Abstract

Refractive indices of various H2SO4–H2O, HNO3–H2O, and H2SO4–HNO3–H2O solutions were measured at four wavelengths in the visible (351.0, 533.5, 632.9, and 782.6 nm) over a temperature range from 30 to -60 °C. The temperature dependence has been determined for the first time to the authors’ knowledge. This dependence is of importance for applications to atmospheric aerosols at low temperatures. In particular, it is shown that (1) the molar refractivity of the solutions is independent of temperature, whereas the temperature dependence of the refractive index arises solely through the temperature dependence of the solution’s mass density, (2) the molar refractivities of H2SO4 and HNO3 in a ternary solution may be calculated as the weighted sum of the molar refractivities of two binary solutions evaluated at a concentration that corresponds to the total acid concentration, and (3) the H2O molar refractivity in the solutions may be taken equal to that of pure water. Although the data for the ternary system have been used for this model verification, data for binary H2SO4–H2O and HNO3–H2O solutions were used to improve the accuracy of the modeled refractive indices to better than 0.0017% or 0.15% for concentrations of 5–70 wt. % and wavelengths from the near ultraviolet to the near infrared (0.25–2 µm).

© 2000 Optical Society of America

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References

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  1. J. E. Dye, B. W. Gandrud, D. Baumgardner, K. R. Chan, G. V. Ferry, M. Loewenstein, K. K. Kelly, J. C. Wilson, “Observed particle evolution in the polar stratospheric cloud of January 24 1989,” Geophys. Res. Lett. 17, 413–415 (1990).
    [CrossRef]
  2. T. Sugita, Y. Kondo, M. Koike, M. Kanada, N. Toriyama, H. Nakajima, T. Deshler, R. Imasu, “Balloon-borne optical counter for in situ aerosol measurements,” J. Amos. Chem. 32, 183–204 (1999).
    [CrossRef]
  3. N. Larsen, I. S. Mikkelsen, B. M. Knudsen, J. Schreiner, C. Voigt, K. Mauersberger, J. M. Rosen, N. T. Kjome, “Comparison of chemical and optical in situ measurements of polar stratospheric clouds,” J. Geophys. Res. 105, 1491–1502 (2000).
    [CrossRef]
  4. D. Baumgardner, J. E. Dye, B. Gandrud, K. Barr, K. Kelly, K. Chan, “Refractive indices of aerosols in the upper troposphere and lower stratosphere,” Geophys. Res. Lett. 23, 749–752 (1996).
    [CrossRef]
  5. E. V. Browell, C. F. Butler, S. Ismail, P. A. Robinette, A. F. Carter, N. S. Higdon, O. B. Toon, M. R. Schoeberl, A. F. Tuck, “Airborne lidar observations in the wintertime Arctic stratosphere: polar stratospheric clouds,” Geophys. Res. Lett. 17, 385–388 (1990).
    [CrossRef]
  6. G. Beyerle, B. P. Luo, R. Neuber, Th. Peter, I. S. McDermid, “Temperature dependence of ternary solution particle volumes by lidar in the Arctic stratosphere during winter 1992/1993,” J. Geophys. Res. 102, 3603–3609 (1997).
    [CrossRef]
  7. M. Pantani, M. Del Guasta, D. Guzzi, L. Stefanutti, “Polar stratospheric clouds of liquid aerosol: an experimental determination of mean size distribution,” J. Aerosol Sci. 30, 559–567 (1999).
    [CrossRef]
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    [CrossRef]
  9. M. E. Hervig, T. Deshler, “Stratospheric aerosol surface area and volume inferred from HALOE, CLAES, and ILAS measurements,” J. Geophys. Res. Atmos. D 103, 25,345–25,352 (1998).
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    [CrossRef]
  12. U. K. Krieger, U. Weers, Th. Peter, “Using Mie-scattering resonances for determination of composition and radii of ternary droplets,” in Polar Stratospheric Ozone, J. A. Pyle, N. R. P. Harris, G. T. Amanatidis, eds., (Commission of the European Communities, Brussels, 1996).
  13. K. S. Carslaw, B. P. Luo, S. L. Clegg, Th. Peter, P. Brimblecombe, P. J. Crutzen, “Stratospheric aerosol growth and HNO3 gas phase depletion from coupled HNO3 and water uptake by liquid particles,” Geophys. Res. Lett. 21, 2479–2482 (1994).
    [CrossRef]
  14. A. Tabazadeh, R. P. Turco, K. Drdla, M. Z. Jacobson, O. B. Toon, “A study of type I polar stratospheric cloud formation,” Geophys. Res. Lett. 21, 1619–1622 (1994).
    [CrossRef]
  15. J. Schreiner, C. Voigt, A. Kohlmann, F. Arnold, K. Mauersberger, N. Larsen, “Chemical analysis of polar stratospheric cloud particles,” Science 283, 968–970 (1999).
    [CrossRef] [PubMed]
  16. K. D. Beyer, A. R. Ravishankara, E. R. Lovejoy, “Measurements of UV refractive indices and densities of H2SO4/H2O and H2SO4/HNO3/H2O solutions,” J. Geophys. Res. Atmos. 101, 14,519–14,524 (1996).
    [CrossRef]
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    [CrossRef] [PubMed]
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  19. A. Haentzsch, F. Dürigen, “Über die chemische Veränderung von Säuren und Salzen in Lösung auf Grund refraktometrischer Daten,” Z. Phys. Chem. 134, 413–452 (1928), Z. Phys. Chem. 136, 1–17 (1928).
  20. R. Lühdemann, “Über die Konzentrationsabhängigkeit der Äquivalentrefraktion einiger Salze und Säuren in wässriger Lösung,” Z. Phys. Chem. B 29, 133–149 (1935).
  21. B. Luo, U. K. Krieger, Th. Peter, “Densities and refractive indices of H2SO4/HNO3/H2O solutions to stratospheric temperatures,” Geophys. Res. Lett. 23, 3707–3710 (1996).
    [CrossRef]
  22. K. N. Marsh, ed., “Recommended Reference Materials for the Realization of Physico-Chemical Properties” (Blackwell Scientific, Oxford, 1987).
  23. I. Thormählen, J. Straub, U. Grigull, “Refractive index of water and its dependence on wavelength, temperature, and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985).
    [CrossRef]
  24. T. E. Smith, R. F. Bonner, “Refractive index-temperature data for anhydrous ethyl alcohol,” Anal. Chem. 24, 517–518 (1952).
    [CrossRef]
  25. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), pp. 803.
  26. A copy of the fortran code (or alternatively a java code) can be obtained from the authors. Send an email to krieger@atmos.umnw.ethz.ch.

