A CRITICAL STUDY OF PRECISIOX CRYOSCOPY: T H E FREEZINGPOINT DEPRESSIOISS O F POTASSIUM COBALTICYAKIDE A S D POTASSIUM FERRICYANIDE* B \ CAMPBELL ROBERTSON
** AND
VICTOR K. L 4 MER
Introduction
1.
The modern uses of exact dilute freezing-point data are concerned with the behavior of the freezing-point depression-concentration relation, a t molalities so low that the unknown factors which enter at concentrations above 0.0j molal are negligible. .it sufficient dilution, it is possible to calculate activities of the solute by thermodynamic methods, without bringing in extensive simplifying assumptions. The accuracy of this procedure is dependent upon the range of dilution into which we may extend measurements without undue sacrifice of experimental precision. In addition to the thermodynamic treatment, we may interpret freezingpoint measurements by introducing kinetic and electrostatic considerations, and evaluating the osmotic properties of the solute on the basis of a physical picture. This approach is exemplified in the Debye-Hdckel theory,l' and its extensions in the work of Gronwall, LaMer, and S a n d r ~ d . ' ~ .it the dilutions necessary, the plot of the observed freezing-point deprewions against concentrat,ion is so nearly linear as to be of no significance. It is essential to use a divergence function which will be sensitive to deviations of the observed measurements from thr ideal behavior. For thi.5 purpose 29 Bjerrum introduced the osmotic coefficient, 4 = -, i.e. the van't Hoff '5,' vXni divided by v, the number of ions. His oeniotic deviation, I - is equivalent t o the Lewis and Randall d'j.'J In the testing of the Debye theory it is conwnient to employ the function
~
e
ZIZ?Ii'
which is proportional to the function
j in' used by Randall and his co-workers. In this expression, z I and z 2 arc 4rSe' '
z;ni,. the valences of the anions and cations respectively, and Ii' = --L1)kT , \Ye shall not enter into extensive mathematical treatment of these theories. The subsequent discussion will accordingly be confined to a feiy points of major importance, which are of especial moment in determining the experimental requirements. * Contrihution S o . 6j.j from the Chandler Laboratories of the Department of Chemistrl-, Columhia University, S e w Tork, S . T. * * Dissertation submitted in partial fulfillmerit of the requirements for the degree of Doctor of Philosophy in the Faculty of Pure Science of Columbia University by Campbell Rohertson, t-niversity Fellow for 1929-30.
I954
CAMPBELL ROBERTSOX A N D VICTOR K . LA MER
The developments of the Debye-Huckel theory by Gronwall, LaMer, and Sandved, indicate that the plot of j/mt against mi (or the analogous plot of
e against mi), should show a definite “hump,” zIzzK a t concentrations of the order of 0.001 molal, the exact concentration depending on the valence type of the salt under consideration. The experimental data of Hausrath,40and of Randall and Scott,8 on barium nitrate, may be interpreted on this basis. The existence and magnitude of the hump are crucial points, and it is unfortunate that it occurs a t just about the concentrations a t which experimental freezing-point results of sufficient precision are difficult tn obtain. The j-function, as before mentioned, is exceedingly sensitive, and its sensitivity to experimental error increases rapidly as measurements are carried below 0.001M. An error of temperature measurement of I X 10-5 degrees, which represents the extreme limit of precision with present methods, corresponds to a change in j of 4% for a uni-univalent salt at 0.001M (one of the least extreme cases). For high valence type salts a t dilutions where the hump is most manifest, the same temperature error may change j by as much as I O or 12%. I t is evident from the foregoing that unambiguous interpretation of freezing-point measurements depends upon reducing the sum of the errors to exceedingly small magnitudes in very dilute solution. The present experimental technique is barely adequate for the purpose, and this study will be primarily devoted to investigating the sources of error, and the limits to which precision may be carried. the Debye development,
2.
~
Experimental Aspects of the Problem
The first methods of R a o ~ l and t ~ ~ of B e ~ k m a n ninvolved ~~ the actual freezing of the solution, with appropriate efforts to measure a static quantity in a continuously changing system. With good technique, the method will yield results of moderate accuracy when the concentrations employed are of the order of 0.1molal, but it can never be considered exact. Twenty years later, it was recognized that inasmuch as the freezing-point of a solution is the point a t which the solution is in equilibrium with the solid phase of the pure solvent, it would be more logical procedure to bring a large quantity of the solid solvent into contact with the given solution and attempt to attain a fairly permanent equilibrium state. Richards, 46,47 in 1903, first realized the marked advantages of the equilibrium method. The subsequent history of the problem has involved the further refinement of this technique.
