Saturation vapor pressure formulations

Holger Vömel
CIRES, University of Colorado, Boulder
 

There is a large number of saturation vapor pressure equations used to calculate the pressure of water vapor over a surface of liquid water or ice. This is a brief overview of the most important equations used. Several useful reviews of the existing vapor pressure curves are listed in the references. Please note the updated discussion of the WMO formulation.

1) Vapor Pressure over liquid water below 0°C

  • Goff Gratch equation

    • (Smithsonian Tables, 1984, after Goff and Gratch, 1946):
  • WMO

    • (Goff, 1957):
  • Hyland and Wexler

    • (Hyland and Wexler, 1983):
  • Buck

    • (Buck Research Manual (1996); updated equation from Buck, A. L., New equations for computing vapor pressure and enhancement factor, J. Appl. Meteorol., 20, 1527-1532, 1981)
    pw = 6.1121  e(18.678 - t / 234.5) t / (257.14 + t)     [1996]                               [4]
    pw = 6.1121 e17.502 t / (240.97 + t)      [1981]                                                [5]
    with t in [°C] and pw in [hPa]
  • Sonntag

    • (Sonntag, 1994)
  • Magnus Teten

    • (Murray, 1967)
  • Bolton

    • (Bolton, 1980)
  • Murphy and Koop

    • (Murphy and Koop, 2005)
  • International Association for the Properties of Water and Steam (IAPWS) Formulation 1995

    • (Wagner and Pruß, 2002) At low temperatures most of these are based on theoretical studies and only a small number are based on actual measurements of the vapor pressure. The Goff Gratch equation [1] for the vapor pressure over liquid water covers a region of -50°C to 102°C [Gibbins 1990]. This work is generally considered the reference equation but other equations are in use in the meteorological community [Elliott and Gaffen, 1993]. There is a very limited number of measurements of the vapor pressure of water over supercooled liquid water at temperatures below °C. Detwiler [1983] claims some indirect evidence to support the extrapolation of the Goff-Gratch equation down to temperatures of -60°C. However, this currently remains an open issue.
    The Hyland and Wexler formulation is used by Vaisala and is very similar to the formula by Sonntag (6). The Magnus Teten formulation [7] is widely used in Meteorology and appeals for its simplicity.
    The comparison for the liquid saturation vapor pressure equations [2]-[8] with the Goff-Gratch equation [1] in figure 1, shows that uncertainties at low temperatures become increasingly large and reach the measurement uncertainty claimed by some RH sensors. At -60°C the deviations range from -6% to +3% and at -70°C the deviations range from -9% to +6%. For RH values reported in the low and mid troposphere the influence of the saturation vapor pressure formula used is small and only significant for climatological studies [Elliott and Gaffen 1993].
    The WMO recommended formula is a derivative of the Goff-Gratch equation, originally published by Goff (1957). The differences between Goff (1957) and Goff-Gratch (1946) are less than 1% over the entire temperature range. The formulation published by WMO (1988) cannot be used due to several typographical errors. The corrected formulation WMO (2000) still differs in the sign of one exponent compared to Goff (1957). This incorrect formulation is in closer agreement with the Hyland and Wexler formulation; however, it is to be assumed that Goff (1957) was to be recommended.

    The most recent review of vapor pressures of ice and supercooled water by Murphy and Kopp (2005) provide a formulation [9] based on recent data on the molar heat capacity of supercooled water. The comparison of the the vapor pressure equations [1] - [8] with the formulation by Murphy and Koop [9] is shown in figure 2.

    The study by Fukuta and Gramada [2003] shows direct measurements of the vapor pressure over liquid water down to -38°C. Their result indicates that at the lowest temperatures the measured vapor pressure may be as much as 10% lower than the value given by the Smithsonian Tables [1], and as shown in figure 1 lower as any other vapor pressure formulation. However, these data are in conflict with measured molar heat capacity data (Muprhy and Koop, 2005), which have been measured both for bulk as for small water droplets.

    Like most other formulations, the IAPWS formulation 1995 (Wagner and Pruß, 2002) are valid only above the triple point. The IAWPS formulation 1995 (Wagner and Pruß, 2002) is valid in the temperature range 273.16 K < T < 647.096 K.

    It is important to note that in the upper troposphere, water vapor measurements reported in the WMO convention as relative humidity with respect to liquid water depend critically on the saturation vapor pressure equation that was used to calculate the RH value.


    Figure 1: Comparison of equations [2]-[10] with the Goff Gratch equation [1] for the saturation pressure of water vapor over liquid water. The measurements by Fukuta et al. [2003] are shown as well.
    (*)WMO (2000) is also shown. This is based on Goff (1957) with the different sign of one exponent, likely due to a typographical error.

    Figure 2: Comparison of equations [1]-[8], [10] with the Murphy Koop equation [9] for the saturation pressure of water vapor over liquid water.
    (*)WMO (2000) is also shown. This is based on Goff (1957) with the different sign of one exponent, likely due to a typographical error.