2000 (1)

N. Larsen, I. S. Mikkelsen, B. M. Knudsen, J. Schreiner, C. Voigt, K. Mauersberger, J. M. Rosen, N. T. Kjome, “Comparison of chemical and optical in situ measurements of polar stratospheric clouds,” J. Geophys. Res. 105, 1491–1502 (2000).
[CrossRef]

1999 (3)

T. Sugita, Y. Kondo, M. Koike, M. Kanada, N. Toriyama, H. Nakajima, T. Deshler, R. Imasu, “Balloon-borne optical counter for in situ aerosol measurements,” J. Amos. Chem. 32, 183–204 (1999).
[CrossRef]

M. Pantani, M. Del Guasta, D. Guzzi, L. Stefanutti, “Polar stratospheric clouds of liquid aerosol: an experimental determination of mean size distribution,” J. Aerosol Sci. 30, 559–567 (1999).
[CrossRef]

J. Schreiner, C. Voigt, A. Kohlmann, F. Arnold, K. Mauersberger, N. Larsen, “Chemical analysis of polar stratospheric cloud particles,” Science 283, 968–970 (1999).
[CrossRef] [PubMed]

1998 (2)

M. E. Hervig, T. Deshler, “Stratospheric aerosol surface area and volume inferred from HALOE, CLAES, and ILAS measurements,” J. Geophys. Res. Atmos. D 103, 25,345–25,352 (1998).
[CrossRef]

R. T. Tisdale, D. L. Glandorf, M. A. Tolbert, O. B. Toon, “Infrared optical constants of low temperature H2SO4 solutions representative of stratospheric sulfate aerosols,” J. Geophys. Res. Atmos. 103, 25,353–25,370 (1998).
[CrossRef]

1997 (1)

G. Beyerle, B. P. Luo, R. Neuber, Th. Peter, I. S. McDermid, “Temperature dependence of ternary solution particle volumes by lidar in the Arctic stratosphere during winter 1992/1993,” J. Geophys. Res. 102, 3603–3609 (1997).
[CrossRef]

1996 (3)

B. Luo, U. K. Krieger, Th. Peter, “Densities and refractive indices of H2SO4/HNO3/H2O solutions to stratospheric temperatures,” Geophys. Res. Lett. 23, 3707–3710 (1996).
[CrossRef]

K. D. Beyer, A. R. Ravishankara, E. R. Lovejoy, “Measurements of UV refractive indices and densities of H2SO4/H2O and H2SO4/HNO3/H2O solutions,” J. Geophys. Res. Atmos. 101, 14,519–14,524 (1996).
[CrossRef]

D. Baumgardner, J. E. Dye, B. Gandrud, K. Barr, K. Kelly, K. Chan, “Refractive indices of aerosols in the upper troposphere and lower stratosphere,” Geophys. Res. Lett. 23, 749–752 (1996).
[CrossRef]

1994 (3)

K. S. Carslaw, B. P. Luo, S. L. Clegg, Th. Peter, P. Brimblecombe, P. J. Crutzen, “Stratospheric aerosol growth and HNO3 gas phase depletion from coupled HNO3 and water uptake by liquid particles,” Geophys. Res. Lett. 21, 2479–2482 (1994).
[CrossRef]

A. Tabazadeh, R. P. Turco, K. Drdla, M. Z. Jacobson, O. B. Toon, “A study of type I polar stratospheric cloud formation,” Geophys. Res. Lett. 21, 1619–1622 (1994).
[CrossRef]

G. K. Yue, L. R. Poole, P.-H. Wang, E. W. Chiou, “Stratospheric aerosol acidity, density and refractive index deduced from SAGE II and NMC temperature data,” J. Geophys. Res. 99, 3727–3738 (1994).
[CrossRef]

1990 (2)

E. V. Browell, C. F. Butler, S. Ismail, P. A. Robinette, A. F. Carter, N. S. Higdon, O. B. Toon, M. R. Schoeberl, A. F. Tuck, “Airborne lidar observations in the wintertime Arctic stratosphere: polar stratospheric clouds,” Geophys. Res. Lett. 17, 385–388 (1990).
[CrossRef]

J. E. Dye, B. W. Gandrud, D. Baumgardner, K. R. Chan, G. V. Ferry, M. Loewenstein, K. K. Kelly, J. C. Wilson, “Observed particle evolution in the polar stratospheric cloud of January 24 1989,” Geophys. Res. Lett. 17, 413–415 (1990).
[CrossRef]

1985 (1)

I. Thormählen, J. Straub, U. Grigull, “Refractive index of water and its dependence on wavelength, temperature, and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985).
[CrossRef]

1975 (1)

1952 (1)

T. E. Smith, R. F. Bonner, “Refractive index-temperature data for anhydrous ethyl alcohol,” Anal. Chem. 24, 517–518 (1952).
[CrossRef]

1935 (1)

R. Lühdemann, “Über die Konzentrationsabhängigkeit der Äquivalentrefraktion einiger Salze und Säuren in wässriger Lösung,” Z. Phys. Chem. B 29, 133–149 (1935).

1928 (1)

A. Haentzsch, F. Dürigen, “Über die chemische Veränderung von Säuren und Salzen in Lösung auf Grund refraktometrischer Daten,” Z. Phys. Chem. 134, 413–452 (1928), Z. Phys. Chem. 136, 1–17 (1928).

Arnold, F.

J. Schreiner, C. Voigt, A. Kohlmann, F. Arnold, K. Mauersberger, N. Larsen, “Chemical analysis of polar stratospheric cloud particles,” Science 283, 968–970 (1999).
[CrossRef] [PubMed]

Barr, K.

D. Baumgardner, J. E. Dye, B. Gandrud, K. Barr, K. Kelly, K. Chan, “Refractive indices of aerosols in the upper troposphere and lower stratosphere,” Geophys. Res. Lett. 23, 749–752 (1996).
[CrossRef]

Baumgardner, D.