Temperature Xeasurement Richards in his I903 paper emphasized the fact that the Beckmann thermometer had insurmountable limitations and that further refinement depended upon the development of better temperature measuring devices.
A CRITICAL STUDY OF PRECISION CRYOSCOPY
I955
There are now two instruments which measure small temperature differences more precisely than the Beckmann thermometer: the multiple junction thermoelement, and the platinum resistance thermometer. The thermoelement was first used for freezing point work by Hausrath,4* and by both in 1902, and later by Jahn,41 D i x ~ n , ~and ' FlugeL3* Adams2 in I 9 I j greatly improved thermoelement technique, and his refinements have been followed by all the subsequent investigators; namely, Harkins, Randall, and their student^,^ 5 8 6 ~ 8 3Hovorka and Rodebush,' Abel, Redlich and v. Lengyel,' and Wesoe.Io The resistance thermometer was first used by G r i f f i t h ~in~ ~1891,and later by Barnes, Archibald and M ~ I n t o s h , ~B' e d f ~ r d ,Elliot,*Z ~ Chadwell,20 and GetniamZ5 Bedford used a differential resistance thermometer, and others a simple single coil. Richards was instrumental in improving the technique of the resistance thermometer, in connection with his calorimetric researches. At the present time it appears that the thermoelement is the superior instrument for freezing-point work. I n general, potential readings present fewer difficulties and errors than resistance readings, particularly a t high sensitivities, where contact resistances are a source of error in bridge methods. I t is not a matter of great difficulty to construct a thermoelement, which, with a fairly sensitive galvanometer, will detect temperature differences of less than I X IO-$ C., while the two most recent investigators using platinum resistance thermometers, Chadwell in 1927, and Getman in 1929, only claim a reading precision of 5 X IO-^ C. Bedford, it is true, claimed for his differential platinum thermometer a reading precision of 6 X IO-^ C.,but since this involved measuring the movement of a sliding contact on a wire to I/Z j o mm., it is doubtful whether his contention deserves much weight.
Concentration Measurement The production of a stable two-phase equilibrium, by using roughly equal quantities of solution and solid solvent, and the measurement of the equilibrium temperature by means of multiple junction thermoelements, solved two of the three major problems in freezing-point determinations. The third is the determination of the concentration of the equilibrium solution, There are two fundamentally different ways of accomplishing this. Either the solution may be introduced in a roughly known concentration, allowed to attain equilibrium, and a portion of it then removed and analyzed, or the solution may be made up in the beginning to an exact concentration, and brought into contact with the solid solvent a t so nearly the final temperature that the concentration change produced by the attainment of equilibrium may be neglected. Adams2 took the first course, using the interferometer for the analysis of his dilute solutions. Hall and Harkins4 made further determinations using . Adams' apparatus. Randall and V a n s e l o ~ ,and ~ later Randall and Scott,8 and Wesoe,lo employed conductivity measurements as means of analysis. Hovorka and Rodebush' introduced the second, or pre-cooling, method.
I956
CAMPBELL ROBERTSON A N D VICTOR K . LA MER
The precision of the interferometer falls off very rapidly in the dilute range,17 and cannot compare with that of the conductivity method. The relative merits of the latter and of the pre-cooling method are more closely balanced, and the decision between them hinges on finer points, such as labor involved, expense of construction, and available apparatus. The conductivity method is more expensive and involves more operations during the course of a determination. In this investigation it was found that the method of precooling was more expeditious and fully as precise. Although conductimetric measurements were used to check the efficiency of the pre-cooling, the former is an unnecessary refinement, if the apparatus is suitably designed for the pre-cooling technique.
