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 ƒ 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 liquid water). Tables of Z, e, and ƒ 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

[1 − (n=1)] P/T.Z.

6

By substituting the appropriate values of P, (101325 P.), T, (288.15K) and Z, (0.99958 from table 1 in the appendix), (14) becomes

[1 −

(n − 1),]

P

(n−1) = 0.0028426

(n-1),, (15)

ᎢᏃ

[1

−

which, when rearranged, becomes

6

(n 1)

(n-1)-6(n-1) + 0.0170556 (n - 1), •

6

= 0.

ᎢᏃ

The appropriate square root of (16) is

(16)

=

and Edlén's formulation is illustrated for T = 293.15K, P = 101325 Pa, U= 50, xco2 = 0.00043, Z = 0.99963 (from table 1), f 1.0041 (from table 2), e, = 2338 Pa (from table 3) and λ = σ ̄1 = 0.6329912714 μm for an iodine stabilized helium-neon laser [8]. Using (20), (n − 1) = 27131.0 × 10-3; using Edlén's formulation (n − 1).ph = 27131.3 × 10-8. For a more extreme case (T = 288.15K, P = 70000 Pa, U 50, Xco2 = 0.00080, Z = 0.99971, f = 1.0030, e, = 1705 Pa, = = (for the same wavelength), (20) gives (n − 1) = 19069.6 × 10, and the Edlen formulation gives (n - 1)ph 19068.1 × 10-8. As will be demonstrated in the next section, the difference between the results for the two formulations is well within the uncertainty of each.

=

(21)

We follow the suggested practice of Eisenhart [9, 10] in stating separately the random and systematic components of the estimated uncertainties. The stated random component is one standard deviation; the stated systematic component is one-third of the half-width of the interval between the bounds on the systematic error.

The uncertainties in calculated (n-1)TP. due to estimated uncertainties [2] in P, T, Z, U, f, e,, and x can be estimated from equation (22). We shall not attempt to estimate the uncertainties in Edlén's [1] dispersion formula

for standard air and his expressions for the effects of CO2 abundance and water vapor partial pressure. The state-ofthe-art in pressure measurement [11] permits the measurement of pressure in a laboratory with a random relative uncertainty of less than 0.02 percent, calibration of pressure measuring instruments against a primary standard of pressure contributes a systematic relative uncertainty of about ± 0.003 percent. The corresponding uncertainties in (n − 1)rp.', in the first example above are ± 5.4 × 10-8 and 0.8 × 10-8.

The measurement of temperature in the air path is potentially as critical as the pressure measurement, in terms of its effect on the uncertainty in the calculated (n−1)TMe'; it is possible to make only a rough estimate of the uncertainty in the temperature measurement. If the vicinity of the path were instrumented with a network of thermopile junctions, the measurements would be expected to have a standard deviation of about ± 0.05K [12] and a systematic uncertainty of the order the± 0.01K. The corresponding uncertainties in (n-1)TP. in the first example are ± 4.6 x 10-8 and ± 0.9 × 10-8.

The estimated systematic relative uncertainty in the compressibility factor, Z, for the first example is ± 0.0017 percent. The corresponding uncertainty in (n−1)πpe is ± 0.5 x 10-8.

The uncertainty in calculated (n-1)TP. due to humidity measurement can be estimated from the second term in (22). The state-of-the-art in humidity measurement [13] permits the measurement of relative humidity, U, with a random uncertainty of ± 0.5 percent relative humidity and a systematic uncertainty of ± 0.3 percent relative humidity. The corresponding uncertainties in (n−1)π. in the first example are ± 0.5 × 10-8 and ± 0.3 × 10-8. The uncertainties contributed by uncertainties in ƒ and e, are negligible [2].