D. Baumgardner, J. E. Dye, B. Gandrud, K. Barr, K. Kelly, K. Chan, “Refractive indices of aerosols in the upper troposphere and lower stratosphere,” Geophys. Res. Lett. 23, 749–752 (1996).
[CrossRef]

J. E. Dye, B. W. Gandrud, D. Baumgardner, K. R. Chan, G. V. Ferry, M. Loewenstein, K. K. Kelly, J. C. Wilson, “Observed particle evolution in the polar stratospheric cloud of January 24 1989,” Geophys. Res. Lett. 17, 413–415 (1990).
[CrossRef]

Beyer, K. D.

K. D. Beyer, A. R. Ravishankara, E. R. Lovejoy, “Measurements of UV refractive indices and densities of H2SO4/H2O and H2SO4/HNO3/H2O solutions,” J. Geophys. Res. Atmos. 101, 14,519–14,524 (1996).
[CrossRef]

Beyerle, G.

G. Beyerle, B. P. Luo, R. Neuber, Th. Peter, I. S. McDermid, “Temperature dependence of ternary solution particle volumes by lidar in the Arctic stratosphere during winter 1992/1993,” J. Geophys. Res. 102, 3603–3609 (1997).
[CrossRef]

Bonner, R. F.

T. E. Smith, R. F. Bonner, “Refractive index-temperature data for anhydrous ethyl alcohol,” Anal. Chem. 24, 517–518 (1952).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), pp. 803.

Brimblecombe, P.

K. S. Carslaw, B. P. Luo, S. L. Clegg, Th. Peter, P. Brimblecombe, P. J. Crutzen, “Stratospheric aerosol growth and HNO3 gas phase depletion from coupled HNO3 and water uptake by liquid particles,” Geophys. Res. Lett. 21, 2479–2482 (1994).
[CrossRef]

Browell, E. V.

E. V. Browell, C. F. Butler, S. Ismail, P. A. Robinette, A. F. Carter, N. S. Higdon, O. B. Toon, M. R. Schoeberl, A. F. Tuck, “Airborne lidar observations in the wintertime Arctic stratosphere: polar stratospheric clouds,” Geophys. Res. Lett. 17, 385–388 (1990).
[CrossRef]

Butler, C. F.

E. V. Browell, C. F. Butler, S. Ismail, P. A. Robinette, A. F. Carter, N. S. Higdon, O. B. Toon, M. R. Schoeberl, A. F. Tuck, “Airborne lidar observations in the wintertime Arctic stratosphere: polar stratospheric clouds,” Geophys. Res. Lett. 17, 385–388 (1990).
[CrossRef]

Carslaw, K. S.

K. S. Carslaw, B. P. Luo, S. L. Clegg, Th. Peter, P. Brimblecombe, P. J. Crutzen, “Stratospheric aerosol growth and HNO3 gas phase depletion from coupled HNO3 and water uptake by liquid particles,” Geophys. Res. Lett. 21, 2479–2482 (1994).
[CrossRef]

Carter, A. F.

E. V. Browell, C. F. Butler, S. Ismail, P. A. Robinette, A. F. Carter, N. S. Higdon, O. B. Toon, M. R. Schoeberl, A. F. Tuck, “Airborne lidar observations in the wintertime Arctic stratosphere: polar stratospheric clouds,” Geophys. Res. Lett. 17, 385–388 (1990).
[CrossRef]

Chan, K.

D. Baumgardner, J. E. Dye, B. Gandrud, K. Barr, K. Kelly, K. Chan, “Refractive indices of aerosols in the upper troposphere and lower stratosphere,” Geophys. Res. Lett. 23, 749–752 (1996).
[CrossRef]

Chan, K. R.

J. E. Dye, B. W. Gandrud, D. Baumgardner, K. R. Chan, G. V. Ferry, M. Loewenstein, K. K. Kelly, J. C. Wilson, “Observed particle evolution in the polar stratospheric cloud of January 24 1989,” Geophys. Res. Lett. 17, 413–415 (1990).
[CrossRef]

Chiou, E. W.

G. K. Yue, L. R. Poole, P.-H. Wang, E. W. Chiou, “Stratospheric aerosol acidity, density and refractive index deduced from SAGE II and NMC temperature data,” J. Geophys. Res. 99, 3727–3738 (1994).
[CrossRef]

Clegg, S. L.

K. S. Carslaw, B. P. Luo, S. L. Clegg, Th. Peter, P. Brimblecombe, P. J. Crutzen, “Stratospheric aerosol growth and HNO3 gas phase depletion from coupled HNO3 and water uptake by liquid particles,” Geophys. Res. Lett. 21, 2479–2482 (1994).
[CrossRef]

Crutzen, P. J.

K. S. Carslaw, B. P. Luo, S. L. Clegg, Th. Peter, P. Brimblecombe, P. J. Crutzen, “Stratospheric aerosol growth and HNO3 gas phase depletion from coupled HNO3 and water uptake by liquid particles,” Geophys. Res. Lett. 21, 2479–2482 (1994).
[CrossRef]

Del Guasta, M.

M. Pantani, M. Del Guasta, D. Guzzi, L. Stefanutti, “Polar stratospheric clouds of liquid aerosol: an experimental determination of mean size distribution,” J. Aerosol Sci. 30, 559–567 (1999).
[CrossRef]

Deshler, T.

T. Sugita, Y. Kondo, M. Koike, M. Kanada, N. Toriyama, H. Nakajima, T. Deshler, R. Imasu, “Balloon-borne optical counter for in situ aerosol measurements,” J. Amos. Chem. 32, 183–204 (1999).
[CrossRef]

M. E. Hervig, T. Deshler, “Stratospheric aerosol surface area and volume inferred from HALOE, CLAES, and ILAS measurements,” J. Geophys. Res. Atmos. D 103, 25,345–25,352 (1998).
[CrossRef]

Drdla, K.

A. Tabazadeh, R. P. Turco, K. Drdla, M. Z. Jacobson, O. B. Toon, “A study of type I polar stratospheric cloud formation,” Geophys. Res. Lett. 21, 1619–1622 (1994).
[CrossRef]

Dürigen, F.

A. Haentzsch, F. Dürigen, “Über die chemische Veränderung von Säuren und Salzen in Lösung auf Grund refraktometrischer Daten,” Z. Phys. Chem. 134, 413–452 (1928), Z. Phys. Chem. 136, 1–17 (1928).

