Water Vapor Pressure Formulations

Saturation vapor pressure formulations

Holger Vömel
CIRES, University of Colorado, Boulder

A large number of saturation vapor pressure equations exists 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 discussion of the WMO formulations.

1) Vapor pressure over liquid water below 0°C

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

Log₁₀ e_w = -7.90298 (373.16/T-1)                                                              [1]
                    + 5.02808 Log₁₀(373.16/T)
                    - 1.3816 10^-7(10^{11.344 (1-}^T^/373.16) -1)
                   + 8.1328 10^-3(10^{-3.49149 (373.16/}^T^-1) -1)
                   + Log₁₀(1013.246)
with T in [K] and e_w in [hPa]

Guide to Meteorological Instruments and Methods of Observation (CIMO Guide)
(WMO, 2008)

e_w = 6.112 e^{(17.62 t/(243.12 +
t))} [2]
with t in [°C] and e_w in [hPa]

WMO
(Goff, 1957):

Log₁₀ e_w = 10.79574 (1-273.16/T)                                                          [3]
                    - 5.02800 Log₁₀(T/273.16)
                    + 1.50475 10^-4(1 - 10^(-8.2969*(^T^/273.16-1)))
                    + 0.42873 10^-3(10^{(+4.76955*(1-273.16/}^T⁾⁾- 1)
                    + 0.78614
with T in [K] and e_w in [hPa]

(Note: WMO based its recommendation on a paper by Goff (1957), which is shown here. The recommendation published by WMO (1988) has several typographical errors and cannot be used. A corrigendum (WMO, 2000) shows the term +0.42873 10^-3(10^{(-4.76955*(1-273.16/}^T⁾⁾- 1) in the fourth line compared to the original publication by Goff (1957). Note the different sign of the exponent. The earlier 1984 edition shows the correct formula.)

Hyland and Wexler
(Hyland and Wexler, 1983):

Log e_w = -0.58002206 10⁴ / T                                                                  [4]
              + 0.13914993 10¹
              - 0.48640239 10^-1 T
              + 0.41764768 10^-4 T²
              - 0.14452093 10^-7 T³
              + 0.65459673 10¹ Log(T)
with T in [K] and e_w in [Pa]

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)

e_w = 6.1121 e^{(18.678 -}^t^/
234.5)^t^{/ (257.14 +}^t⁾ [1996] [5]
e_w = 6.1121 e^17.502^t^{/ (240.97 +}^t⁾ [1981] [6]
with t in [°C] and e_w in [hPa]

Sonntag
(Sonntag, 1994)

Log e_w = -6096.9385 / T                                                                             [7]
                 + 16.635794
                 - 2.711193 10^-2 * T
                 + 1.673952 10^-5 * T²
                 + 2.433502 * Log(T)
with T in [K] and e_w in [hPa]

Magnus Tetens
(Murray, 1967)

e_w = 6.1078 e^{17.269388 *}^(T-273.16)^/
(^{T –
35.86}⁾ [8]
with T in [K] and e_w in [hPa]

Bolton
(Bolton, 1980)

e_w = 6.112 e^{17.67 *}^t^/
(^t^+243.5) [9]
with t in [°C] and e_w in [hPa]

Murphy and Koop
(Murphy and Koop, 2005)

Log e_w = 54.842763
                - 6763.22 / T
                - 4.21 Log(T)
                + 0.000367 T
                + Tanh{0.0415 (T - 218.8)}
                · (53.878 - 1331.22 / T - 9.44523 Log(T) + 0.014025 T)         [10]

with T in [K] and e_w in [Pa]

International Association for the Properties of Water and Steam (IAPWS) Formulation 1995
(Wagner and Pruß, 2002)

Log (e_w/22.064e6) = 647.096/T * ((-7.85951783 ν
                                                            + 1.84408259 ν^1.5
                                                             - 11.7866497 ν³
                                                            + 22.6807411 ν^3.5
                                                             - 15.9618719 ν⁴
                                                            + 1.80122502 ν^7.5))                        [11]

with T in [K] and e_w in [Pa] and ν = 1 - T/647.096

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 WMO Guide to Meteorological Instruments and Methods of Observation (CIMO Guide, WMO No. 8) formulation [2] is widely used in Meteorology and appeals for its simplicity. Together with the formulas by Bolton [9] and Buck [6] it has the same mathematical form as older the Maguns Tetens [8] formula and differs only in the value of the parameters.
The Hyland and Wexler formulation is used by Vaisala and is very similar to the formula by Sonntag [7]. The comparison for the liquid saturation vapor pressure equations [2]-[11] 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 (WMO No. 49, Technical Regulations) recommended formula [3] 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) 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 review of vapor pressures of ice and supercooled water by Murphy and Kopp (2005) provides a formulation [10] based on recent data on the molar heat capacity of supercooled water. The comparison of the the vapor pressure equations with the formulation by Murphy and Koop 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]-[11] 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)