Secondary Factors influencing Design The Injuence o j Dissolced Air. The solubility of air in water a t 0°C. is I O + mols per liter, hence it is obvious that air constitutes a source of great potential errors in dilute solutions. The use of the thermoelement as a measuring instrument involves the immersion of one of its legs in the mixture of solution and solid solvent, and the other in a mixture of the two phases of the pure solvent alone. Thus as a necessary condition for precise measurements the contents of the two vessels must be identical in all points except for the dissolved substance being measured; either we must h a w ri(1 forPipn material, such as air, in either vessel, or, alternatively, JW n;ust have it present in the same concentration in each. At first glance the simplest way of fulfilling the atiejvc condition ~vould appear to be the evacuat,ion of both vessels. Thr rxperimcntal difficulties, however, are very great, due to thc tentl~ilcyof dissolved gases to remain supersaturated, and to the difficulty of moiiitaining an :microbic condition in the apparatus during the manipulatiuns. lt:~ndi~!l and T.:lnselon's npparatus embodies provision for evacuation, tiit at the cost of consitierahle experimental complexity. Even then, they are iihligeti to rclrase the vaciiuni during t'he manipulations. The other means of attaining idontic:il conditions is t o pass air into both vessels and keep them conipletel~Lsaturatrd. This procedure, first usrd by Hovorka and Itodebush, possesse~the inhrxrent advantage that, although the last traces of air are difficult to remove f l o r i l solution, t h e solution pro rapid. The process of solutic~nof air in water has been intensively stutiitd'"19~?3.."6 in connection with the purification of water aupplies, and it has been definitely established that thc solution of thcl gas in the surface layer of liquid is practically instantaneous, so that only d e q u a t e mixing of the liquid is needed for quiiik arid complete total saturation. JVe hiivc :tccortiingly adopted the method of saturating with air at atmospheric pressure. Errors of the Air-salurcitzorz 'I'echr~i'y~te.The systenin tic errors of the air saturation method are smali and of definitely calcuhhle magnitude. The solubility of the air will be changrd by the slightly lowcr temperature of the vessel which contains the ice and solution, a n d by the prcsence of the dissolved 1.28 X
A CRITICAL STGDY OF PRECISION CRTOSCOPP
1957
salt. These errors give a constant percentage error in t'he observed freezingpoint lowering, since the effects on air solubility are, nearly enough, linear functions of salt concentration. From the measurements of Fox,24'summarized by Coste,*l we may calculate both errors for the case of sodium chloride, and safely assume that they will be of the same order of magnitude for other salts. Dissolved sodium chloride decreases the solubility of the air to the extent of introducing an error of 0 , 0 2 6 7 ~in the observed freezing-point depression, while the lower temperature in the solution vessel will tend to increase the solubi1it.y to give an error of o.oo4y0 in the other direction. The pressure of air over the liquid in the two vessels must be equal. A difference in pressure of 2 cm. of water will change the solubility of the air by 2.5 X IO-^ mols per liter, introducing a 0 . 1 3 7 ~error for the case of a 0.001 molal KC1 solution. This percent error will be greater for more dilute and less for stronger solutions. If the air-stream used for stirring and saturating is not pre-cooled to exactly o°C, it will melt ice and cause dilution. If it is not saturated with water vapor a t 0°C i t will evaporate some of the water in the solution vessel as it passes through and thus concentrate the solution. These contingencies are guarded against by passing the air stream through a series of wash bott'les filled with ice and water. which simultaneously cool it, and saturate it with water vapor a t 0°C. h slight amount of water will be condensed in the solution vessel on account of this treatment,, since the solution has a lower vapor pressure. This error, however, is only of the order of 0.0027~. Some ice will be melted in the solution vessel, since the incoming air is warmer than the solution by an amount equal to the freezing-point lowering, but this error is, under the most adverse conditions, less than o.oz%. The calculations for the foregoing error figures are given in detail in the section on errors. The State of Equilibrium. The term "equilibrium state'' describes an ideal, which rarely, if ever, exists in the physical world. I t is often approximated to a point beyond the detecting sensitivity of instruments, but in freezing-point work on dilute solutions considerable care is required to bring about even this close an approach to the ideal condition. It is fairly easy to maintain a solid-liquid equilibrium when the two phases have the same composition, but where N solution is in contact with the solid phase of the pure solvent, it is more difficult. To define the difficulty, let us consider a static system consisting of ice, solution, a containing vessel, and the leg of a thermoelement. Let it be assumed that at some given instant a state of temperature equilibrium exists in this ressel, and that the walls of the vessel are perfect heat insulators, so that the only thermal flow is through the thermoelement. Since the outer end of t,he thermoelement is at a slightly higher temperature than this leg, a quantity of heat will be continually flowing in through the wires and the casing. If the element is to indicate the temperature, which has been assumed momentarily to exist, this heat must be absorbed as fast
1958
CAMPBELL ROBERTSON A N D VICTOR K. LA MER
