JOURNAL OF RESEARCH of the National Bureau of Standards - A. Physics and Chemistry Vapor Pressure Equation for Water Arnold Wexler and Lewis Greenspan Institute for Basic Standards, National Bureau of Standards, Washington, D.C. 20234 (February 19, 1971) Some precise measurements of the vapor pressure of liquid water at seven temperatures in the range 25 to 100 °C were reported recently by H. F. Stimson of NBS. These measurements have an estimated standard deviation of 20 ppm or less, except at 25 °C where the estimated standard deviation is 44 ppm. We have derived a formula which yields computed values of vapor pressure agreeing with Stimson's measurements to within 7 ppm. We integrated the Clausius-Clapeyron equation using the accurate calorimetric data of Osborne, Stimson, and Ginnings and the Goff and Gratch formulations for the virial coefficients of water vapor to obtain an equation that has a rational basis. This equation was then adjusted to bring it into closer accord with Stimson's pressure measurements. Two tables are given of the vapor pressure, expressed in pascals, as a function of temperature at 0.1-deg intervals over the range 0 to 100 °C, one on the International Practical Temperature Scale of 1948 and the other on the International Practical Temperature Scale of 1968. Key words: Clausius-Clapeyron equation; saturation vapor pressure over water; vapor pressure; vapor 1. Introduction In establishing and maintaining humidity standards, in calibrating hygrometers, and in making precision humidity measurements, an accurate equation for the vapor pressure of water is essential. Such an equation also plays a vital role in steam power technology. It is important in the fields of meteorology and air conditioning where precise calculations relating to the water vapor content of atmospheric air are often made. It is employed in chemical thermody namics, where the vapor pressure of the pure water substance is used as a reference standard in calibrating boiling point apparatus, in comparative vapor pres sure measurements, in ebulliometry and in evaluating vapor pressure data of organic liquids. It has been the practice at NBS to use the Goff and Gratch vapor pressure formulation [1] in all work per taining to hygrometry. Recently, H. F. Stimson [2] published the results of some precise measurements of the vapor pressure of water at seven temperatures from 25 to 100 °C. These vapor pressures, obtained with the steam boiler and precision manometer used at NBS in the 1940's to calibrate standard platinum resistance thermometers [3], have an estimated standard deviation of 20 ppm or less except at 25 °C where the estimated standard deviation is 44 ppm. Although vapor pressures calculated using the Goff and Gratch formulation are in reasonably good accord with Stim Figures in brackets indicate the literature references at the end of this paper. where p is the pressure of the saturated vapor, v is the specific volume of the saturated vapor, T is the absolute thermodynamic temperature, y is an experimentally measured calorimetric quantity, and dp/dT is the derivative of the vapor pressure with respect to the absolute thermodynamic temperature. The quantity y [4] is the heat supplied to water to evaporate a unit mass of water isothermally and withdraw its vapor from the calorimeter. It differs from the latent heat of vaporization, L, by a small quantity, ẞ, that is, and v' is the specific volume of the saturated liquid. The quantity ẞ may be pictured as the heat necessary to vaporize water to fill the space no longer filled with liquid in the y experiment. The virial equation of state for water vapor, expressed as a power series in p, (5) The derivation of eq (10) is similar in some respect to that of Goff and Gratch [5, 6]. 1 In 1939, Osborne, Stimson and Ginnings [7] re ported weighted mean values of y from 0 to 200° based on precise calorimetric measurements in th range 0 to 100 °C and on similar measurements mad in the same laboratory during 1930-32 in the rang (4) 50 to 270 °C [8] and in 1937 in the range 100 to 374 ° [9]. We fitte eq (8) to these weighted means by th method of least squares, retaining the units internal tional joules per gram for y and temperature t in de grees C based on the International Temperatur Scale of 1927 (ITS-27) but letting the absolute tem perature T48=t+273.15. In our range of interest, 0 t 100 °C, t has the same numerical value on the Inter national Temperature Scale of 1927 (ITS-27) [10] the International Temperature Scale of 1948 (ITS-48 [11], and the International Practical Temperatur Scale of 1948 (IPTS-48) [12]. Therefore, we wil use tinterchangeably