Dye, J. E.

D. Baumgardner, J. E. Dye, B. Gandrud, K. Barr, K. Kelly, K. Chan, “Refractive indices of aerosols in the upper troposphere and lower stratosphere,” Geophys. Res. Lett. 23, 749–752 (1996).
[CrossRef]

J. E. Dye, B. W. Gandrud, D. Baumgardner, K. R. Chan, G. V. Ferry, M. Loewenstein, K. K. Kelly, J. C. Wilson, “Observed particle evolution in the polar stratospheric cloud of January 24 1989,” Geophys. Res. Lett. 17, 413–415 (1990).
[CrossRef]

Ferry, G. V.

J. E. Dye, B. W. Gandrud, D. Baumgardner, K. R. Chan, G. V. Ferry, M. Loewenstein, K. K. Kelly, J. C. Wilson, “Observed particle evolution in the polar stratospheric cloud of January 24 1989,” Geophys. Res. Lett. 17, 413–415 (1990).
[CrossRef]

Gandrud, B.

D. Baumgardner, J. E. Dye, B. Gandrud, K. Barr, K. Kelly, K. Chan, “Refractive indices of aerosols in the upper troposphere and lower stratosphere,” Geophys. Res. Lett. 23, 749–752 (1996).
[CrossRef]

Gandrud, B. W.

J. E. Dye, B. W. Gandrud, D. Baumgardner, K. R. Chan, G. V. Ferry, M. Loewenstein, K. K. Kelly, J. C. Wilson, “Observed particle evolution in the polar stratospheric cloud of January 24 1989,” Geophys. Res. Lett. 17, 413–415 (1990).
[CrossRef]

Glandorf, D. L.

R. T. Tisdale, D. L. Glandorf, M. A. Tolbert, O. B. Toon, “Infrared optical constants of low temperature H2SO4 solutions representative of stratospheric sulfate aerosols,” J. Geophys. Res. Atmos. 103, 25,353–25,370 (1998).
[CrossRef]

Grigull, U.

I. Thormählen, J. Straub, U. Grigull, “Refractive index of water and its dependence on wavelength, temperature, and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985).
[CrossRef]

Guzzi, D.

M. Pantani, M. Del Guasta, D. Guzzi, L. Stefanutti, “Polar stratospheric clouds of liquid aerosol: an experimental determination of mean size distribution,” J. Aerosol Sci. 30, 559–567 (1999).
[CrossRef]

Haentzsch, A.

A. Haentzsch, F. Dürigen, “Über die chemische Veränderung von Säuren und Salzen in Lösung auf Grund refraktometrischer Daten,” Z. Phys. Chem. 134, 413–452 (1928), Z. Phys. Chem. 136, 1–17 (1928).

Hellwege, A. M.

K. H. Hellwege, A. M. Hellwege, “Optische Konstanten,” in Chemie, Astronomie, Geophysik, Technik, Vol. II8 of Landolt–Börnstein Zahlenwerte und Funktionen für Physik (Springer-Verlag, Berlin, 1962), p. 901.

Hellwege, K. H.

K. H. Hellwege, A. M. Hellwege, “Optische Konstanten,” in Chemie, Astronomie, Geophysik, Technik, Vol. II8 of Landolt–Börnstein Zahlenwerte und Funktionen für Physik (Springer-Verlag, Berlin, 1962), p. 901.

Hervig, M. E.

M. E. Hervig, T. Deshler, “Stratospheric aerosol surface area and volume inferred from HALOE, CLAES, and ILAS measurements,” J. Geophys. Res. Atmos. D 103, 25,345–25,352 (1998).
[CrossRef]

Higdon, N. S.

E. V. Browell, C. F. Butler, S. Ismail, P. A. Robinette, A. F. Carter, N. S. Higdon, O. B. Toon, M. R. Schoeberl, A. F. Tuck, “Airborne lidar observations in the wintertime Arctic stratosphere: polar stratospheric clouds,” Geophys. Res. Lett. 17, 385–388 (1990).
[CrossRef]

Imasu, R.

T. Sugita, Y. Kondo, M. Koike, M. Kanada, N. Toriyama, H. Nakajima, T. Deshler, R. Imasu, “Balloon-borne optical counter for in situ aerosol measurements,” J. Amos. Chem. 32, 183–204 (1999).
[CrossRef]

Ismail, S.

E. V. Browell, C. F. Butler, S. Ismail, P. A. Robinette, A. F. Carter, N. S. Higdon, O. B. Toon, M. R. Schoeberl, A. F. Tuck, “Airborne lidar observations in the wintertime Arctic stratosphere: polar stratospheric clouds,” Geophys. Res. Lett. 17, 385–388 (1990).
[CrossRef]

Jacobson, M. Z.

A. Tabazadeh, R. P. Turco, K. Drdla, M. Z. Jacobson, O. B. Toon, “A study of type I polar stratospheric cloud formation,” Geophys. Res. Lett. 21, 1619–1622 (1994).
[CrossRef]

Kanada, M.

T. Sugita, Y. Kondo, M. Koike, M. Kanada, N. Toriyama, H. Nakajima, T. Deshler, R. Imasu, “Balloon-borne optical counter for in situ aerosol measurements,” J. Amos. Chem. 32, 183–204 (1999).
[CrossRef]

Kelly, K.

D. Baumgardner, J. E. Dye, B. Gandrud, K. Barr, K. Kelly, K. Chan, “Refractive indices of aerosols in the upper troposphere and lower stratosphere,” Geophys. Res. Lett. 23, 749–752 (1996).
[CrossRef]

Kelly, K. K.

J. E. Dye, B. W. Gandrud, D. Baumgardner, K. R. Chan, G. V. Ferry, M. Loewenstein, K. K. Kelly, J. C. Wilson, “Observed particle evolution in the polar stratospheric cloud of January 24 1989,” Geophys. Res. Lett. 17, 413–415 (1990).
[CrossRef]

Kjome, N. T.

N. Larsen, I. S. Mikkelsen, B. M. Knudsen, J. Schreiner, C. Voigt, K. Mauersberger, J. M. Rosen, N. T. Kjome, “Comparison of chemical and optical in situ measurements of polar stratospheric clouds,” J. Geophys. Res. 105, 1491–1502 (2000).
[CrossRef]

Knudsen, B. M.