as it leaks in. The only place where it can be absorbed without raising the temperature of the liquid is a t the ice-solution interface. However, as soon as a minute quantity of ice melts, a film of water in contact with the ice is formed, and the temperature of equilibrium for this contact is not the freezingpoint of the solution, but that of water. Hence, if there is no mixing whatever in the vessel, the presence of a thermoelement will continually tend to destroy the equilibrium temperature it is meant to measure. Counteracting this tendency we have convection currents due to the slight temperature differences set up, and whatever artificial stirring which may be introduced. It cannot be anticipated that convection effects will be efficient considering the minute temperature differences involved. Hence it is evident that there must be continual and very efficient stirring if we are to successfully measure the temperature equilibrium condition with an instrument which is a conductor of heat. It is important to minimize the amount of heat conduction along the thermoelement, but unless it is rendered absolutely zero, the stirring cannot be omitted. The foregoing discussion has been postulated on the perfect heat insulating properties of the containing vessels. Thermal leakage through the vessel walls of course exists, concurrently with that through the thermoelement, but its effect on the stability of the measured equilibrium is less, because in this case the absorption of heat occurs at a relatively remote region of the vessel. Another consideration, as well as the above, dictates thorough stirring. !Phk pressure coefficient of the freezing-point of water is o.0075"C. per atmosphere. Therefore if we have a vertical vessel 14cm. deep, filled with a mixture of ice and water, the equilibrium temperature for the bottom will be o . o o o ~ ~ C . lower than that for the top. A state of perfect freezing-point equilibrium cannot exist in a vessel of finite size, (unless we have only a plane horizontal contact surface between the phases) and efficient stirring is imperative if we are to approximate the mean temperature to an equal extent in both containers. It is best to have the temperature of the outside thermostatic bath approximate that of the vessel which contains the ice and the solution. By doing this, a higher thermal head is put on the vessel which contains the ice and water, but here there is not the disability existing where the two phases differ in composition. 3. The Apparatus and Procedure Outline of Apparatus The apparatus is illustrated in Figs. I and 2. Fig. I shows that part of the assembly which is contained within the ice-water thermostat. Fig. 2 shows the electrical system. The assembly consists of the following chief parts: ( I ) A double-walled copper tank (Fig. I,A) of about IOO L. capacity, equipped with air-jet circulators (not shown), which, filled with ice and water, serves as a thermostat. Two 500 cc silvered Dewar vessels (Fig. I,B), supported in a brass (2) frame in the thermostat. The vessels are provided with air circulators, and with lead tubes to permit emptying and filling without disturbing the assembly.
A CRITICAL STUDY OF PRECISION CRYOSCOPY
I959
(3) A rack holding ten joo cc bottles, clamped into the bottom of the thermostat. The bottles are each equipped with a lead tube and vent tube opening above the surface of the bath, and contain the series of concentrations on which measurements are being made. ( 4 ) A 48 junction copper-nickel thermoelement (Fig. I,D) the legs encased in seamless silver tubes, one leg sealed into each Dewar vessel. ( 5 ) An electrical measuring system, (Fig. 2 ) consisting of a standard I-ohm coil (A) kept a t z~OC., a high sensitivity galvanometer (B), and a special form of potentiometer circuit.
FIG.I Freezing-Point Apparatus
Procedure i n making Measurements. As the first step, all glassware was cleaned with chromic acid. It was then steamed out for four hours, and kept in contact with distilled water for 2 4 hours more to remove adsorbed material from the cleaning mixture. Dilute solutions of desired concentrations were made up by weight from the dried salt. Usually five concentrations were prepared, and two reservoir bottles filled with each. The bottles were then assembled in the rack, and the latter clamped in place in the bottom of the thermostat tank. The stoppers carrying the tubes leading from the bottles were sealed with “picein” cement as an extra precaution against leaks. One hundred and fifty pounds of cracked ice were placed in the tank, enough water run in to make a mush, the circulators started, the tank covered,
1960
CAMPBELL ROBERTSOS A S D VICTOR IC. LA MER
and the assembly left over night to allow the contents of the bottles to reach o°C. Several measurements of the rate of cooling of these bottles were made, using an auxiliary thermoelement, the average half-time being 18 minutes. On this basis about 4 hours should be sufficient to cool the bottle contents fyom zg"C. to 0 . 0 0 3 ~ C .In the mornings the auxiliary thermoelement reading was never greater than 0.01 degree and the average was nearer o.oo4OC. At the start of the actu:tl run, the series of three gas-washing bottles, which served to chill the air used for saturating and stirring the contents of the freezing-point vessels, were half-filled with cracked ice and water and put in place inside the thermostat.
I
.";$-::-%".3
-
1
I
ri
I
The two freezing-point vessels were filled with sifted and washed cracked ice, the stoppers, carrying inlet and exit tubes, stirrers, and thermoelement, were inserted, sealed into place with "picein" cement, the seals tested f o r tightness with compressed air, and the frame carrying the vessels lowered into place in the thermostat tank. The thrrmoelement lead wires were connected to the potentiometer circuit, the vessels filled with previously chilled distilled water, and a zero thermoelement reading taken. The procedure in making a measiiremr~ntfor a given concentration w3.j then as follows: the contents of the solution vessel were transferred by suction to a waste container, and the vessel filled from bottle S o . I , by applying air pressure through the vent tube of the bottle. The small stirrer and saturator inside the vessel was run for three minutes. Then, with the circulator running, this first filling was removed in the same
A CRITICAL STUDY OF PRECISIOK CRYOSCOPY
1961
manner as before. The whole operation was then repeated twice more in the same way, except that the third filling was from bottle No z and was circulated for five minutes. The third filling was removed, and the final filling run in from bottle S o 2. The circulator was again started, and a t intervals of ten minutes, readings of the thermoelement electromotive forces aero taken. If the so-minute and 30-minute readings agreed to within j X IO& degree, their mean was then recorded as the reading, and the entire process repeated for the next concentration. If the solution was slow in coming to equilibrium, a reading was taken at the end of 40 minutes.
