__Saturationvapor pressure formulations__

HolgerVömel

CIRES,University of Colorado, Boulder

A large number of saturation vaporpressure equations exists to calculate the pressure of water vaporover a surface of liquid water or ice. This is a brief overview ofthe most important equations used. Several useful reviews of theexisting vapor pressure curves are listed in the references. Pleasenote the discussion of the WMO formulations.

#### 1) Vapor pressure over liquidwater below 0°C

Goff Gratch equation

*(Smithsonian Tables, 1984, after Goff andGratch, 1946)*:

Log_{10} *e*_{w}= -7.90298(373.16/*T*-1)[1]

+ 5.02808 Log_{10}(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)

+ Log_{10}(1013.246)

with *T* in [K] and *e*_{w}in [hPa]

Guide to MeteorologicalInstruments 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):*

Log_{10} e_{w}= 10.79574(1-273.16/*T*)[3]

- 5.02800 Log_{10}(*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 itsrecommendation on a paper by Goff (1957), which is shown here. Therecommendation published by WMO (1988) has several typographicalerrors 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 publicationby Goff (1957). Note the different sign of the exponent. The earlier1984 edition shows the correct formula.)

Hyland and Wexler

*(Hylandand Wexler, 1983):*

Log *e*_{w} =-0.58002206 10^{4} /*T*[4]

+ 0.13914993 10^{1}

- 0.48640239 10^{-1} *T*

+ 0.41764768 10^{-4} *T*^{2}

- 0.14452093 10^{-7} *T*^{3}

+ 0.65459673 10^{1} Log(*T*)

with *T* in [K]and *e*_{w} in [Pa]

Buck

*(BuckResearch Manual (1996); updated equation from Buck, A. L., Newequations 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]

withtin [°C] ande_{w}in[hPa]

Sonntag

*(Sonntag,1994)*

Log e_{w} =-6096.9385 /*T*[7]

+ 16.635794

- 2.711193 10^{-2} * *T*

+ 1.673952 10^{-5} * *T*^{2}

+ 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

*(Murphyand Koop, 2005)*(Video) Formulas of Volume Pressure Water Vapor

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 forthe Properties of Water and Steam (IAPWS) Formulation 1995

*(Wagnerand Pruß, 2002)*

Log (e_{w}*/22.064e6*)= 647.096/T * ((-7.85951783 ν

+ 1.84408259 ν^{1.5}

- 11.7866497 ν^{3}

+ 22.6807411 ν^{3.5}

- 15.9618719 ν^{4}

+ 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 theseare based on theoretical studies and only a small number are based onactual measurements of the vapor pressure. The Goff Gratch equation[1] for the vapor pressure over liquid water covers a region of -50°Cto 102°C [Gibbins 1990]. This work is generally considered thereference equation but other equations are in use in themeteorological community [Elliott and Gaffen, 1993]. There is a verylimited number of measurements of the vapor pressure of water oversupercooled liquid water at temperatures below °C. Detwiler[1983] claims some indirect evidence to support the extrapolation ofthe Goff-Gratch equation down to temperatures of -60°C. However,this currently remains an open issue.

The WMO Guide toMeteorological Instruments and Methods of Observation (CIMO Guide,WMO No. 8) formulation [2] is widely used in Meteorology and appealsfor 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.

TheHyland and Wexler formulation is used by Vaisala and is very similarto the formula by Sonntag [7]. The comparison for the liquidsaturation vapor pressure equations [2]-[11] with the Goff-Gratchequation [1] in figure 1 shows that uncertainties at low temperaturesbecome increasingly large and reach the measurement uncertaintyclaimed 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%. ForRH values reported in the low and mid troposphere the influence ofthe saturation vapor pressure formula used is small and onlysignificant for climatological studies [Elliott and Gaffen 1993].

