Vapor Pressure Formulation for Water in Range 0 to 100 °C. A Revision

In 1971 Wexler and Greenspan published a formulation for the vapor pressure of water encompassing the temperature range 0 to 100 °C. In this paper a revision is made of that earlier formulation to make it consistent with the definitive experimental value of the vapor pressure of water at its triple point recently obtained by Guildner, Johnson, and Jones. The two formulations are essentially identical at temperatures from 25 to 100 °C. For temperatures below 25 °C the new formulation predicts values that are higher than the 1971 formulation. At the triple point, the vapor pressure given by the new formulation is 611.657 Pa whereas the value given by the 1971 formulation is 611.196 Pa. A table is given of the vapor pressure as a function of temperature at 0.1-deg intervals over the range 0 to 100 °C on the International Practical Temperature Scale of 1968, together with values of the temperature derivative at 1-deg intervals.


Derivation
In 1971 , Wexler and Greenspan [1]1 derived an equation for computing the vapor pressure of water over the temperature range 0 to 100°C. They integrated the Clapeyron equation, using the accurate calorimetric data of Osborne, Stimson and Ginnings [2], and the Goff and Gratch formulation [3] for the virial coefficients of water vapor, to obtain a smoothing function that has a rational b asis. Calculated values of vapor pressure agreed with th e very precise m easurements of Stimson [4] to within 7 ppm from 25 to 100°C. Comparable m easurem ents below 25°C were not available for comparison. R ecently, Guildner, Johnson, and Jones [5] completed a series of highly accurate m easurem ents of the vapor pressure of water at its triple point. They obtained the definitive value of 611.657 Pa 2 with an es timated total uncertainty at 99 percent confidence limits (3 sigmas plus the estimated sys tematic errors) of ± 0.010 Pa (± 16 ppm). The 1971 equa tion predicts a vapor pressure at the triple point of 611.196 Pa, a valu e which is lower by 0.461 P a (754 ppm ) . The 1971 formulation, therefore, was r eexamined and revised so that it is now consistent with this new experimental triple point value as well as with the older Stimson measurements. By using new gas thermometry da ta [6] it was possible to derive an equ a tion for vapor pressure as a function of thermodynamic temperature and also of the International Practical Temperature Scale of 1968.
A modified ver sion of the Clapeyron equation [7] is chosen as the starting point: 1 Figures in brackets indicate the literature references at t he end of t his paper. , 1 Pa = l N jm'= JO-' bar = 1O-2 mb = 7.50062XlO-J mm JIg. dp 'Y dT= Tv (1) where p is the pressure of the saturated v apor, v is the specific volume of the saturated vapor, T is the absolu te th ermodynamic temperature, ~/ is an experimentally m easured calorimetric qu~ntity not substantially different from the latent heat of vaporization [7], and dp/dT is the derivative of the vapor pressure with respect to the absolute temper ature . The specific volume, obtained from the virial eq uation of state for water vapor, is v= RTZ= RT (l+B'p+O'p2+ ... ) (2) P P where R is the gas constant for water v apor, Z is the compressibility factor, B' is the second pressureseries virial coeffi cient and 0 ' is th e third press ureseries virial coeffi cient. When eq (2) is substituted into eq (1) it follows that dp 'Y p = RT 2Z dT (3) After performing several simple mathematical manipulations and integrating, eq (3) becomes (P JT 'Y IT 'Y (Z-l) J 110 d(lnp) = ToRTz dT -To RTz ---z-dT (4) where Po and p are the initial and final vapor pressures corresponding to temperatures To and T respectively. ' The quantity 'Y is represented by the polynomial equation 'Y=ao+aIT+a2T2+asTs (5) where ao, aI, a2 and as are constants. Inserting eq (5) into the first integral on the right-hand side of eq (4) yields The constants in eq (5) were obtained by fitting the Osborne, Stimson, and Ginnings weighted mean values of 'Y [2] from 0 to 150°C by the method of least squares after converting the reported temperatures, given on the International Temperature Scale of 1927 (ITS-27), to absolute thermodynamic temperatures and the reported heat units of international joules to (absolute) joules. 3 The conversion of temperatures on ITS-27 to thermodynamic temperatures will be reserved for later discussion. The coefficients of eq (5) have the following values: ao=3423.8440, at= -5.2277204, a2=0.9855719 X lO-2~ and as= -.11305118XlO- 4 • The triple point vapor pressure Po = 611.657 Pa and the absolute temperature at the triple point T o= 273. 16 K [5] were selected as lower limits of integration for substitution into eq (6). The gas constant for water vapor, R, is 0.461520 joules per gram kelvin and was derived from the CODATA r ecommended value [8] of 8.31441 joules per mole kelvin for the univeral gas constant and 18.01528 grams for the molecular weight of naturally occurring water on the unified carbon-12 scale. 4 The Keys equation [12] was used for the second pressure-series virial coefficient. When converted to SI units, compatible with eq (2), it takes the form where B' is in units of reciprocal pressure, (Pa)-t. From the experimental vapor pressure data of 3  Stimson [4] and of Guildner, Johnson, and Jones [5], and the calorimetric data of Osborne, Stimson, and Ginnings [2], the saturated specific volumes of water vapor were calculated using eq (1). These volumes, when inserted in to eq (2), together with B' from eq (7), yielded values of an effective third pressureseries virial coefficient 0'. These computed values of 0' were fitted by the method of least squares to the equation (8) yielding do=0.311018X10 s , d1 =-0.349634X10 6 , d2=0.116994X 10 9 , and ds= -0.126779X 1011. Equation (8) is valid only from the triple point to the steam point and is expressed in 81 units of the square of the reciprocal pressure, (Pa) -2, compatible with eq (2).