Detaded Description of the Apparatus (A) The ice thermostat. The thermostat consisted of a double-walled copper tank, inside dimensions 80 x 60 x 30 cm, the gross capacity being 9 j liters. The j cm. space in the hollow walls was filled with ice-box felt, to minimize the heat losses. When the tank was allowed to stand filled with ice without disturbing the cover, the last of the ice did not melt for about four days. The tank was covered with a frame carrying two sheets of glass about j cm apart, allowing inspection of the interior without removal of the cover, and a t the same time diminishing the heat losses. To eliminate stratification, four air-jet circulators were placed on the sides of the tank, lifting 36 liters of water per minute from the bottom of the tank, and distributing it over the ice at the top. They were made from pieces of one-inch conduit, about 2 5 inches long, and extended from the bottom to the top of the tank, the upper end being bent around on a 6 inch radius to a horizontal delivery opening a t the level of the liquid in the tank. Large diameter glass tubes are fastened to the backs of the pipes, and a t the bottom are bent around in goose-neck shape, terminating in a vertical jet projecting one inch into the bottom orifice. With circulators of the type described, no variations of greater than a few thousandths of a degree were ever observed on auxiliary thermoelements inserted in various parts of the bath. (B) The freezing-point vessel assembly. The freezing-point vessels were high-vacuum silvered Dewar flasks with a capacity of about qjo cc. They were rigidly fastened about 3 cm. apart in a frame made of heavy brass. With the frame and flasks in position, the tops of the flasks were 18 cm. below the surface of the bath. The flasks were closed with large rubber stoppers. From the stoppers were suspended miniature air-jet circulators, similar in principle to those which were found to work well in the outside tank. They were made of glass, the construction being evident from the diagram. These small circulators lifted about 2 0 0 cc. per minute, a volume equal to the total liquid contents of each vessel during measurements. The stoppers carried thin glass drain tubes, extending exactly to the center of the rounded bottom of each flask, SO that when suction was applied to this tube and the liquid drained, less than 0.3 cc. remained in the bottom. The flasks were also equipped with vent tubes extending about 14 cm. below the lower surface of the stopper, the ex-
1962
CAMPBELL ROBERTSON AND VICTOR K . LA MER
tension being long enough so that the contents of the flasks never came in contact with the rubber. The legs of the thermoelement extended to within j cm. of the bottoms of the flasks. The thermoelement leads were carried to the top of the bath through an 8 mm glass tube filled with paraffin. (C) The solution bottle assembly. The solution bottles were ten 500 cc bottles rigidly held in a frame, the latter carrying two brass rods extending to the top edge of the tank, and there being clamped under projecting lugs, to hold the rack in place as the bottles were gradually drained. The lead tubes, extending from the bottoms of the bottles, and the vent tubes, extending from the shoulders, were single lengths of small diameter glass tubing. The lead tubes were carried to the center front of the rack, where they were bent up to the top of the tank. The vent tubes were likewise carried above the surface, five a t each side. The lead and vent tubes were capped to prevent the accidental ingress of water.
The Design of Sensztzae Thermolements By increasing the number of junctions, the theoretical sensitivity, attainable with a thermoelement, increases almost without limit. In the design of a specific instrument for a given problem, however, there are binding restrictions, and for any set of conditions there is a definite optimum design. The thermolement is an instrument of low electromotive force and relatively high resistance, so that the greatest degree sensitivity, while nominally a question of electromotive force per degree, is ultimately, where galvanometers are involved, a question of current per degree. The design is also limited by the allowable heat conductivity and by the necessity of having the element of such form and size that all the junctions a t each end may be assuredly a t the temperature they are meant to measure. The weight to be assigned to each of these factors varies acccording to the problem in hand, but in general for the type of element where the junctions are bundled together and enclosed in a case, the first and third are the most important. Thus the allowable bulk of the completed instrument is the chief limiting factor and the following development is based on this assumption. The most desirable metal pair, fulfilling these criteria, is one which has the maximum ratio of thermal electromotive force to total resistance. It is of no value to select a couple of high electromotive force per degree if the resistance is simultaneously increased in like or greater proportion. This governing ratio has been ascertained for a number of combjnations which seemed promising. When the thermoelectric potentials of the various metals are plotted on a linear scale, it is a t once seen that most of the common metals lie close together, and since this group includes copper, the others in it may be neglected, as having no advantage, since silver alone has a lower resistance than copper. The available metals away from the central group on the scale, are iron and antimony, on one side, nickel, cobalt, bismuth, and constantan, on the other. Thus there are some ten combinations for which the electromotive forceresistance ratio may profitably be calculated.
