Humidity Fixed Points of Binary Saturated Aqueous Solutions

An evaluated compilation of equilibrium relative humidities in air versus temperature from pure phase to approximately 105 pascal (1 atm) in pressure is presented for 28 binary saturated aqueous solutions. The relative humidities of the solutions range from about 3 to 98 percent. Using a data base from 21 separate investigations comprising 1106 individual measurements, fits were made by the method of least squares to regular polynomial equations with two through four coefficients. Equations and tables are presented along with the estimated uncertainties in the correlated results.


Introduction
Research, hyg romete r calibration, testing a nd material conditioning oft en req uire th e accurate control of humidity in a working space. The co mm on me th ods of controlling th e humidity accurately use either a humidity generator [lA]t or th e equilibrati on of a closed space with a che mi cal system [lB] whi c h produces th e desired equilibrium vapor pressure .
Humidity ge nerators te nd to be ex pe ns ive a nd complex wh ereas equilibrati on with c he mi cal syste ms th at provid e fix ed points is a relati vely in expe nsiv e a nd s im ple method of humidity control. Among th e c hemi cal sys tems used for thi s purpose are aqueous sulphuric ac id solutions, glyce rine and water soluti ons a nd single a nd binary salt solutions . Eac h such solution offe rs a degree of humidity adjus tm e nt that can be achieved by cha nging its concentra tion. On th e other hand, special problems are associated with th e use of solutions because their concentrations mu st be mea sured and controlled. Not only must the concentration of the solution be determined initially but the presence of any humidity sources or sinks in th e controlled space and even the initial equilibration process of the space can alter the solution concentration.
An especially useful method of humidity control by chemical system involves th e use of binary saturated aqueous solutions (primarily of sin gle salts) in which the solute is highl y non-volatile.
At any temperature , th e co ncentrati on of a saturated solution is fi xed and does not ha ve to be dete rmined . By providing excess solute, th e soluti on will re ma in saturated even in the presence of modest sources or s inks . Whe re th e solute is a solid in the pure ph ase, it is easy to de termine that th ere is indeed saturation. Du e to th e ease of its use, thi s is a popul a r method of humidity co ntrol.
Since a given saturated salt soluti on provides only on e relative humidity (RH) at any desired tempe rature, a different relative humidity must be achi e ved by selecting a nother appropriate salt. Though mu c h data on saturated salt solu-1 Figures in brac kets indicate the lite rat ure re ferencL'S al the end of this pape r.
tions have been produced and ma ny compilations of th e equilibrium relative humidities of selected saturated salt soluti ons exi st, th e re are no compilations for whi ch the d ata have been criti cally analyzed a nd estimates of th e un certa inti es involved give n, a step whi ch is abolutely essential to th e implime nta ti on of th e concept of fi xed po ints.
We ha ve moved to fill thi s gap by compiling, from th e lite ra ture, data on a suffi c ie nt va ri ety of saturated salt soluti ons to cover th e e ntire ra nge of relativ e humidity at reasonably close interv als. We have adjus ted th ese da ta [1 -21] to be consiste nt with te mpe ra tures on IPTS-68 a nd th e mos t recent equati ons for th e va por pressure of water [22] . We have also a nalyzed th e ex pe rim ental techniques used in obta ining th e ori ginal da ta a nd have made estim ates of th e un certa inti es in the ori ginal data . We have th e n used th ese da ta to calcul a te " best" values of relative humidity in air as a fun c ti on of temperature from pure phase to approxim ately 10 5 pascal (1 atm) in pressure for th ese saturated soluti ons .