N. Larsen, I. S. Mikkelsen, B. M. Knudsen, J. Schreiner, C. Voigt, K. Mauersberger, J. M. Rosen, N. T. Kjome, “Comparison of chemical and optical in situ measurements of polar stratospheric clouds,” J. Geophys. Res. 105, 1491–1502 (2000).
[CrossRef]

Kohlmann, A.

J. Schreiner, C. Voigt, A. Kohlmann, F. Arnold, K. Mauersberger, N. Larsen, “Chemical analysis of polar stratospheric cloud particles,” Science 283, 968–970 (1999).
[CrossRef] [PubMed]

Koike, M.

T. Sugita, Y. Kondo, M. Koike, M. Kanada, N. Toriyama, H. Nakajima, T. Deshler, R. Imasu, “Balloon-borne optical counter for in situ aerosol measurements,” J. Amos. Chem. 32, 183–204 (1999).
[CrossRef]

Kondo, Y.

T. Sugita, Y. Kondo, M. Koike, M. Kanada, N. Toriyama, H. Nakajima, T. Deshler, R. Imasu, “Balloon-borne optical counter for in situ aerosol measurements,” J. Amos. Chem. 32, 183–204 (1999).
[CrossRef]

Krieger, U. K.

B. Luo, U. K. Krieger, Th. Peter, “Densities and refractive indices of H2SO4/HNO3/H2O solutions to stratospheric temperatures,” Geophys. Res. Lett. 23, 3707–3710 (1996).
[CrossRef]

U. K. Krieger, U. Weers, Th. Peter, “Using Mie-scattering resonances for determination of composition and radii of ternary droplets,” in Polar Stratospheric Ozone, J. A. Pyle, N. R. P. Harris, G. T. Amanatidis, eds., (Commission of the European Communities, Brussels, 1996).

Larsen, N.

N. Larsen, I. S. Mikkelsen, B. M. Knudsen, J. Schreiner, C. Voigt, K. Mauersberger, J. M. Rosen, N. T. Kjome, “Comparison of chemical and optical in situ measurements of polar stratospheric clouds,” J. Geophys. Res. 105, 1491–1502 (2000).
[CrossRef]

J. Schreiner, C. Voigt, A. Kohlmann, F. Arnold, K. Mauersberger, N. Larsen, “Chemical analysis of polar stratospheric cloud particles,” Science 283, 968–970 (1999).
[CrossRef] [PubMed]

Loewenstein, M.

J. E. Dye, B. W. Gandrud, D. Baumgardner, K. R. Chan, G. V. Ferry, M. Loewenstein, K. K. Kelly, J. C. Wilson, “Observed particle evolution in the polar stratospheric cloud of January 24 1989,” Geophys. Res. Lett. 17, 413–415 (1990).
[CrossRef]

Lovejoy, E. R.

K. D. Beyer, A. R. Ravishankara, E. R. Lovejoy, “Measurements of UV refractive indices and densities of H2SO4/H2O and H2SO4/HNO3/H2O solutions,” J. Geophys. Res. Atmos. 101, 14,519–14,524 (1996).
[CrossRef]

Lühdemann, R.

R. Lühdemann, “Über die Konzentrationsabhängigkeit der Äquivalentrefraktion einiger Salze und Säuren in wässriger Lösung,” Z. Phys. Chem. B 29, 133–149 (1935).

Luo, B.

B. Luo, U. K. Krieger, Th. Peter, “Densities and refractive indices of H2SO4/HNO3/H2O solutions to stratospheric temperatures,” Geophys. Res. Lett. 23, 3707–3710 (1996).
[CrossRef]

Luo, B. P.

G. Beyerle, B. P. Luo, R. Neuber, Th. Peter, I. S. McDermid, “Temperature dependence of ternary solution particle volumes by lidar in the Arctic stratosphere during winter 1992/1993,” J. Geophys. Res. 102, 3603–3609 (1997).
[CrossRef]

K. S. Carslaw, B. P. Luo, S. L. Clegg, Th. Peter, P. Brimblecombe, P. J. Crutzen, “Stratospheric aerosol growth and HNO3 gas phase depletion from coupled HNO3 and water uptake by liquid particles,” Geophys. Res. Lett. 21, 2479–2482 (1994).
[CrossRef]

Mauersberger, K.

N. Larsen, I. S. Mikkelsen, B. M. Knudsen, J. Schreiner, C. Voigt, K. Mauersberger, J. M. Rosen, N. T. Kjome, “Comparison of chemical and optical in situ measurements of polar stratospheric clouds,” J. Geophys. Res. 105, 1491–1502 (2000).
[CrossRef]

J. Schreiner, C. Voigt, A. Kohlmann, F. Arnold, K. Mauersberger, N. Larsen, “Chemical analysis of polar stratospheric cloud particles,” Science 283, 968–970 (1999).
[CrossRef] [PubMed]

McDermid, I. S.

G. Beyerle, B. P. Luo, R. Neuber, Th. Peter, I. S. McDermid, “Temperature dependence of ternary solution particle volumes by lidar in the Arctic stratosphere during winter 1992/1993,” J. Geophys. Res. 102, 3603–3609 (1997).
[CrossRef]

Mikkelsen, I. S.

N. Larsen, I. S. Mikkelsen, B. M. Knudsen, J. Schreiner, C. Voigt, K. Mauersberger, J. M. Rosen, N. T. Kjome, “Comparison of chemical and optical in situ measurements of polar stratospheric clouds,” J. Geophys. Res. 105, 1491–1502 (2000).
[CrossRef]

Nakajima, H.

T. Sugita, Y. Kondo, M. Koike, M. Kanada, N. Toriyama, H. Nakajima, T. Deshler, R. Imasu, “Balloon-borne optical counter for in situ aerosol measurements,” J. Amos. Chem. 32, 183–204 (1999).
[CrossRef]

Neuber, R.

G. Beyerle, B. P. Luo, R. Neuber, Th. Peter, I. S. McDermid, “Temperature dependence of ternary solution particle volumes by lidar in the Arctic stratosphere during winter 1992/1993,” J. Geophys. Res. 102, 3603–3609 (1997).
[CrossRef]

Palmer, K. F.

Pantani, M.

M. Pantani, M. Del Guasta, D. Guzzi, L. Stefanutti, “Polar stratospheric clouds of liquid aerosol: an experimental determination of mean size distribution,” J. Aerosol Sci. 30, 559–567 (1999).
[CrossRef]

Peter, Th.