A CRITICAL STUDY OF PRECISION CRTOSCOPY
I963
The following data are from the International Critical Tables. Pair
Ratio 1.39 0.81 0.6s
Iron-copper Antimony-copper Bismuth-copper Constantan-copper Nickel-copper
Pair
Iron-nickel Iron-constantan Antimony-nickel Antimony-constantan Antimony-bismuth
0.77 2.47
Ratio 2.18 0.90 1.17
0.83 0.66
From these ratios, which are the true measure of the relative sensitivity to be obtained from an element of given cross-section, it is evident that the thermal electromotive force done is not a reliable guide. Several combinations are superior to the copper constantan commonly employed, coppernickel particularly so, for this type of work.* Ratio of the Two Metals. For a given pair of metals, there is an optimum ratio of cross-sections which gives the minimum resistance. If A represents the total cross-section of a pair of wires M and R, and p is the total resistance of the two, P = p,,!Ax
+ plZlfA(1-x)
(1)
where x is the fraction of cross-section represented by M. Solving for the minimum value of p in terms of x,
@ 6X
=
-pw/Ax2
x =
-
PM
+ pR/A(r-x)* *d p m Pn
,
Pn
- Pw
The real root gives the optimum fraction of cross-section for M. For zopper-nickel elements, this ratio is I :z.I , which is most nearly satisfied by a difference of three sizes in the B. & S. gauge.
Number of Junctions. The final point of design is the choice of specific sizes and number of junctions, as functions of the metal pair, the allowable cross-section, and the galvanometer characteristics.
Let: D = scale divisions per amp. A
allowable cross section. K res. const. of metal pair. N = number of junctions R, = galv. res. (plus potentiometer = =
Rj E
resistance per junc. e.m.f. per degree per junc. C cross sec. per junc. y scale divisions per "C. res. if appreciable.) = = = =
From these quantities we may obtain the expression for scale divisions per degree, the criterion of sensitivity.
+
y = DNE/(R, RjN), (4) or scale divisions per degree equal scale divisions per microampere times the ratio of total electromotive force to total resistance. *%e footnote at bottom of page 1976.
I944
CAMPBELL ROBERTSON AND VICTOR K . LA MER
Rut, N
=
A/C and It, = K/C
Substituting, y = DAEC/(R,C2
+ AK)
(5) (4)
Solving for the maximum,
+
dy/dC = DAE(AK - RgC2)/(RaC2 AK)'
(7)
Equating the derivative to zero, we get,
Thus for any allowable cross-section selected for the element, two wire sizes may be found which will give the maximum sensitivity. To make the numerical calculation from equations (8) and ( 9 ) , it is necessary to have a value for K. This is obtained from the expression.
where L is the length of one junction, and x and ( I -x) are the fractions of cross-section of the respective metals. K has the dimensions of ohms X cm2, and has the value for copper-nickel elements 35 cm. long, of 5 . 2 9 X IO-I. The condition for greatest sensitivity is that the resistance of the element equal the external resistance. This may be shown by transposing equation(;).
R, = AK/C'
=
R,N
Insulation. When the thermoelement is immersed in the solution, the condition for precision is that the temperatures of the wire junctions and the outside of the casing shall differ only by a negligible amount. The total heat conductivity from the junctions out through the insulating and embedding material, and casing, must be much greater than that along the thermoelement wires and the casing from the outside environment, which is a t some differing temperature. The first obvious requisite is that the bundled thermoelement wires should, as nearly as possible, entirely fill the casing, t o obtain the shortest possible heat path from the solution to the tips of the junctions. This requires very thin, but electrically effective, insulation between the junctions In the course of this work, several sorts of insulating compound were used. The thin rubber coat, described by Adams2 and by White3Lwas first tried, but with little success, due to the difficulty of getting an even coat of the rubber over the tips. The solution of rubber in carbon disulphide tended to gather into globules over the heavy parts of the junctions and to pull away entirely from the sharper bends of the metal, producing short circuits. Bakelite varnish and a special compound of gutta percha and rosin were tried, but did not prove as satisfactory as shellac. Shellac can be evenly applied with a fine camel's hair brush, dries readily and does not crack or come off unless the wire is bent very sharply.
