equation, Edlén's dispersion formula for standard air, and Edlen's empirically. For air with a CO2 abundance of x by volume, Edlen derived derived expressions for the effects of Co, abundance and water vapor partial pressure on refractivity, and in so doing (n − 1), = (1 + 0.540 (x – 0.0003)] (n 1), (5) to develop a simpler formulation and to estimate uncertainties in the calculated refractivity.

and, The Edlén 1966 [1] dispersion formula for standard air is

niph

Mopp = -h (5.7224 0.0457 0) x 10-8 (6) (n-1), x 108 = 8342.13 + 2406030 (130-02)-1 + 15997 (38.9 - 02)“,

(1) for the difference in refractive index of moist air holding h

torr of water vapor at a total pressure p. (To avoid using the where n is the refractive index, o is the vacuum wave num- same symbol for two different quantities, in the present ber, (1//vac), in um- and standard air is dry air at 288.15K, work h has been substituted for Edlén's A.

From (4) and the relation • Center for Continuous Process Technology Programs, National Engineering Laboratory 'Figures in brackets indicate literature references at the end of this paper

(n − 1)ep = (n − 1), D./D,,

(7)

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where P is the pressure in Pa, M, is the apparent molecular weight of dry air, R is the universal gas constant, T is the temperature in kelvins, Z is the compressibility factor (the non-ideality of the air-water vapor mixture is reflected in the departure of Z from 1), U is the relative humidity in percent, and f is the enhancement factor (a factor which expresses the fact that the effective saturation vapor pressure of water in air is greater than the saturation vapor pressure, e,, of pure phase over a plane surface of pure ordinary liq. uid water). Tables of Z, e, and f are provided in the appendix of the present paper.

The Lorentz-Lorenz (4,5] formulation of the ClausiusMossotti (6,7) equation can be expressed as

(n-1) = 3 - {9 – 0.0028426 (n-1), •

(17) [6 – (n − 1)] 7. We shall return now to Edlén's development and combine (2) with (3) (n-1).p = K, D.p =

Kvp (1 + ep)

(18) (1 + at) [1 - (n=1).]

(1 + E.p) is recognized to be 1/2, (1 + at) = T1273.15, and p = 760 P/101325; therefore,

1

no-1 na + 2

(n-1).p =

(10)

760 X 273.15 KP

101325 TZ

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= C-le

=C the left side of which can be approximated [1] by ž (n − 1).

[1 - (n-1)] 1

M.

[1-(n − 1)/6). Therefore,

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where Qand M, are the density and apparent molecular weight, respectively, of dry air and C and C' are constants. Since Q. = PM,/RTZ (2), (11) becomes

С'Р

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(n-1) =

(12)

RTZ [1 - (07)]'

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where e, is in Pa. Equation (20) corresponds to (8) combined with (5) and (6), i.e. Edlen's formulation [1]. The agreement between the refractivity of moist air calculated using (20)

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