The WMO (WMO No. 49, Technical Regulations) recommendedformula [3] is a derivative of the Goff-Gratch equation, originallypublished by Goff (1957). The differences between Goff (1957) andGoff-Gratch (1946) are less than 1% over the entire temperaturerange. The formulation published by WMO (1988) cannot be used due toseveral typographical errors. The corrected formulation WMO (2000)differs in the sign of one exponent compared to Goff (1957). Thisincorrect formulation is in closer agreement with the Hyland andWexler formulation; however, it is to be assumed that Goff (1957) wasto be recommended.

The review of vapor pressures ofice and supercooled water by Murphy and Kopp (2005) provides aformulation [10] based on recent data on the molar heat capacity ofsupercooled water. The comparison of the the vapor pressure equationswith 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 liquidwater down to -38°C. Their result indicates that at the lowesttemperatures the measured vapor pressure may be as much as 10% lowerthan the value given by the Smithsonian Tables [1], and as shown infigure 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 asfor small water droplets.

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

It is important to note that in theupper troposphere, water vapor measurements reported in the WMOconvention as relative humidity with respect to liquid water dependcritically on the saturation vapor pressure equation that was used tocalculate the RH value.

Figure 1: Comparison of equations[2]-[11] with the Goff Gratch equation [1] for the saturationpressure of water vapor over liquid water. The measurements by Fukutaet al. [2003] are shown as well. ^{(*)}WMO(2000) is also shown. This is based on Goff (1957) with the differentsign of one exponent, likely due to a typographical error.

Figure 2: Comparison of several equations with the equation by Sonntag [7] for the saturation pressure of water vapor over liquid water.

The equations by Hyland and Wexler [4], the nearly identical equation by Wexler (1976, see reference below) and the equation by Sonntag [7] are the most commonly used equations among radiosonde manufacturers and should be used in upper air applications to avoid inconsistencies.

#### 2) Vapor pressure over ice

Goff Gratch equation

*(Smithsonian Tables, 1984)*:

Log_{10} e_{i}= -9.09718 (273.16/*T* -1)[12]

- 3.56654 Log_{10}(273.16/ *T*)

+ 0.876793 (1 - *T*/ 273.16)

+ Log_{10}(6.1071)

with *T* in [K] and e_{i}in [hPa]

Hyland and Wexler

*(Hylandand Wexler, 1983.):*

Log e_{i} =-0.56745359 10^{4} /*T*[13]

+ 0.63925247 10^{1}

- 0.96778430 10^{-2} *T*

+ 0.62215701 10^{-6} *T*^{2}

+ 0.20747825 10^{-8} *T*^{3}

- 0.94840240 10^{-12} *T*^{4}

+ 0.41635019 10^{1} Log(*T*)

with *T* in [K]and e_{i} in [Pa]

Guide to MeteorologicalInstruments and Methods of Observation (CIMO Guide)

*(WMO,2008)*

*e*_{i} =6.112 e^{(}^{22.46 t/(272.62 +t}^{))}[14]

with *t* in [°C] and e_{i} in [hPa]

Magnus Teten

*(Murray,1967)*

*e*_{i} =6.1078 e^{21.8745584 * }^{(T-273.16)}^{ /(}^{T –7.66}^{)}[15]

with *T* in [K] and e_{w} in [hPa]

Buck

*(BuckResearch Manual, 1996)*

e_{i}= 6.1115 e^{(23.036 - }^{t}^{ /333.7) }^{t}^{ / (279.82 + }^{t}^{)}[1996][16]e_{i}= 6.1115 e^{22.452 }^{t}^{/ (272.55+}^{t}^{)}[1981][17]

withtin [°C] and e_{i}in [hPa]

Marti Mauersberger

*(Martiand Mauersberger, 1993)*

Log_{10}e_{i}= -2663.5 /T+12.537[18]

withTin [K] and e_{i}in [Pa]

Murphy and Koop

*(Murphyand Koop, 2005)*

Log e_{i}=9.550426

- 5723.265/T

+ 3.53068 Log(T)

- 0.00728332T[19]

withTin [K] and e_{i}in [Pa]

The Goff Gratch equation [11] forthe vapor pressure over ice covers a region of -100°C to 0°C.It is generally considered the reference equation; however, otherequations have also been widely used. The equations discussed hereare mostly of interest for frost-point measurements using chilledmirror hygrometers, since these instruments directly measure thetemperature at which a frost layer and the overlying vapor are inequilibrium. In meteorological practice, relative humidity is givenover liquid water (see section 1) and care needs to be taken toconsider this difference.