A CRITICAL STUDY OF PRECISIOX CRYOSCOPY
196s
I t is necessary to fill the space between the bundle of wires and the casing with some suitable heat conducting imbedding material. Wood’s metal is unsuitable, for although it is not difficult to secure adequate insulation between the bundled thermoelement junctions, it is quite another matter to sufficiently insulate the junctions to allow insertion in a conducting mass wit,hout the advent of short circuits. After several unsuccessful trials, the metal-imbedding idea was abandoned in favor of paraffin. The thermal conductivity of paraffin is considerably greater than that of air, though of course much less than that of any metallic mass. Whites1 recommends the use of naphthalene, which has a slightly higher thermal conductivity than paraffin, but the difference was not thought to he of sufficient moment to offset the ease of handling of paraffin. The thermal properties of elements as ordinarily constructed can he greatly improved by using silver, or other metal, tubes instead of glass. The improvement in the use of silver is not entirely a function of the thermal conductivities, though that of silver is much greater than that of glass. The important point is that the silver tube may he very thin-walled, not over 0.1.; mm. in thickness, while a glass tube of such wall-thickness would be too fragile for use.
Construction and Calibrntion of the Thermoelements. Two multi-junction thermoelements were constructed. The first, a z 5-junction copper-constantan element, enclosed in glass, was used for the first freezing-point measurements, and subsequently as an auxiliary element for ascertaining the stability and attainment of the temperature equilibrium between the ice bath and the solutions in the reservoir bottles. The second, a &junction copper-nickel element, enclosed in silver tubes, was used for the freezing-point measurements here reported. The copper-constantan element was made from KO. 30 double cotton copper covered copper wire, and X o . 28 double cotton covered constantan wire. The wires were given several coats of shellac, cut to 3 j cm. length, and the ends bared for about 7 mm. The thermoelement was assembled on a board 3 2 cm. in width, along the edges of which were bolted tnxo strips containing grooves on the side in contact with the board. Each strip had more than enough grooves for the entire number of wires in the thermoelement. The wires were inserted through these grooves in the proper order, and the projecting ends twisted together. The junctions were soldered by successive dips into small crucibles of melted rosin and solder, the temperature of the solder being such that only the thinnest of coats remained on the junction. The enclosure for the first thermoelement consistcd of glass tubes of 8 mm. outside diameter. The central portion of the element was enclosed in a semicircular glass tube of 11mni. diameter, bent to a radius of 70 mm. After the shellac insulation on the junctions had dried, the wires were bundled together into a U-shape, thrust through the center enclosure, and the two lead wires brought out through a side tube. The glass legs were then poured partly full of paraffin, the bundle of junctions inserted, the legs
I 966
CAMPBELL ROBERTSON AKD VICTOR K . LA MER
fastened into the central bend, and the remaining interior poured full of paraffin as a precaution against the intrusion of moisture. Since this element was not intended to be used for other than a relatively rough instrument, it was calibrated against two Beckmann thermometers, which had previously been compared with each other along the entire length of their scales. The electromotive force of the copper-constantan element as thus obtained was 956 microvolts per degree. Its resistance at o°C. was 9 2 . 5 ohms. The Copper-Nickel Element. The copper-nickel thermoelement embodied the findings of the previous section as to wire sizes, and the improvement of making the legs of the enclosure of silver instead of glass. Insulated nickel wire is not an article of commerce, but a quantity of KO. 28 high purity nickel wire was specially double silk insulated for us by Driver, Harris & Co., of Kewark. The copper wire was No. 32 double silk covered. The methods of assembling, soldering, and insulating the tips, were identical with those previously described, except that on account of the larger number of junctions, the wires were cut in lengths of 33. j , 34.j , 3 j . j, and 36. j cm., in order to stagger the junctions and avoid undue bulk a t any one point. The silver tubes were seamless spun of C.P. silver, the dimensions being I j centimeters length by 8 mm. outside diameter. The average wall thickness was 0.I 5 mm. There were no appreciable corrosion effects. For the calibration of the copper-nickel thermoelement, the equation E = Eo - aTn, first suggested for thermoelements by Rodebush,l6 and subsequently used by Randall and V a n ~ e l o w ,was ~ taken as the calibration function. One leg of the element was kept a t o°C. in a mush of melting ice, and the other was immersed successively in a mush of redistilled mercury, in melting ice, and in a thermostat a t 2 5 O C . , the exact temperature of the latter being checked with a Bureau of Standards calibrated thermometer. The results are given in the following table: E, T,OK. Ice-mercury 234.2 -26775 0 Ice-ice 2-73 ’ 1 298. I - 19708 Ice-25 C. From these three points on the curve, the values for the constants a and n, in the above equation, were derived. The values were: a = 0.5887 n =2.1388 The values for the derivatives a t 273.1 K, are:
(g)
273.1 =
748.67
The electromotive force of this element a t 0°C. is accordingly 748.67 microvolts per degree. The deviation from linearity is only 0.02% for a measured temperature difference of 0.I O C . , and the calibration can hence be regarded as linear for the small temperature differences measured in this work.