G. Beyerle, B. P. Luo, R. Neuber, Th. Peter, I. S. McDermid, “Temperature dependence of ternary solution particle volumes by lidar in the Arctic stratosphere during winter 1992/1993,” J. Geophys. Res. 102, 3603–3609 (1997).
[CrossRef]

B. Luo, U. K. Krieger, Th. Peter, “Densities and refractive indices of H2SO4/HNO3/H2O solutions to stratospheric temperatures,” Geophys. Res. Lett. 23, 3707–3710 (1996).
[CrossRef]

K. S. Carslaw, B. P. Luo, S. L. Clegg, Th. Peter, P. Brimblecombe, P. J. Crutzen, “Stratospheric aerosol growth and HNO3 gas phase depletion from coupled HNO3 and water uptake by liquid particles,” Geophys. Res. Lett. 21, 2479–2482 (1994).
[CrossRef]

U. K. Krieger, U. Weers, Th. Peter, “Using Mie-scattering resonances for determination of composition and radii of ternary droplets,” in Polar Stratospheric Ozone, J. A. Pyle, N. R. P. Harris, G. T. Amanatidis, eds., (Commission of the European Communities, Brussels, 1996).

Poole, L. R.

G. K. Yue, L. R. Poole, P.-H. Wang, E. W. Chiou, “Stratospheric aerosol acidity, density and refractive index deduced from SAGE II and NMC temperature data,” J. Geophys. Res. 99, 3727–3738 (1994).
[CrossRef]

Ravishankara, A. R.

K. D. Beyer, A. R. Ravishankara, E. R. Lovejoy, “Measurements of UV refractive indices and densities of H2SO4/H2O and H2SO4/HNO3/H2O solutions,” J. Geophys. Res. Atmos. 101, 14,519–14,524 (1996).
[CrossRef]

Robinette, P. A.

E. V. Browell, C. F. Butler, S. Ismail, P. A. Robinette, A. F. Carter, N. S. Higdon, O. B. Toon, M. R. Schoeberl, A. F. Tuck, “Airborne lidar observations in the wintertime Arctic stratosphere: polar stratospheric clouds,” Geophys. Res. Lett. 17, 385–388 (1990).
[CrossRef]

Rosen, J. M.

N. Larsen, I. S. Mikkelsen, B. M. Knudsen, J. Schreiner, C. Voigt, K. Mauersberger, J. M. Rosen, N. T. Kjome, “Comparison of chemical and optical in situ measurements of polar stratospheric clouds,” J. Geophys. Res. 105, 1491–1502 (2000).
[CrossRef]

Schoeberl, M. R.

E. V. Browell, C. F. Butler, S. Ismail, P. A. Robinette, A. F. Carter, N. S. Higdon, O. B. Toon, M. R. Schoeberl, A. F. Tuck, “Airborne lidar observations in the wintertime Arctic stratosphere: polar stratospheric clouds,” Geophys. Res. Lett. 17, 385–388 (1990).
[CrossRef]

Schreiner, J.

N. Larsen, I. S. Mikkelsen, B. M. Knudsen, J. Schreiner, C. Voigt, K. Mauersberger, J. M. Rosen, N. T. Kjome, “Comparison of chemical and optical in situ measurements of polar stratospheric clouds,” J. Geophys. Res. 105, 1491–1502 (2000).
[CrossRef]

J. Schreiner, C. Voigt, A. Kohlmann, F. Arnold, K. Mauersberger, N. Larsen, “Chemical analysis of polar stratospheric cloud particles,” Science 283, 968–970 (1999).
[CrossRef] [PubMed]

Smith, T. E.

T. E. Smith, R. F. Bonner, “Refractive index-temperature data for anhydrous ethyl alcohol,” Anal. Chem. 24, 517–518 (1952).
[CrossRef]

Stefanutti, L.

M. Pantani, M. Del Guasta, D. Guzzi, L. Stefanutti, “Polar stratospheric clouds of liquid aerosol: an experimental determination of mean size distribution,” J. Aerosol Sci. 30, 559–567 (1999).
[CrossRef]

Straub, J.

I. Thormählen, J. Straub, U. Grigull, “Refractive index of water and its dependence on wavelength, temperature, and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985).
[CrossRef]

Sugita, T.

T. Sugita, Y. Kondo, M. Koike, M. Kanada, N. Toriyama, H. Nakajima, T. Deshler, R. Imasu, “Balloon-borne optical counter for in situ aerosol measurements,” J. Amos. Chem. 32, 183–204 (1999).
[CrossRef]

Tabazadeh, A.

A. Tabazadeh, R. P. Turco, K. Drdla, M. Z. Jacobson, O. B. Toon, “A study of type I polar stratospheric cloud formation,” Geophys. Res. Lett. 21, 1619–1622 (1994).
[CrossRef]

Thormählen, I.

I. Thormählen, J. Straub, U. Grigull, “Refractive index of water and its dependence on wavelength, temperature, and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985).
[CrossRef]

Tisdale, R. T.

R. T. Tisdale, D. L. Glandorf, M. A. Tolbert, O. B. Toon, “Infrared optical constants of low temperature H2SO4 solutions representative of stratospheric sulfate aerosols,” J. Geophys. Res. Atmos. 103, 25,353–25,370 (1998).
[CrossRef]

Tolbert, M. A.

R. T. Tisdale, D. L. Glandorf, M. A. Tolbert, O. B. Toon, “Infrared optical constants of low temperature H2SO4 solutions representative of stratospheric sulfate aerosols,” J. Geophys. Res. Atmos. 103, 25,353–25,370 (1998).
[CrossRef]

Toon, O. B.

R. T. Tisdale, D. L. Glandorf, M. A. Tolbert, O. B. Toon, “Infrared optical constants of low temperature H2SO4 solutions representative of stratospheric sulfate aerosols,” J. Geophys. Res. Atmos. 103, 25,353–25,370 (1998).
[CrossRef]

A. Tabazadeh, R. P. Turco, K. Drdla, M. Z. Jacobson, O. B. Toon, “A study of type I polar stratospheric cloud formation,” Geophys. Res. Lett. 21, 1619–1622 (1994).
[CrossRef]

E. V. Browell, C. F. Butler, S. Ismail, P. A. Robinette, A. F. Carter, N. S. Higdon, O. B. Toon, M. R. Schoeberl, A. F. Tuck, “Airborne lidar observations in the wintertime Arctic stratosphere: polar stratospheric clouds,” Geophys. Res. Lett. 17, 385–388 (1990).
[CrossRef]

Toriyama, N.