Background
The methods used by investi ga tors to d etermine th e water vapor in equilibrium with saturated salt soluti ons a re dive rse. A short description of the vari ous me th ods used in th e referenced papers is of interest.
(1) The direct measurement of the vapor press ure. A chamber containing a saturated salt solution a t a co ntroll ed temperature is first evac uated to remove all gases. E vapora ti on from the solution is th en all owed to proceed until th e a mbi e nt vapor, essentially all wate r, has come to equilibrium with th e solution and a direct de te rmination of th e total pressure within th e cha mber is mad e by conv e ntional pressure measure me nt techniques.
(2) Dew point measurement. The dew point of th e gas within a c hamber co nta ining a saturated salt solutio n at co ntroll ed temperature is measured by means of a cooled mirror within the c ha mber. Us in g vapor pressure ta bles or equa ti ons, this dew point is conv erted to the va por pressure of wate r.
(3) Isopiestic vapor pressure measurement. The va por pres-s ure of a saturated salt solution in one cell or chamber is allowed to come to equilibrium with a cell or chamber containing a reference solution at a fi xed temperature. The refe re nce solution must be well characterized as to vapor pressure as a function of concentration at the reference te mperature. Under the equilibrium co ndition, the equilibrium vapor pressure of the saturated salt solution is ide ntical to the equilibrium vapor pressure of the reference solution .
After the tw o cells have reached equilibrium, the conce ntration of the refe re nce solution is dete rmined (usually by weighing) and th e vapor pressure is calc ulated .
(4) Relative vapor press ure measurement. A c hamber conta ining a saturated salt solution a nd a chamber containing pure water or other well c harac terized solution are each evacuated to re move all non-water vapor gases. The two c hambers are maintain ed at the same te mperature and the absolute pressure of the saturated salt solution is measured as in the first me thod. In addition the pressure diffe rence betw een the two chambe rs and/or the pressure of the refere nce solution is de termined. The ratio of the vapor pressure of the saturated salt solution to th e vapor pressure of th e wate r is the activity (or re lative humidity) of th e satura ted salt solution.
(5) Measurement with a calibrated humidity senso r. A c hambe r conta ining a saturated salt solution a nd a humidity sensor are brought to equilibrium at a controlled te mpe rature. Calibra ti on of th e sensor before or/and after the measureme nt provid es the means of determining the eq uilibrium vapor pressure.
(6) Gravimetric determination . Dry gas is passed through the binary satura ted solution at a fi xed te mpe rature. The wate r va por in th e efflue nt gas is absorbed by a desiccant and measured by weighing. The volume of the gas is also de termined. From these the vapor pressure or the mixing ra ti o can be d etermined.
As one would imagin e , the e rrors associated with th ese me thods diffe r as to so urce and magnitude. The en'ors in any of the methods a re also fun c ti ons of the le vel of vapor pressure be ing measured as well as the tempe rature of the saturated salt solution. The re is, the refore, probably no one method that gives a best measureme nt under all conditions.

Method
We have accumula ted experime ntal data from various researc he rs  and calc ulated " best" values of re la tive humidity a nd the associated uncerta inties of those values. Typical me thods of calc ulating or recalculating the rela ti ve humidity and associated uncerta inties for th e vari ous investigations are giv e n in the Appendix. Our data base consists of 21 investi gations and includes some of the most cited work in the fi eld . In total , 1106 individual calculations of relative humidities and associated un certainties were made whic h involved 89 saturated solutions. Not all data nor all satura ted solutions in this s tudy we re found sati s factory for use.
The original data were corrected to be consistent with te mperature on IPTS-68, with the mos t recent formulation for th e vapor pressure of water [22] and with the most recent equations for the e nhancement of water vapor in air [23]. The computed relative humidity data we re the n collated and fitted by the me thod of leas t squares to regular polynomials as a function of te mperature in degrees Celsius (IPTS-68). In the fitting process, each datum was we ighted inversely proportional to the estimated uncertainty of the datum. The order of the polynomial used in the fit was determined by an F-test or b y a nalysis of the result of fits to various orde rs. An arbitrary d ecision was made not to use any order higher than 3. Also, no data at temperatures below 0 °C or above 104°C we re used in the fits .
In the fitting process, the standard deviation of the predic ted value was computed for each datum. These s ta ndard deviations were themselv es fitt ed to a quadratic equation , as a function of te mperature, by the method of least squares. At any desired te mpe ra ture for a given saturated salt solution, the standard deviation of th e predicted value was calculated using the appropriate quadra ti c equation. T hree times thi s value was the n assigned as the estima te d uncertainty for the corresponding value of rela tive humidity, wi th certain exception s discussed below. This is the value whi c h appears in table 2.
Where a numbe r of investi gation s of the same solution exis ted and the re la tive humidity vs te mpe rature results of one investigation we re comple te ly inco nsis te nt with the results of th e other investi ga ti ons, th e data of the deviant in vestigation were eliminated and a ne w fit made .
The data used in this pape r me t one of the following crite ria: (1) a large numbe r of inves ti gations we re included and exhibited a s ma ll residual standa rd d ev iation of th e relative humidity vs te mpera ture fit s; (2) although fe w investi gati ons we re included, the me thod of measurement was judged to be s upe ri or a nd estima tes of th e un certainties of th e ori ginal measure me nts the mselves we re s mall ; a nd (3) the data were in a relative humidity ra nge which was not a pprox imated by a ny of the oth e r binary saturated solutions. Table 1 conta ins coeffi cie nts for the data of the selected salts fitted to an equati on of th e form:

Results
where RH is in percent a nd t is in °C (IPTS-68). The salts a re lis ted in ascending ord e r of RH a t 25°C. Also in cluded in table 1 is the residual s ta ndard de viati on of th e fit , the range of te mpe rature over whi c h th e fi t was pe rformed a nd refere nces for the fundame ntal data th a t were involved in that partic ular fit.
Ta ble 2 gives the calc ul a ted rela tive humid ities for eac h of the binary saturated solutions at 5-degree inte rvals along with the estimated un certainties in rela tive humidity at each of the tempera tures . The saturated salt solutions are present ed in the same order as in ta ble 1.