Because eq (6) is an implicit function in p, calculations of p were made by iteration. The integral on the right was evaluated numerically at 20 mK intervals by means of the trapezoidal rule [13]. Iteration at each interval was terminated when successive values of p differed by less than 0.1 ppm. Fifty-one numerical values of the right-hand integral of eq (6), at 2-kelvin intervals starting at the triple point, were fitted by the method of least squares to the equation +bs(T2-To2)+bt ln f (9) yielding bo= -0.13750137 X 10\ bl = -0.14185668X 10 2 , b2= 0.49593509 X 10-1 , and b3= -O.29488830 X 10-4 • It should be noted that the b' s are replacements for the parameters of equations (7) and (8). By combining terms on the right-hand side of eq (6), and then integrating the lefthand side, an explicit equation was obtained, namely, which, with the appropriate constants, reduces to (11) where Co= -0 .604 36 117 X 10\ cI=O.189318833X10 2 , C2= -0.28238594 X 10-r, cs=0.17241129XlO-\ and c4=0.2858487X 10 1 • At the steam point, the value of the vapor pressure given by eq (ll) is greater than the defin ed value of 101325 Pa by 3.4 Pa (38 ppm). By introducing an arbitrary but minor change in the coeffi cients Cil C2, and C3 , the equation was adjusted to pass through 101325 P a with negligible effect on the intermediate vapor pressures. The three adjusted coeffi cients now have th e followin g values : cl = 0.1 893292601 X 10 2 , C2= -0.28244925 X IO-t, and c3 = 0.1725033 1X 10-4 •

Conversion to IPTS-68
Over the range 0 to 100 °C, the temperature in degrees Celsius has the same numerical value on the International T emp erature Scale of 19 2.. (ITS-27), the International T emperature Scale of ·1948 (ITS-48), and the International Practical T emperature Scale of 1948 (IPTS-48). On the other hand , over the same range, the temperature on the In ternational Practical T emperature Scale of 1968 (IPTS-68) differs from that of ITS-27, ITS-48 and IPTS-48. Using the corrections given by Riddl e, Furukawa and Plumb [14], temperatures on these latter three scales were converted to IPTS-68.
Guildner and Edsinger [6] have made a series of measurements on the realization of the thermodynamic temperature scale (TTS) from 273.2 to 730 .44 K by means of gas thermometry. They fitted their data to an equation of the form (12) wh ere T68 is the absolu te temperature in kelvins on IPTS-68. They obtained the following values for the coefficients : iXo = 0.1l92951052 X 106, iX, = -0.1l99-170ll X 10\ iX2 = 0.427014907 X 10', iX3 = -0.63794-2023 X 10-2 , and iX4 = 0.353749196 X 10-s. The residual standard devia tion of the fit was 1.57 m K .
Their data were refitted up to 472.78 K to an equ ation of the form (13) which imposed the constraint that t68= t at the triple point and where t68 and t are the temperatures in degrees Celsius on IPTS-68 and TTS, respectively. This equation was th en converted to absolute temper atures, yielding (14) where Po= 0.4949479, p,= -0.46352557 X lO -2 , P2= 0.13852 156 X 10-4 and P3= -0.12872954 X lO -7 . Over the r ange from 273.15 to 373. 16 Ie (the range of interest here), the temperatu res calculated by eq (14) do no t differ from those calculated by eq (12) by more than 0.79 mE:; the standard deviation of the difference between T68 as calcul ated by eq (14) and T68 as meas ured by Guildner and Ed singer is 1.5 mK.
In the range from the triple point to the steam point, the numerical values on IPTS-68 become progressively larger than those on TTS at identical temperatures. At the steam point T 68 is grea ter than T by abou t 25 mK ('" 67 ppm).
One way of calculating the vapor pressure is to convert IPTS-68 to TTS temperatures via eq (14) and then to insert these computed thermodynamic temperatures into eq (11). Alternatively, eq (11) can be transformed to IPTS-68 by substituting eq (14) into eq (ll). This algebraic manipulation yields (15) The coeffi cients are given in table 1.
This algebraic conversion increased the number of terms from fiv e in eq (ll) to eigh t in eq (15). The feasibili ty of simplifying eq (15) was investigated. The procedure adopted was Lo fit by the method of least squares, 102 values of vapor pressure, generated by eq (15) at one-kelvin intervals star ting at the ice point, to an equation of t he form n In p= ~ g i T~82 + gll +l In T68 (16) i = ! for 3 <n < 6 with and wi thout the In T68 term.