    2) Vapor Pressure over ice

  • Goff Gratch equation

    • (Smithsonian Tables, 1984):
  • Hyland and Wexler

    • (Hyland and Wexler, 1983.):
  • Magnus Teten

    • (Murray, 1967)
  • Buck

    • (Buck Research Manual, 1996)
    pi = 6.1115  e(23.036 - t / 333.7) t / (279.82 + t)     [1996]                                 [14]
    pi = 6.1115 e22.452 t / (272.55+t)      [1981]                                                    [15]
    with t in [°C] and pi in [hPa]
  • Marti Mauersberger

    • (Marti and Mauersberger, 1993)
    Log10 pi = -2663.5 / T + 12.537                                                                   [16]
    with T in [K] and pi in [Pa]
  • Murphy and Koop

    • (Murphy and Koop, 2005)
    Log pi = 9.550426 - 5723.265/T + 3.53068 Log(T) - 0.00728332 T      [17]
    with T in [K] and pi in [Pa]
    The Goff Gratch equation [11] for the vapor pressure over ice cover a region of -100°C to 0°C. It is generally considered the reference equation; however, other equations have also been widely used. The equations discussed here are mostly of interest for frost-point measurements using chilled mirror hygrometers, since these instruments directly measure the temperature at which a frost layer and the overlying vapor are in equilibrium. In meteorological practice, relative humidity is given over liquid water (see section 1) and care needs to be taken to consider this difference.
    Buck Research, which manufactures frost-point hygrometers, uses the Buck formulations in their instruments. These formulations include an enhancement factor, which corrects for the differences between pure vapor and moist air. This enhancement factor is a weak function of temperature and pressure and corrects about 0.5% at sea level. For the current discussion it has been omitted.
    The Marti Mauersberger equation is the only equation based on direct measurements of the vapor pressure down to temperatures of 170 K.
    The comparison of equations 12-17 with the Goff Gratch equation (figure 3) shows, that with the exception of the Magnus Teten formula, the deviations in the typical meteorological range of -100°C to 0°C are less than 2.5%, and smaller than typical instrumental errors of frost-point hygrometers of 5-10%.
    Not shown is the WMO recommended equation for vapor pressure over ice, since it is nearly identical with the Goff-Gratch equation [11].


    Figure 3: Comparison of equations [12]-[17] with the Goff Gratch equation [11] for the saturation pressure of water vapor over ice.

    3) References

    Bolton, D., The computation of equivalent potential temperature, Monthly Weather Review, 108, 1046-1053, 1980..
    Buck, A. L., New equations for computing vapor pressure and enhancement factor, J. Appl. Meteorol., 20, 1527-1532, 1981.
    Buck Research Manuals, 1996
    Detwiler, A., Extrapolation of the Goff-Gratch formula for vapor pressure over liquid water at temperatures below 0°C, J. Appl. Meteorol., 22, 503, 1983.
    Elliott, W. P. and D. J. Gaffen, On the utility of radiosonde humidity archives for climate studies, Bull. Am. Meteorol. Soc., 72, 1507-1520, 1991.
    Elliott, W. P. and D. J. Gaffen, Effects of conversion algorithms on reported upper air dewpoint depressions, Bull. Am. Meteorol. Soc., 74, 1323-1325, 1993.
    Fukuta, N. and C. M. Gramada, Vapor pressure measurement of supercooled water, J. Atmos. Sci., 60, 1871-1875, 2003.
    Gibbins, C. J., A survey and comparison of relationships for the determination of the saturation vapour pressure over plane surfaces of pure water and of pure ice, Annales Geophys., 8, 859-886, 1990.
    Goff, J. A., and S. Gratch, Low-pressure properties of water from -160 to 212 F, in Transactions of the American society of heating and ventilating engineers, pp 95-122, presented at the 52nd annual meeting of the American society of heating and ventilating engineers, New York, 1946.
    Goff, J. A. Saturation pressure of water on the new Kelvin temperature scale, Transactions of the American society of heating and ventilating engineers, pp 347-354, presented at the semi-annual meeting of the American society of heating and ventilating engineers, Murray Bay, Que. Canada, 1957.
    Hyland, R. W. and A. Wexler, Formulations for the Thermodynamic Properties of the saturated Phases of H2O from 173.15K to 473.15K, ASHRAE Trans, 89(2A), 500-519, 1983.
    Marti, J. and K Mauersberger, A survey and new measurements of ice vapor pressure at temperatures between 170 and 250 K, GRL 20, 363-366, 1993
    Murphy, D. M. and T. Koop, Review of the vapour pressures of ice and supercooled water for atmospheric applications, Quart. J. Royal Met. Soc, 131, 1539-1565, 2005.
    Murray, F. W., On the computation of saturation vapor pressure, J. Appl. Meteorol., 6, 203-204, 1967.
    Smithsonian Met. Tables,  5th ed., pp. 350, 1984.
    Sonntag, D., Advancements in the field of hygrometry, Meteorol. Z., N. F., 3, 51-66, 1994.
    Wagner W. and A. Pruß, The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use, J. Phys. Chem. Ref. Data, 31, 387-535, 2002.
    World Meteorological Organization, General meteorological standards and recommended practices, Appendix A, WMO Technical Regulations, WMO-No. 49, 1988.
    World Meteorological Organization, General meteorological standards and recommended practices, Appendix A, WMO Technical Regulations, WMO-No. 49, corrigendum, August 2000.

    IDL source for all equations

    27 November 2006
    Holger.Voemel@Colorado.edu