on these scales as may prov convenient. However, the absolute temperature T48 due to the choice of 273.15 conforms to IPTS-48 The coefficients of eq (8) have the following numerica a=3.4660697 × 103, (6) Z is the compressibility factor, R is the gas constant for water vapor, B' is the second pressure-series virial coefficient, and C' is the third pressure-series virial coefficient. For temperatures up to 100 °C, fourth and higher virial coefficients affect Z by less than 12 ppm and therefore will be neglected. Performing some simple mathematical manipulations and integrations, eq (5) becomes = T γ dᎢ γ (1) dT (7) ['" d(In p) = [[ RT dT - [ RT (2 z 1) dr where po and p correspond to vapor pressures at temperatures To and T, respectively. The first integral on the right-hand side of the equation provides the major contribution to the vapor pressure. The second integral on the right-hand side of the equation accounts for the deviation of water vapor from ideal gas behavior. We shall show later that the quantity y can be represented with high precision by the polynomial equation values: b=-5.6067899 c = 1.0963233 × 10-2, and d=-1.2366148 × 10-5. Th residual standard deviation is 0.231 internationa joules per gram, about 1 part in 10,000. Virial coefficients for steam are usually derived from p-v-t measurements [13, 14, 15, 16]. Unfor tunately, there are no p-v-t measurements for steam (water vapor) in the range 0 to 100 °C. The early experiments of Knoblauch, Linde an Klebe [17], which were performed at temperature from 100 to 180 °C, are suspect because of the stati method used which leads to systematic errors due t the adsorption of water vapor on the walls of the con tainer [13]. The experiments of Keyes, Smith, and Gerry [13], made by a continuous flow-method, cove the range 195 to 460 °C. These, too, are suspect below the critical temperature because of possible adsorp tion effects [16]. Other investigators have provided data at higher temperatures and pressures. It is possi ble to extrapolate the virial coefficients obtained a higher temperatures to lower temperatures [13, 14] but such extrapolations do not give values with the requisite accuracy for our purpose. Alternately, calori metric data can be employed to obtain enthalpy co efficients, which, when integrated with respect to temperature, yield virial coefficients. This procedure was employed by Goff and Gratch [5, 6], who used the 38 to 125 °C throttling experiment data of Collins and Keyes [18] and the y data of Osborne, Stimson and Ginnings [7] to derive second and third pressure series virial coefficients valid for the range 0 to 100 2 The residual standard deviation is given by 1/2 °C [5, 6, 19]. Because the Goff and Gratch empirical relationships for the pressure-series virial coefficients are based on experimental data in the range of temperaures of interest to us, and because, of the various virial coefficients we tried, the Goff and Gratch virial oefficients yielded values of vapor pressure in closest greement with Stimson's experimental values, we hose to use them. We converted the Goff and Gratch elationships [19] to SI units compatible with eqs (4) nd (6) to obtain 72000 × 10 48 3 × 10-8 - 12.150799, C=-2.1671578, p is expressed in pascals, T48t48 +273.15 in kelvins, and t48 is in degrees Celsius on IPTS-48. Because eq (14) is implicit in p we calculated p by iteration, numerically evaluating the integral at 1/4-deg intervals by means of the trapezoidal rule [22]. Iteration at each interval was terminated when successive values of p differed by less than one ppm. Evaluating the integral at 1/4-deg intervals was con(12) sidered satisfactory. A check calculation at 80 °C showed that decreasing the interval to 0.025 deg produced a change in p of much less than one ppm from that obtained using 1/4-deg intervals. (13) here B' is in units of reciprocal pressure, 1/pascal,3 is in units of the square of the reciprocal pressure, pascal), and T48 is based on IPTS-48. The equations 3 derived by Goff and Gratch presumably were in rms of absolute temperature on ITS-27. Over the inge 0 to 100 °C, the errors arising in B', C', Z, and -1 from using T48 without conversion are negligible. The Clausius-Clapeyron equation is an exact thermonamic expression in which T is the absolute thermonamic temperature. It therefore follows that the mperature T in eq (10), which was derived from the ausius-Clapeyron equation, is also the absolute ermodynamic temperature. We will assume that 8 is a reasonably close approximation to the absolute ermodynamic temperature T and reserve for later scussion the reasons for this assumption. We