Equation (16) i analogous to eq (15) except for the number of term s. For n = 4 and including th e In T 68 term, eq (16) yields valu es of vapor pressure which differ from those calculated using eq (15) by 0.4 ppm or less. For n = 4 but without the In T68 term, eq (16) yield s valtlCs which differ from t hose calcu lated using eq (15) by 20 ppm or less . For convenience th ese two versions of eq (16) will be designated eq (16a) an d eq (16b) respectively. The coefficients are given in table 1. The use of more terms does not improve the agreement whereas decreasing n to 3 degrades the agreement by an order of magnitude or more.  A comparison between this formulation, u sing eq (15) as a base line, and the 1971 formulation is shown in table 3. The two are in substantial agreement (<: 37 ppm) from 100 to 25°C. Below 25 °C the difference between the two formulations increases from 37 ppm, reaching 754 ppm at the triple poin\'.
Because the numerical values of temperatures on IPTS-68 are gre' ater than those on TTS, it follows that the vapor pressures calculated on IPTS-68 with eq (15) are smaller than those calculated on TTS with eq (11) when using the same numerical values for temperature. As shown in table 4, for identical numerical values from 0 to 100°C, eq (11 ) on TTS yields vapor pressures that increasingly exceed those calculated with eq (15) on IPTS-68 until , at 100°C, the former is larger than the latter by 901 ppm. It is obvious that a substantial error will result unless the temperature is expressed on the appropriate scale for each equation.
A comparison of this formulation, using eq (15) as the base lin e, with several other formulations [15][16][17][18][19][20] 5 in common use is shown in figures 1 and 2, with the differences given in pascals and parts per million, respectively. Appropriate temperature scale adjustments have been made to these formulations so that the calculated vapor pressures are on IPTS-68. The important feature of this comparison is that as the temperature decreases below about 20°C, these formulations predict values of vapor pressure that are consistently smaller than those obtained with eq (15). At the triple and ice points, the differences reach magnitudes of the order of 700 to 900 ppm.
In the earlier paper the 1971 formulation was compared with experimental data at and below the steam point. This will not be repeated except to note that, because this and the 1971 formulations  yield essentially the same values of vapor pressure above 25°C, the degree of accord with this formulation will be comparable. There are two sets of modern vapor pressure measurements of . water in the temperature range from 25°C and below, those of Douslin [23] and Besley and Bottomley [24]. Differ en ces between these data and eq (15) are given in tables 5 and 6. Dou slin used an inclined de ad weight piston page to m ake his measurements. He reported that his estimated sys tematic error varied from 0. 31 Pa at -2.5 °C (609 ppm) to 0.81 Pa at 20.0 °C (346 ppm). His values are higher than those predicted by eq (15). Besley and Bottomley used a mercury manometer to make their series of measurements which they fitted to an empirical equation. They give no estimate of the overall systematic error of their measurements; rather they reported that the standard deviation of the fit wa s 1.7 mtorr (0.23 Pa) and used this as an estimate of their experimental impre?ision. T~eir cor~elated value at the triple pomt 1S smaller m magl1ltude by about 906 ppm than that calculated with eq (15). Their values gradually approach those obtained with eq (15) until, at about 13 .5 °C, the two agree. At 25°C, the Besley and Bottomley values are higher by 57 ppm.
Using eq (15), vapor pressures in pascals were computed, as a function of temperature in degrees Celsius, on IPTS-68 from 0 to 100°C. These computed values, as well as the derivative with respect to temperature, are given in table 7. In the 1971 paper, an analysis was presented of the uncertainties in such quantities and cons tan ts as /" R, and Z and the contributions these uncertainties make in the calculation of p. A similar analysis will not be repeated here. Although in this work the computation was modified by using the triple point, rather than the steam point, as the lower limit of integration, substituting different virial coefficients for those of Goff and Gratch, and using the Guildner and Edsinger data for converting between TTS and IPTS-68, the conclusions of the earlier analysis are still valid: If the parameters entering into the computation are completely independent, then they must be known to an accuracy that is 1 to 2 orders of magnitude better than they are now known for thermodynamic calculations of vapor pressure to have an uncertainty comparable to the measurements of Stimson and of Guildner, Johnson and Jones. Equations (11) and (15) are presented, therefore, not as accurate theoretical representations of the properties of water but as smoothing functions that have a rational basis. Because of this rational basis, it is believed that the formulation has more validity for predicting (interpolating) vapor pressures at temperatures where  corrob ora~ing exp erimen tal data of high acc uracy do not eXIst than does one t hat is purely empirical.
Where vapor pressures are desired in terms of TTS, eq (11) should be used. Wh ere vapor pressures are desired in terms of IPTS-68, th en eq (15) or (16) should be used. We believe eq (15) is preferable because it has a thermodynamic basis whereas eqs (16a) and (16b) are empirical. However, eq (16b) without the In T 68 term h as four coefficien ts and eq (16a) with the In T6 8 term h as five coefficients compared to the eigh t in eq (15). Therefore, where a reduction in coeffi cien ts is desirable, ei ther version of eq (16) may be used. Both eqs (15) and (16) show comparable agreement with experimental data.