Buck Research, which manufacturesfrost-point hygrometers, uses the Buck formulations in theirinstruments. These formulations include an enhancement factor, whichcorrects for the differences between pure vapor and moist air. Thisenhancement factor is a weak function of temperature and pressure andcorrects about 0.5% at sea level. For the current discussion it hasbeen omitted.

The Marti Mauersberger equation is the onlyequation based on direct measurements of the vapor pressure down totemperatures of 170 K.

The comparison of equations 12-17 with theGoff Gratch equation (figure 3) shows, that with the exception of theMagnus Teten formula, the deviations in the typical meteorologicalrange of -100°C to 0°C are less than 2.5%, and smaller thantypical instrumental errors of frost-point hygrometers of 5-10%.

Notshown is the WMO recommended equation for vapor pressure over ice,since it is nearly identical with the Goff-Gratch equation [12].

Figure 3: Comparison of equations[13]-[18] with the Goff Gratch equation [12] for the saturationpressure of water vapor over ice.

#### 3) References

Bolton, D., Thecomputation of equivalent potential temperature, Monthly WeatherReview, 108, 1046-1053, 1980..

Buck, A. L.,New equations for computing vapor pressure and enhancement factor, J.Appl. Meteorol., 20, 1527-1532, 1981.

BuckResearch Manuals, 1996

Detwiler, A.,Extrapolation of the Goff-Gratch formula for vapor pressure overliquid 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 reportedupper 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., Asurvey and comparison of relationships for the determination of thesaturation vapour pressure over plane surfaces of pure water and ofpure ice, Annales Geophys., 8, 859-886, 1990.

Goff,J. A., and S. Gratch, Low-pressure properties of water from -160 to212 F, in Transactions of the American society of heating andventilating engineers, pp 95-122, presented at the 52nd annualmeeting of the American society of heating and ventilating engineers,New York, 1946.

Goff, J. A. Saturationpressure of water on the new Kelvin temperature scale, Transactionsof the American society of heating and ventilating engineers, pp347-354, presented at the semi-annual meeting of the American societyof heating and ventilating engineers, Murray Bay, Que. Canada, 1957.

Hyland, R. W. and A. Wexler, Formulations for theThermodynamic Properties of the saturated Phases of H2O from 173.15Kto 473.15K, ASHRAE Trans, 89(2A), 500-519, 1983.

Marti,J. and K Mauersberger, A survey and new measurements of ice vaporpressure at temperatures between 170 and 250 K, GRL 20, 363-366, 1993

Murphy, D. M. and T. Koop, Review of the vapourpressures 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, TheIAPWS formulation 1995 for the thermodynamic properties of ordinarywater substance for general and scientific use, J. Phys. Chem. Ref.Data, 31, 387-535, 2002.

Wexler, A., Vapor Pressure Formulation for Water in Range 0 to 100°C. A Revision, Journal of Research of the National Bureau of Standards, 80A, 775-785, 1976.

World MeteorologicalOrganization, General meteorological standards and recommendedpractices, Appendix A, WMO Technical Regulations, WMO-No. 49, Geneva1988.

World Meteorological Organization,General meteorological standards and recommended practices, AppendixA, WMO Technical Regulations, WMO-No. 49, corrigendum, Geneva August2000.

World Meteorological Organization, Guide to MeteorologicalInstruments and Methods of Observation, Appendix 4B, WMO-No. 8 (CIMOGuide), Geneva 2008.

