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Vapor Pressure Formulation for Ice

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Vapor Pressure Formulation for Ice JOURNAL OF RESEARCH of the Notiona l Burea u of Standards-A. Ph ys ics a nd Chemi stry Vol. B1A, No. 1, Janua ry-February 1977 Vapor Pressure Formulation for Ice Arnold Wexler Institute for Basic Standards, National Bureau of Standards, Washignton, D.C. 20234 (September 23, 1976) A new fonn ulation is presenl ed for Ih e vapor pressure of ice from Ihe triple point to - 100°C based on thermodynamic calculations. Use is made of Ih e definilive ex pe rimenlal value of the vapor pressure of waler a t its triple poinl recently oblained by Cuildner, Johnson, and Jones. A table is given of Ihe vapor pressure as a function of le mperature at O.I-degree inl e rv a ls over Ih e range 0 10 - 100 °C, logel he r wilh Ihe values of Ih e lemperature derivative at I-degree inl e rvals . T he formulaLion is compa red with published experime nlal measurement s and vapor pressure equaLi ons. It is eslima led Ihat Ihi s formulalion predicls Ihe vapor pressure of ice wit h an overall uncertainly Ihal varies from 0.016 percenl al the Iriple poinl 10 0.50 percenl a l - 100 °C. K ey wo rd s: Clausius- Clapey ron equation; sa turation vapor press ure over ice; th erm al properties of ice; vapor pressure; vapor pressure at th e triple point; vapor pressure of ice; waler vapor. 1. Introduction 2 . Derivation In meteorology, air conditioning, and hygrometry, partic u­ The Clausius-Clapeyron equation, when applied to the larly in the maintenance and use of standards and generators solid-va por phase transition for the pure water-substance, in calibrations and in precision measureme nts, accurate val­ may be wrillen ues of the vapor pressure of the pure wa ter-s ubstance are essential. Because of this Wexler and Greenspan [1]1 re­ dp cently published a ne w vapor pressure formulation for the (1) pure liquid phase, based on th erm odynamic calculations, dT T(v - v") whic h is in excell ent agreement from 25 to 100 DC with the precise measureme nt s of Stimson [2]. Wexler [3] s ubse­ where p is the pressure of the saturated vapor, v is the quently revised this formulation to make it consiste nt with the specifi c volume of th e saturated vapor, v" is the specifi c defi nitive experimental value of the vapor pressure of water at volume of the saturated ice, T is the absolute th e rmodynamic its triple point obtained by Guildner, Johnson, and Jones. temperature, L is th e latent heal of sublimati on, a nd dpldT is The purpose of this present paper is to apply a similar method the derivative of the vapor pressure with respect to tempe ra­ of calculation to the pure ice phase and derive a ne w formula­ ture. The latent heat of sublimation is give n by tion for temperatures down to -100 DC. This new formulation for ice is constrained to yield the id entical value of vapor L=h- h" (2) pressure at the triple point as that given by the revised formulation for the liquid phase. where h is the specific enthalpy of saturated water vapor at A critical examination of the experimental vapor-pressure temperature T a nd h" is the specific e nthalpy of saturated ice data of ice discloses the disconcerting fact that the dispersion at the same temperature T. among those values far exceeds modern accuracy require­ The equati on of state for saturated water vapor may be ments. This dispersion arises, in part, from the inhere nt expressed by difficulties experi e nced by investigators in making precision measureme nts of these low pressures and from th e ambigui­ pv = ZRT (3) ties in the temperature scale used in th e early 1900's when several major series of determinations were made. Thermody­ where Z is the compressibility factor and R is the specific gas namic calculations, based on accurate thermal data, provide constant. When eq (3) is substituted into eq (1) the result is an alternate route to th e determination of vapor pressure. It is therefore not surprising that s uch calculations have been dp L ( -=-- 1 + VII)- dT (4) made repeatedly for ice with varying degrees of success. It is P ZRT2 v interesting to note that these calculations have been preferred over the existent experimental vapor pressures, primarily where highe r order terms of v"lv are neglected because v"lv because the calculations appear to yield less uncertainty than « 1. On integrating, eq (4) becomes the measurements. P 9 I ( ") j - dOn p) = 11'2--2 1 + ~ dT (5) I Figures in brac ke ts indi