T. Sugita, Y. Kondo, M. Koike, M. Kanada, N. Toriyama, H. Nakajima, T. Deshler, R. Imasu, “Balloon-borne optical counter for in situ aerosol measurements,” J. Amos. Chem. 32, 183–204 (1999).
[CrossRef]

Tuck, A. F.

E. V. Browell, C. F. Butler, S. Ismail, P. A. Robinette, A. F. Carter, N. S. Higdon, O. B. Toon, M. R. Schoeberl, A. F. Tuck, “Airborne lidar observations in the wintertime Arctic stratosphere: polar stratospheric clouds,” Geophys. Res. Lett. 17, 385–388 (1990).
[CrossRef]

Turco, R. P.

A. Tabazadeh, R. P. Turco, K. Drdla, M. Z. Jacobson, O. B. Toon, “A study of type I polar stratospheric cloud formation,” Geophys. Res. Lett. 21, 1619–1622 (1994).
[CrossRef]

Voigt, C.

N. Larsen, I. S. Mikkelsen, B. M. Knudsen, J. Schreiner, C. Voigt, K. Mauersberger, J. M. Rosen, N. T. Kjome, “Comparison of chemical and optical in situ measurements of polar stratospheric clouds,” J. Geophys. Res. 105, 1491–1502 (2000).
[CrossRef]

J. Schreiner, C. Voigt, A. Kohlmann, F. Arnold, K. Mauersberger, N. Larsen, “Chemical analysis of polar stratospheric cloud particles,” Science 283, 968–970 (1999).
[CrossRef] [PubMed]

Wang, P.-H.

G. K. Yue, L. R. Poole, P.-H. Wang, E. W. Chiou, “Stratospheric aerosol acidity, density and refractive index deduced from SAGE II and NMC temperature data,” J. Geophys. Res. 99, 3727–3738 (1994).
[CrossRef]

Weers, U.

U. K. Krieger, U. Weers, Th. Peter, “Using Mie-scattering resonances for determination of composition and radii of ternary droplets,” in Polar Stratospheric Ozone, J. A. Pyle, N. R. P. Harris, G. T. Amanatidis, eds., (Commission of the European Communities, Brussels, 1996).

Williams, D.

Wilson, J. C.

J. E. Dye, B. W. Gandrud, D. Baumgardner, K. R. Chan, G. V. Ferry, M. Loewenstein, K. K. Kelly, J. C. Wilson, “Observed particle evolution in the polar stratospheric cloud of January 24 1989,” Geophys. Res. Lett. 17, 413–415 (1990).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), pp. 803.

Yue, G. K.

G. K. Yue, L. R. Poole, P.-H. Wang, E. W. Chiou, “Stratospheric aerosol acidity, density and refractive index deduced from SAGE II and NMC temperature data,” J. Geophys. Res. 99, 3727–3738 (1994).
[CrossRef]

Anal. Chem. (1)

T. E. Smith, R. F. Bonner, “Refractive index-temperature data for anhydrous ethyl alcohol,” Anal. Chem. 24, 517–518 (1952).
[CrossRef]

Appl. Opt. (1)

Geophys. Res. Lett. (6)

B. Luo, U. K. Krieger, Th. Peter, “Densities and refractive indices of H2SO4/HNO3/H2O solutions to stratospheric temperatures,” Geophys. Res. Lett. 23, 3707–3710 (1996).
[CrossRef]

J. E. Dye, B. W. Gandrud, D. Baumgardner, K. R. Chan, G. V. Ferry, M. Loewenstein, K. K. Kelly, J. C. Wilson, “Observed particle evolution in the polar stratospheric cloud of January 24 1989,” Geophys. Res. Lett. 17, 413–415 (1990).
[CrossRef]

D. Baumgardner, J. E. Dye, B. Gandrud, K. Barr, K. Kelly, K. Chan, “Refractive indices of aerosols in the upper troposphere and lower stratosphere,” Geophys. Res. Lett. 23, 749–752 (1996).
[CrossRef]

E. V. Browell, C. F. Butler, S. Ismail, P. A. Robinette, A. F. Carter, N. S. Higdon, O. B. Toon, M. R. Schoeberl, A. F. Tuck, “Airborne lidar observations in the wintertime Arctic stratosphere: polar stratospheric clouds,” Geophys. Res. Lett. 17, 385–388 (1990).
[CrossRef]

K. S. Carslaw, B. P. Luo, S. L. Clegg, Th. Peter, P. Brimblecombe, P. J. Crutzen, “Stratospheric aerosol growth and HNO3 gas phase depletion from coupled HNO3 and water uptake by liquid particles,” Geophys. Res. Lett. 21, 2479–2482 (1994).
[CrossRef]

A. Tabazadeh, R. P. Turco, K. Drdla, M. Z. Jacobson, O. B. Toon, “A study of type I polar stratospheric cloud formation,” Geophys. Res. Lett. 21, 1619–1622 (1994).
[CrossRef]

J. Aerosol Sci. (1)

M. Pantani, M. Del Guasta, D. Guzzi, L. Stefanutti, “Polar stratospheric clouds of liquid aerosol: an experimental determination of mean size distribution,” J. Aerosol Sci. 30, 559–567 (1999).
[CrossRef]

J. Amos. Chem. (1)

T. Sugita, Y. Kondo, M. Koike, M. Kanada, N. Toriyama, H. Nakajima, T. Deshler, R. Imasu, “Balloon-borne optical counter for in situ aerosol measurements,” J. Amos. Chem. 32, 183–204 (1999).
[CrossRef]

J. Geophys. Res. (3)

N. Larsen, I. S. Mikkelsen, B. M. Knudsen, J. Schreiner, C. Voigt, K. Mauersberger, J. M. Rosen, N. T. Kjome, “Comparison of chemical and optical in situ measurements of polar stratospheric clouds,” J. Geophys. Res. 105, 1491–1502 (2000).
[CrossRef]