1 December 2011

Holger.Voemel@Colorado.edu

## FAQs

### Water Vapor Pressure Formulations? ›

**simple_pressure = e^(20.386 - (5132 / (temperature + 273))** , where vapor pressure is expressed in mmHg and temperature in kelvins.

### How do you calculate saturated water vapor pressure? ›

Moisture comes in several forms for materials exposed outdoors: humidity, condensation (dew), and rain. Relative humidity (RH) is the ratio of the ambient vapor pressure of water to the saturation water vapor pressure: **RH=p _{w}/p_{s}**.

### How many types of vapor pressure are there? ›

Substance | Vapor pressure | Temperature (°C) |
---|---|---|

(Pa) | ||

Xenon difluoride | 600 Pa | 25 |

Water (H_{2}O) | 2.3 kPa | 20 |

Propanol | 2.4 kPa | 20 |

### How do you calculate water vapor? ›

The water vapor mixing ratio, w, is typically at most about 40 g kg^{–}^{1} or 0.04 kg kg^{–}^{1}, so even for this much water vapor, **q = 0.040/(1 + 0.040) = 0.038 or 38 g kg ^{–}^{1}**. Thus, water vapor mixing ratio and specific humidity are the same to within a few percent.

### What is the standard vapor pressure of water? ›

Boiling Point

For water, the vapor pressure reaches the standard sea level atmospheric pressure of **760 mmHg at 100°C**.

### What is the difference between vapour pressure and saturated vapour pressure? ›

Saturation pressure is defined as the temperature at which the liquids start to boil at the given temperature. Vapour pressure is the measure of the tendency of the liquid to change into a vapour. It is the pressure exerted by the vapour in thermodynamic equilibrium on its condensed phase at a given temperature.

### How do you calculate vapor pressure? ›

In chemistry, vapor pressure is the pressure that is exerted on the walls of a sealed container when a substance in it evaporates (converts to a gas). To find the vapor pressure at a given temperature, use the Clausius-Clapeyron equation: **ln(P1/P2) = (ΔH _{vap}/R)((1/T2) - (1/T1))**.

### Which method is used for determination of vapour pressure? ›

Two major experimental methods are used for the measurement of vapor pressure at high temperatures: **the boiling point method and the transpiration method**. Pressure equalization occurs very rapidly in the boiling point method, whereas thermal transport is rather slow.

### Which factors affect vapor pressure? ›

Vapor pressure is the pressure caused by the evaporation of liquids. Three common factors that influence vapor press are **surface area, intermolecular forces and temperature**. The vapor pressure of a molecule differs at different temperatures.

### How do you calculate water vapor mixing ratio? ›

The mixing ratio of water vapor q is defined by **q=ρw/ρ**, where ρw is the density of water vapor and ρ is the density of dry air.

### What are the units of vapor pressure? ›

The most common unit for vapor pressure is the torr. **1 torr = 1 mm Hg (one millimeter of mercury)**. Most materials have very low vapor pressures.

### At what pressure will water be a vapor at 0 C? ›

At 0° C the molecules of liquid water move slowly, their escaping tendency is small, and the equilibrium vapor pressure above the liquid is only **4.6 mm Hg**.

### Can you calculate water pressure from flow rate? ›

Find the square of volumetric flow rate. Find the square of flow factor. Divide the square of volumetric flow rate by the square of flow factor. Multiply the resultant with the specific gravity of the fluid to obtain the differential pressure.

### How do you calculate water pressure in a water tank? ›

water pressure calculator at height - YouTube

### Does vapor pressure increase with temperature? ›

As the temperature of a liquid increases, the kinetic energy of its molecules also increases and as the kinetic energy of the molecules increases, the number of molecules transitioning into a vapor also increases, thereby increasing the vapor pressure.

### What happens to saturation vapor pressure when temperature increases? ›

It is important to understand that saturation vapor pressure (or the maximum amount of water vapor that can be in the air) **increases exponentially** as temperature increases. The sharp increase in saturation vapor pressure with increasing temperature continues beyond 40°C (104°F).