The uncertainty in calculated (n-1). due to a variation in CO2 abundance, x, can be estimated from (5). In the first example, a variation in x of ± 0.0001 corresponds to a systematic uncertainty in (n - 1)TРe of ± 1.5 × 10-8.

determination of the air density in a balance case, thus avoiding the uncertainties in the parameters and environmental variables in air density calculations. A similar scheme will be used in the transfer of the mass unit [15]. Having estimated the uncertainty in calculated (n − 1)TMPe' due to the uncertainties in the various variables to be about 1 X 10-7 at the level of the equivalent of 1 standard deviation, it is of interest to estimate how much improvement would result from the direct determination of air density, Q, if practicable. From (9),

dard deviation. The major contributors to the uncertainty in refractivity are the uncertainties in the measurements of pressure and temperature. The magnitude of the uncertainty due to variation in CO, concentration can approach that of the uncertainties due to the pressure and temperature measurements. Therefore, the CO, concentration should be treated as a variable and should be observed.

If it were practicable to make a direct measurement of air density representative of the air path, the uncertainty in calculated refractivity due to the uncertainties in the various variables and parameters would be reduced by a factor of about 2.5.

0.00033); recalling that is the density of moist air. By substituting (25) in ୧

The uncertainties in the various parameters in (25), other than o and (n-1),, are taken from [2]. The resulting overall uncertainty in the calculated (n-1)TP. are 1.9 × 10-8 random and ± 1.8 x 10-8 systematic. The uncertainty due to the effect on M. of a variation of xco2, 1.1 × 10-8 per 0.0001, has necessarily not been included. It can be concluded that even it the uncertainty in a direct determination of o were negligible, the uncertainty in (n-1)TP. due to the uncertainties in the various variables and parameters would be reduced by a factor of about 2.5. The major contributors to the uncertainty in (n - 1)rpe are the uncertainties in R, M. and U.

a

5. Conclusions

Jones's air density equation [2], Edlén's [1] 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 simple refractivity of air equation, and estimates have been made of uncertainties in calculated refractivity.

The general equation is (22), which is valid in the visible region; tables of Z, ƒ and e, have been included in the appendix of this paper. The overall estimated uncertainty is about 1 x 10' at the level of the equivalent of 1 stan

The author is pleased to express his thanks to John S. Beers at whose suggestion this work was undertaken, and to Catherine DeLeonibus for typing the manuscript.

6. References

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[5] Lorenz, L. Ueber die Refractions constante. Ann. Physik u. Chemie. N. F. 11: 70-103; 1880.

[6] Clausius, R. Die Mechanische Warme theorie, Vol. 2. Braunschweig: Friedrich Vieweg und Sohn; 1879. 62-97.

[7] Mossotti, O. F. Memorie di matematica e di fisica della Societa Italiana delle Scienze Residente in Modena. (26)(2): 49-74; 1850.

[8] Schweitzer, W. G.; Kessler, E. G., Jr.; Deslattes, R. D.; Layer, H. P.; Whetstone, J. R. Description, performance, and wavelengths of iodine stabilized lasers. Appl. Opt. 12(12): 2927-2938; 1973.

[9] Eisenhart, C. Expression of the uncertainties of final results. Science. 160: 1201-1204; 1968.

[10] Eisenhart, C. Realistic evaluation of the precision and auccuracy of instrument calibration systems. J. Res. Nat. Bur. Stand. (U.S.). 67C: 161-187; 1963 April-June.

[11] Heydemann, P. L. M. Private communication.

[12] Roeser, W. F.; Wensel, H. T. Temperature, its measurement and control in science and industry. New York: Reinhold Publ. Corp.; 1941. p.

311.

[13] Wexler, A. Measurement of humidity in the free atmosphere near the surface of the earth. Meteorological Monographs. 11(33): 262-282; 1970. [14] Bowman, H. A.; Schoonover, R. M.; Jones, M. W. Procedure for high precision density determinations by hydrostatic weighing. J. Res. Nat. Bur. Stand. (U.S.). 71C(3): 179-198; 1967 July-September. [15] Schoonover, R. M. Private communication. [16] Hyland, R. W. A correlation of the second virial coefficients and enhancement factors for moist air. J. Res. Nat. Bur. Stand. (U.S.). 79A(4): 551-560; 1975 July-August.

[17] Wexler, A.; Greenspan, L. Vapor pressure equation for water in the range 0 to 100 °C. J. Res. Nat. Bur. Stand. (U.S.). 75A(3): 213-230; 1971 May-June.