G. Beyerle, B. P. Luo, R. Neuber, Th. Peter, I. S. McDermid, “Temperature dependence of ternary solution particle volumes by lidar in the Arctic stratosphere during winter 1992/1993,” J. Geophys. Res. 102, 3603–3609 (1997).
[CrossRef]

G. K. Yue, L. R. Poole, P.-H. Wang, E. W. Chiou, “Stratospheric aerosol acidity, density and refractive index deduced from SAGE II and NMC temperature data,” J. Geophys. Res. 99, 3727–3738 (1994).
[CrossRef]

J. Geophys. Res. Atmos. (2)

R. T. Tisdale, D. L. Glandorf, M. A. Tolbert, O. B. Toon, “Infrared optical constants of low temperature H2SO4 solutions representative of stratospheric sulfate aerosols,” J. Geophys. Res. Atmos. 103, 25,353–25,370 (1998).
[CrossRef]

K. D. Beyer, A. R. Ravishankara, E. R. Lovejoy, “Measurements of UV refractive indices and densities of H2SO4/H2O and H2SO4/HNO3/H2O solutions,” J. Geophys. Res. Atmos. 101, 14,519–14,524 (1996).
[CrossRef]

J. Geophys. Res. Atmos. D (1)

M. E. Hervig, T. Deshler, “Stratospheric aerosol surface area and volume inferred from HALOE, CLAES, and ILAS measurements,” J. Geophys. Res. Atmos. D 103, 25,345–25,352 (1998).
[CrossRef]

J. Phys. Chem. Ref. Data (1)

I. Thormählen, J. Straub, U. Grigull, “Refractive index of water and its dependence on wavelength, temperature, and density,” J. Phys. Chem. Ref. Data 14, 933–945 (1985).
[CrossRef]

Science (1)

J. Schreiner, C. Voigt, A. Kohlmann, F. Arnold, K. Mauersberger, N. Larsen, “Chemical analysis of polar stratospheric cloud particles,” Science 283, 968–970 (1999).
[CrossRef] [PubMed]

Z. Phys. Chem. (1)

A. Haentzsch, F. Dürigen, “Über die chemische Veränderung von Säuren und Salzen in Lösung auf Grund refraktometrischer Daten,” Z. Phys. Chem. 134, 413–452 (1928), Z. Phys. Chem. 136, 1–17 (1928).

Z. Phys. Chem. B (1)

R. Lühdemann, “Über die Konzentrationsabhängigkeit der Äquivalentrefraktion einiger Salze und Säuren in wässriger Lösung,” Z. Phys. Chem. B 29, 133–149 (1935).

Other (6)

K. H. Hellwege, A. M. Hellwege, “Optische Konstanten,” in Chemie, Astronomie, Geophysik, Technik, Vol. II8 of Landolt–Börnstein Zahlenwerte und Funktionen für Physik (Springer-Verlag, Berlin, 1962), p. 901.

U. K. Krieger, U. Weers, Th. Peter, “Using Mie-scattering resonances for determination of composition and radii of ternary droplets,” in Polar Stratospheric Ozone, J. A. Pyle, N. R. P. Harris, G. T. Amanatidis, eds., (Commission of the European Communities, Brussels, 1996).

World Meteorological Organization, “Scientific Assessment of Ozone Depletion: 1998,” (WMO, Geneva, 1999).

K. N. Marsh, ed., “Recommended Reference Materials for the Realization of Physico-Chemical Properties” (Blackwell Scientific, Oxford, 1987).

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), pp. 803.

A copy of the fortran code (or alternatively a java code) can be obtained from the authors. Send an email to krieger@atmos.umnw.ethz.ch.

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Figures (6)

Fig. 1
Fig. 1

Deviation from the measured refractive index compared with the literature value versus temperature: △, H2O (534 nm); ▽, H2O (633 nm); □, H2O (783 nm); ⊗ methycyclohexane (534 nm); ⊙ methycyclohexane (633 nm), ×, trimethylpentane (534 nm); +, trimethylpentane (633 nm); ● ethanol (589 nm).

Fig. 2
Fig. 2

Schematic of Setup 2 for refractive-index measurements. For a detailed description, see text.

Fig. 3
Fig. 3

Refractive index versus temperature measurement of the binary solution with 35.09-wt. % H2SO4. Open and filled symbols mark two independent measurements: △, ▲, 351 nm; ○, ●, 534 nm; □, ■, 633 nm; ▽, ▼, 783 nm. Solid lines, the corresponding model calculations; for comparison, the dashed and dotted lines show the old model21 calculation based on the Palmer–Williams data17 for 351 and 783 nm, respectively.

Fig. 4
Fig. 4

Refractive index versus temperature of the ternary solution with 17.41-wt. % H2SO4 and 17.41-wt. % HNO3. All open symbols show measurements with setup 1: △, 351 nm; ○, 534 nm; □, 633 nm; ▽, 783 nm. Filled squares, 633 nm measured with setup 2. Lines show the corresponding model calculations.

Fig. 5
Fig. 5

Deviation of all measurements from the improved model prediction (Δn = n Experiment - n Model) versus refractive index. Top, uncorrected values; bottom, values for which the measurements with setup 1 have been corrected (see text). ○, Binary solutions measured with setup 1; ⊛, ternary solutions measured with setup 1; ●, measurements with setup 2.

Fig. 6
Fig. 6

○, Comparison of model prediction with the measurements of Pantani et al.7 All data fall within a ±0.0025 deviation from with the model predictions, except the data for the 60-wt. % HNO3 solution (see text). As a reference, Lühdemann’s data20 for a 61.89-wt. % HNO3 solution (●) and the data of this study for the 65.10-wt. % HNO3 solution (+) are plotted as well.

Tables (3)

Tables Icon

Table 1 Refractive Indices of the Calibration Liquids

Tables Icon

Table 2 Concentrations of the Samples Measured

Tables Icon

Table 3 Results of the Measurements

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

n2λ, Ws, Wn, T=1+2aλ, Ws, Wn, T1-aλ, Ws, Wn, T,
aλ, Ws, Wn, T=i NiTAiλ, Ws, Wn,
NiT=ρTWiMi,
Aiλ, Ws, Wn=Aibλ, Ws+Wn,
Aibλ, W=aibT0, λ, W-NwT0AwλNiT0,
aibT0, λ, W=nibT0, λ, W2-1nibT0, λ, W2+2.

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