in- rted into eq (10) the coefficients of eq (8) on the basis T48, and the virial coefficients given by eqs (12) d (13) as functions of T48. We selected as lower limits integration for substitution into eqs (10) and (11) e pressure Po=101325.0 pascals, which is one andard atmosphere, and the temperature To=373.15 which is the absolute temperature assigned to the eam point at one standard atmosphere on T48. The s constant for water vapor, R, is 0.46151 joules per am kelvin and was derived from the NAS-NRC commended value [20] of 8.3143 joules per mol lvin for the universal gas constant and 18.01534 ams for the molar mass of water vapor on the unified rbon-12 scale. The units of the quantity y were ide consistent with R by the conversion factor 1 ernational joule 1.000165 (absolute) joule [21]. ter inserting the appropriate constants and convern factor, eq (10) becomes 3 = The calculation of p by eq (14) is best accomplished with the aid of a high-speed digital computer. For desk-type calculations, an explicit equation without the integral is desirable. Equation (14) was converted into such an explicit form. A polynomial of the fourth degree was fitted, by the method of least squares, to numerical values of the integral at 1-deg intervals from 0 to 100°C to give which is identical to eq (16), except that the numerical values of the coefficients E, through Es are slightly different. The coefficients are listed in table 1. We investigated the feasibility of simplifying eq (17). Numerous formulas have been proposed for empirically representing the functional relationship between vapor pressure and temperature [23, 24, 25, 26]. A procedure often followed is to select the formula with the least number of terms that not only best fits the vapor pressure data but also yields derivatives that are smooth and regular. We chose the equation because it is analogous in form to our derived eq (17) and fitted it for 2 n 4 by the method of least squares to values of vapor pressure generated at 1-deg intervals from 0 to 100 °C by eq (17). The coefficients are listed in table 1. For convenience in identifying the several versions of eq (18), each has been designated with a letter a through c corresponding to n = 2, 3, and 4, respectively. 3. Results The measurements made at the PhysikalischTechnische Reichsanstalt by Holborn and Henning [27] in 1908 from 50 to 200 °C and by Scheel and Heuse [28] in 1910 from 0 to 50 °C have, until now, constituted the major sources of data for the vapor pressure of water in the 0 to 100 °C range. In 1919, PTR published revised values of the vapor pressure [29] and these have served as the input data for many empirical formulations [30, 31]. The steam point has been used as one of the defining fixed points for the several successive practical temperatures scales that have been adopted internationally [10, 11, 12, 32]. Because of this, the vapor pressure near the steam point has been investigated in great detail [33, 34, 35, 36, 37, 38]. Most of these measurements were made over narrow temperature spans around 100 °C. The largest temperature range, 73 to 130 °C, was covered in 1939 by Moser and Zmaczynski [37] who used both a static and a dynamic method in their determinations. Beattie and Blaisdell in 1937 [36] and Michels, Blaisse, Ten Seldar Wouters in 1943 [38] limited their observations several degrees below and above 100 °C. Douslin and McCullough in 1963 [39] and D in 1970 [40], using an inclined dead-weight piston gam made vapor pressure measurements from 0 to 201 The above series of measurements, plus the men urements of Stimson, comprise the most impc determinations covering the 0 to 100 °C span. We compared our formulation with these eve mental data and with other formulations in order obtain a general perspective on the degree of a achieved. However, because of the high precision internal consistency of the Stimson measuremer judged the efficacy of our vapor pressure equ." primarily on their agreement with the Stimson des 3.1. Comparisons With Stimson's Measurements Differences in vapor pressure between Stirmeasurements and eqs (14), (16), (17), and the given in table 2. The standard deviation of Stirvalues are also shown. Equation (14) yields 1. which agree with Stimson's measurements to one standard deviation of the latter, except at 80 °C where the agreement is within two start deviations. The maximum difference is 32 ppm occurs at 25 °C. The conversion of eq (14), an im equation, to eq (16), an explicit equation, has effect on the differences. Equation (17), which wa TABLE 2. Comparison between calculated vapor pressures and Stimson's measurements on IPTS-46 |