Discussion
Although the me thod used for fittin g the data gave no proble ms, the assignment of weights to eac h datum re quired some judgment. Three methods of we ighting were conside re d: (1) weights were assigned inve rsely proportional to the variance of the individual datum where the variance wa s ta ke n as the square of th e total un certainty; (2) weights we re assigned inversely proportional to the estimated total uncertainty of the individual datum; and (3) weights of unity we re assigned to all data.
All of the data were fitted three times, once for each type of we ighting. The results were assembled into three tables of "

93
relative humidity at 5-degree intervals . Eac h calculated value of relative humidity was assigned an uncertainty equal to three times the standard deviation of the predicted value. As might be expected, th e calc ulated re lative humidities and the corresponding uncertainties d iffered for each of the three weightings. For the saturated solutions chosen for presentation in this paper, it was noted with so me sati sfaction that all relative humidities calcul ated from the three differently weighted fits agreed with each oth er to within the assigned uncertainty for eac h. A we ighting inversely proporti ona l to the square of the estimated total un certainty for each datum was judged to be inappropriate . Although it is common to assign we ights proportional to the inverse of s igma squared such an approach is usuall y based on a s igma whi c h is stati sti call y dete rmined. This is not th e case he re. The method used to obta in the estimated total uncerta inty is given in the Appe ndix. It was felt tha t th e use of the square of th e estima ted un certainty would have placed a n un acceptabl y hi gh va lu e on the author's estim ate of th e errors co ntributing to th e total un certainty. So me investi gators did not provide suffi c ie nt inforrn ati on in th e ir publications to make poss ible compl e te ly obj ective es timates of the ir errors. [n those cases, the estimated total un cert a inty included components based on the au thor's s ubj ective judgme nts .
A we ighting of unity was like wi se unsati s factory s in ce it would in no wa y take into accou nt th e innate diffe re nce in un certainty due to me th od, temperature a nd rela ti ve humidity range, nor wo uld it place a ny re li ance on th e a uthor's judgme nt of th e quality of th e research. A we ighting proporti onal to the inverse of the estima ted un certainty a ppeared to be a reasonable compromise betw ee n the othe r ex tremes and all data presented in this publication were processed us in g th at we ighting me th od .
Wh e re th e data for a partic ular saturated salt solution in clud ed a numbe r of in ves ti gation s , three times th e standard devi ati ons of the computed valu es were accepted as the estimated uncertainty. Whe re th e data were based onl y on one or tw o investi gati ons it is evid e nt that self co nsiste nt data, though quite in accurate, co uld giv e small estimated sta ndard de viations of the computed values. It is al so evid e nt th at such standard deviations a re not a valid estimate of un cert a inty. Und er those c ircumsta nces wh ere th e results from fittin g th e polynomial equation to th e ori ginal data for any saturated salt solution gave values for three times the standa rd de viation of the predic ted value that we re less than th e estimated total uncertainty of the original data, it was the estimated total uncertainty of the original data whi c h was used as the final estimate of uncertainty for the calc ulated " best" value of relative humidity.
The data presented in table 2 are given at 5°C intervals over the temperature range of the original data with ex trapolations beyond these ranges never exceeding 2.5 0c. All calculated values of relative humidity are give n to 0.01 percent relative humidity. This does not in any way impl y a n accuracy of 0.01 percent. The designated estimated un certainties still give the bes t predi ction of accuracy . It was fe lt that to fail to give the relative humiditi es to .01 pe rcent would be discarding information , imprecise as it mi ght be . Sin ce th e estimated uncertainties are given, we see no proble m with presenting the values of relative humidity with fi gures far beyond their estimated un certainties.
The uncertainties presented do not include uncertainties in the vapor pressure equation [22] or e nhancement equations [23] used . The "esults presented are therefore for the exact valu es of references [22] and [23]. The enhancement fa c tor for a satura ted salt solution in air is not known precisely . Analys is of th e fa ctors involved indicate that at one a tm osphere pressure or less, the difference between the enhancement fa ctor over a saturated salt solution and over pure water is negli gible . That is not the case at high pressures. The data presented are therefore considered valid near or below on e atmosphere total pressure. If saturation vapor pressure values other than those given by Wexler [22] are used, the relative humidities should be multiplied by the ratio of these saturation vapor pressures to those of Wexler. Many compilations of non-critically e valuated data on th e equilibrium humidity of saturated salt solutions ex is t [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. Table 3 is a comparison of values from thi s work and corresponding values tak e n from fiv e of these othe r compilations at four te mpe ratures . Of th e li sted compilations, o nl y this work (column a) and Hic kman's work (co lumn d) give th e so urces of the data. Hi c km a n's values (in column d) we re directl y copied fr om hi s c it ed refe re nces without modifi cation. None of the compilations oth e r th an ours (column a) gives estimates of unce rtaint y. The re for e, one would logicall y co nclude that th e authors of those compilations cons id e r their valu es to be uncertain onl y in th e las t fi g ure presented. It is also likely that some of th e va lues in one compilatio n came from the same sources as th e values in oth e r compilationssuch a relationship a ppears to exist betw een column b a nd column d.
If we assume an uncertainty of 1/2 of th e last di git in th e values giv e n in these oth e r co mpilations, and if we add that un cert a inty to th e estim ated un certainty for th e corresponding values in column a , we find that th e valu es in co lumn a (the results of this work) agree with the values in at least o ne of th e other compilations to within this composite un certainty at all points, exc e pt for:

Potassium carbonate at 10°C
Sodium bromide at 20 °C Ammonium c hloride at 30°C Potassium bromide at 10 °C, 20°C, and 30 °C Potassium chloride at 10°C, and 20 °C It should be noted that this comparison of compilations is over a limited temperature range and for onl y 17 of the 28 salt solutions evaluated and collated in this pape r.

Appendix
In all cases, the mo s t fundame ntal measure me nts presented we re used to calc ulate the actual relative humidity obtained by ea ch investigator for each datum. No atte mpt was made to evaluate purity of water or solute o r its e ffect in any investigation.
As a first s tep, 'Ill temperatures we re conve rted from the temperature scale in whic h the data we re presented into IPTS-68 temperature equivale nts. Where the temperature scales were not given, a judgment was made as to the most likely temperature scale used, based on the date of the researc h.
Likewise, where vapor pressures based on vapor pressure equations or tables were given, these were converted to new vapor pressures based on the Wexle r formulation. In the case of reported relative humidities based on de w-point meas ureme nts, the dew-point te mperature wa s reconstruc ted fr om a knowledge of the vapor pressure equation used. From th e reported control temperature and the recons tru cted dew-poi nt temperature a new relative humidity was calculated us in g th e Wexler and Gree nspan equations for vapor pressures and enhancements factors, respectively.
Whe re the isopiestic method was used with sulfuric acid as the isopiestic solution, the values of Shankman [39] for sulfuric acid activity were used to determine th e relative humidity of the saturated salt solution. This was done (1) for co nsis te nc y, because many of the researchers had done likewise; (2) because Shankman described his expe rimental work in suffi cien t detail to e nable us'to judge its quality and to estim a te the un certainty in his work ; and (3) hi s values appeared to be the most accurate available.
In dete rmining estimates of total un certainty for each datum, the un cert a inty was ta ke n as the square root of th e sum s of individual uncertainties (in te rm s of relative humidit y) squared as described by Ku [40]. Individual un certainties involved in th e individual measure me nt s we re obtain ed from th e inves ti gators' own estima tes wh e re these see med reasonable . Wh e re th e in ves ti gator did not present a reasonable estimate of un certa int y for a partic ula r paramete r, thi s a utho r made hi s own estimate of th e uncert ain ty of that parame te r based on hi s judgme nt of th e in vestigator's work a nd hi s estim a te of th e s ta te of th e art at th e time of th e investi gati on. The relative humidity uncert a inty assoc ia ted with each of th e parame te r un certainties was obtained by calc ulating the re lative humidity with and without the un certainty added to th e re lated parame te r, th e diffe re nce be ing the relative humidity un certainty [or th at parti c ular parame ter.
In so me cases the individual paramete r un certainties are not ind e pe nd e nt in the ir effect on th e relative humidity un certainty. A case in point is th e relative vapor pressure meas ure me nt method. In this technique, the individual te mpe rature and pressure measure me nt uncertainties are of no great co nseque nce, it is the estimates of th e te mperature diffe re nce and th e press ure difference in the two pressure measure me nt s that are significant. In addition, an estimate of th e degree of equilibrium achieved is of significance . In th ese types of s ituations, estimates of the differences were used in li e u of estimates of th e individual measureme nts.
In th e case of the relative humidity sensor calibration technique, an estimate of the calibration uncertainty as well as te mpe rature uncertainty were used. In the isopiestic technqiue, the relevant uncertainties are the te mperature difference, the concentration determination, the uncertainty in equilibrium and the uncertainty in the refere nce solution data.
Composite uncertainties for each datum based on the square root of the sum of the individual paramete r uncertainties squared were thus obtained.
As stated earlier, these estimates of uncertainties are the result of subjective judgments as well as objective estimates. For the great preponderance of data presented in this paper, these judgments have a minor effect on the relative humidity values as well as the total uncertainty, as was shown by the small difference obtained for the three differe nt methods of weighting.