cate the literature references at the end of Ihis pUpt'L PI 1'1 ZRT v 5 where PI and P2 are the saturation vapor pressures at temper­ approaches its limiting value of unity as the temperature atures TI and T 2 , respectiv ely. Suitable functions will be drops. Saturated water vapor, therefore, tends to be have more sought for Z , v, v" and l in order to complete the integration of and more like an ideal gas as the temperature decreases, eq (5). thereby reducing the effect of errors in B I. Functions for the compressibility factor Z and the specific Table 1 shows a comparison of Z from the triple point to volume of saturated water vapor v will be based on a virial -100°C, for water vapor saturated with respect to ice, equation of state expressed as a power series in p. A function calculated using the empirical second virial coefficient equa­ for the specific volume of saturated ice v" will be developed tions of Goff and Gratch [12, 13, 14], Keyes [15, 16], and from experimental data for the coeffi cient of linear expansion Juza as given by Bain [1 7]. The maximum difference in Z , as and the densitv at 0 0c. A function for the latent heat of well as v, is 118 ppm and occurs at 0.01 0c. This can be used sublimation l \;ill be derived from the specific enthalpies h" as an indication of uncertainty alt hough the actual error is and h of saturated ice and saturated water vapor, respec­ indete rminate. The differences decrease as the temperature tively. Use will be made of measurements of the specific heat decreases. At -70°C and below, th e diffe re nces are equal of ice to obtain h" whereas statistical mechanical calculations to , or less than, one ppm since, at such temperatures, the of the ideal-gas (zero-pressure) specific heat of water will second virial coeffi cient makes a negligible contr.ibution toZ. serve as input data for establishing an expression for h. TABLE 1. Compressibility factor for saturated waler vapur 2.1 Temperature over tee Temperature Cornpressibilily factor" Guildner and Edsinger [5] have recently made measure­ ments on the realization of the thermodynamic temperature Goff °C Keyes Keves a nd scale from 273.16 to 730 K by means of gas thermometry. 1969 '1 1947' Bain d Grate h e Unfortunately there are no similar high precision measure­ Z Z Z Z ments below 273.16 K. Therefore, it will be assumed that the 0.01 0.999624 0.999501 0.999529 0.999506 International Practical Temperature Scale of 1968 (JPTS-68) o .999624 .999501 .999529 .999506 [6] is a sufficiently close approximation to the absolute - 10 .999907 .999726 .999747 .999730 thermodynamic temperature so that the thermal quantities - 20 .999958 .999856 .999871 .9998.59 -30 .999982 .999928 .999938 .999930 given in terms of IPTS-68 can be used in eq (5). From the - 40 .999993 .999966 .999972 .999967 triple point to - 100°C the temperature t in degrees Celsius -50 .999998 .999984 .999986 .999985 has the same numerical values on the International Tempera­ - 60 .999999 .999994 .999996 .999994 ture Scale of 1927 (ITS-27) [7], the International Tempera­ - 70 1.000000 .999997 .999999 .999998 - 80 1.000000 .999999 1.000000 .999999 ture Scale of 1948 (ITS-48) [8] and the International Practi cal - 90 1.000000 1.000000 1.000000 1.000000 Temperature Scale of 1948 (IPTS-48) [9]. However, the ice - 100 1.000000 1.000000 LOOOOOO 1.000000 point on IPTS-48 is defined as equal to 273.15 kelvins a Calculaled bv eq (6) using B' given by the inciicaleri in vesti gator. whereas on ITS-27 and ITS-48 it is defined as equal to b Ref [16]. 273.16 kelvins. The difference T68 - T 48 , wh ere T68 and T48 c Ref [1 5]. are the kelvin temperatures on IPTS-68 and IPTS-48, respec­ " Ref [17]. e Ref [12- 14]. tively, in the range of interest reaches a maximum of 0.0336 kelvin at 200 K [10, 11]. Using th e corrections given by Riddle, Furukawa, and Plumb [11], temperatures on ITS-27, The 1969 second virial coefficient of Keyes [16] will b e ITS-48 and IPTS-48 have been converted to IPTS-68 where used in order to be consistent with the earlier use of this same needed in the calculations. coefficient [3]. His virial coeffi cie nt equation, conv erted to SI units compatible with eq (6) , is 2.2. Specific Volume of Saturated Vapor Equation (3) is used to calculate the specific volume of B' = [0.44687 _ ( 565.965) saturated water vapor v. The compressibility factor Z IS T ]'2 expressed as a power series in P IOOMUO ] XIO (7) Z = 1 + B'p + C'p2 + (6) wh e re B I is in units of reciprocal pressure (Pa)-I.
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