### What is the relationship between temperature and saturation vapor pressure? ›

**As the temperature rises, the saturated vapour pressure increases rapidly**, and so does the density of the vapour, approaching that of the liquid. At a certain temperature the density of the vapour becomes equal to that of the liquid, and the vapour and liquid become indistinguishable.

### How is vapor composition calculated? ›

️ Calculating the Vapor Pressure of a Two-Component Solution ...

### What is example of vapor pressure? ›

It is important to note that when a liquid is boiling, its vapor pressure is equal to the external pressure. For example, **as water boils at sea level, its vapor pressure is 1 atmosphere** because the external pressure is also 1 atmosphere.

### How do you calculate vapor pressure using Raoult's Law? ›

Using Raoult's Law to calculate the vapor pressure of a component

### Which solution has the highest vapour pressure? ›

- Glucose in the equimolar mixture has higher vapour pressure.
- Hence, option A is correct.

### How do you calculate vapor pressure from evaporation? ›

Evaporation and Vapor Pressure - YouTube

### What does the Clausius-Clapeyron equation represent? ›

The Clapeyron equation (also called the Clausius-Clapeyron equation) **relates the slope of a reaction line on a phase diagram to fundamental thermodynamic properties**.

### What two factors determine the vapor pressure of a solution? ›

Learning Objectives. To know how and why the vapor pressure of a liquid varies with temperature. To understand that the equilibrium vapor pressure of a liquid depends on **the temperature and the intermolecular forces present**.

### Does vapor pressure depend on volume? ›

Vapor pressure depends on the composition of the liquid and the temperature. **Vapor pressure neither depend on the total surface area of the liquid, nor on the volume of the liquid in the container, nor on the volume of container itself**, just as long as some liquid remain when equilibrium is reached.

### Does vapor pressure depend on viscosity? ›

Answer: **The vapour pressure of a liquid depends upon the nature of liquid and temperature**.

### How do you calculate mix ratios? ›

**Divide 1 by the total number of parts (water + solution)**. For example, if your mix ratio is 8:1 or 8 parts water to 1 part solution, there are (8 + 1) or 9 parts. The mixing percentage is 11.1% (1 divided by 9). Need another example?

### How do you calculate water vapor pressure from temperature? ›

Simple formula

**simple_pressure = e^(20.386 - (5132 / (temperature + 273))** , where vapor pressure is expressed in mmHg and temperature in kelvins.

### What's the mix ratio? ›

What is the mixing ratio? When we talk about mixing ratio, we mean **the amount of each substance in a mixture compared to the total amount of the mixture**. The amount of a given substance can be expressed as a percent of the full 100% of a mixture.

### Why the vapor pressure of water is important? ›

Water's vapor pressure is very important to life forms on Earth because **its value is sufficiently high to let the process of evaporation to occur, but sufficiently low to also allow water to exist in liquid and solid forms**.

### How do you calculate vapor pressure? ›

In chemistry, vapor pressure is the pressure that is exerted on the walls of a sealed container when a substance in it evaporates (converts to a gas). To find the vapor pressure at a given temperature, use the Clausius-Clapeyron equation: **ln(P1/P2) = (ΔH _{vap}/R)((1/T2) - (1/T1))**.

### How do you calculate saturation vapor density? ›

For low temperatures (below approximately 400 K), SVD can be approximated from the SVP by the ideal gas law: **P V = n R T** where P is the SVP, V is the volume, n is the number of moles, R is the gas constant and T is the temperature in kelvins.

### What is saturated water vapor? ›

Definition of saturated vapor

: **vapor at the temperature of the boiling point corresponding to its pressure and so incapable of being compressed or cooled without condensing** — compare equilibrium sense 1c.

### How do you calculate saturated temperature? ›

So, if you are able to determine the pressure at any of these points (evaporator, condenser, or receiver), you can easily determine the “saturation” temperature by **finding the measured pressure on the P-T card and reading the corresponding temperature**.