A CRITICAL S T U D Y OF PRECISION CRYOSCOPY
The Electrical Measuring System The thermoelectromotive forces measured ranged from two to IOO microvolts, and it was necessary to measure them with a precision of a t least 0.1%. Fig. 3 shows an auxiliary circuit, resembling that used by Hovorka and Rodebush,' which, when used in conjunction with a Type K potentiometer, allows the rapid measurement of electromotive forces from 0.1 microvolt to 500 microvolts, with a precision limited only by the sensitivity of the galvanometer in the thermoelement circuit.
10,000
ohmr
FIG.3 The Electrical Measuring Circuit
Descriptzon. The unknown electromotive force of the thermoelement is balanced against a potential drop produced across a calibrated I-ohm copper kept in a thermostat a t zs0C. coil R1, The potential drop El is varied by adjusting the variable resistances R, and Rs,until a balance is secured. Its magnitude is then easily calculated from the simple relations of the circuit. Current from an auxiliary battery B flows through the circuit, and this current Iz divides itself between the two branches of the circuit containing R1 and R Prespectively. The total current If is measured with the Type K potentiometer, in terms of the potential drop across the io00 ohm coil R3. This measured electromotive force ES,is then related to El, the electromotive force balanced against the thermoelement, by the following functions :
Or, combining,
Thus El is an exact function of ESand a factor made up of the resistance values of the coils employed.
1968
C A N P B E L L ROBERTSON A l i D VICTOR K. LA MER
Miscellaneous Observations on the Electrical System The high sensit,ivity galvanometer used was a Leeds 8: Xorthrup Co., Type HS, having a coil resistance of 14.8 ohms, period 7.i seconds, external critical damping resistance 20 ohms, and an approximate sensitivity of j o mm. scale per microvolt at the reading distance of 5 meters. The galvanometer was supported on a weighted platform, the latter suspended by three heavy ropes from a ceiling beam. From the underside of the platform extended a heavy brass rod, carrying three heavy sheet steel vanes, which dipped into a vessel filled with heavy lubricating oil. The effectiveness of this suspension is evidenced by the fact that the vibrating movement of the reflected beam on the scale was on the average less than 0.2 mni. a t five meters scale distance, although when the galvanometer was placed on the stonetopped desk in the same laboratory, the vibration at one-meter scale distance was such as to render reading almost impossible. It was found that the direction and force of the wind made some difference in the behaviour of the instrument, which was located on the sixth floor of the Chandler laboratories, and on very windy days it was impossible to make precision readings. nee boxes indicated on the diagram were conThe coils and variable re nected throughout with KO. I 8 rubber insulated copper wire. The electrical system was supported entirely on metal, and the parts of the metal supporting system connected, as recommended by lVhite.30 This procedure eliminates the danger of stray potential differences from the I I O or 2 2 0 volt supply mains leaking into the sensitive circuit,., Purasific Currents. Ti1 g t - ~ m di:, may br said that there will always be some stray thermoelectric f o ~ , ~A . ioltage may even be detected between two pieces of copper, due to differences i n hardness or surface condition, and a soldered joint may be a gruvp source of error, if any appreciable thermal gradient exists through the solder. Precision circuits should include only one metal, preferably copper, and if dissimilar metals must be joined, the junction should be lagged, if there is any possibility of its acquiring a thermal gradient. Fortunately for the operation of thermoeicment systems, it is always possible to ascertain and roughly measure the magnitude of the parasitics. If a copper coil of exactly the same resistance be substituted for the thermoelement,, the rest of the circuit remaining untouched, the galvanometer will register zero, unless parasitics arc present, JThere they exist, they are ordinarily fairly constant over periods as long as I O or 12 hours, and a false aero setting may therefore be used to compensate. Preparation of Materials. The potassium cobalticyanide and potassium ferricyanide were C.P. salts, recrystallized three times and dried. The solutions of known concentration were made up directly by weight in quantity sufficient to assure a precision of better than 0.1%. In the early part of the work, “conductivity water” was prepared after the usual method, but it was subsequently found that the distilled water
A CRITICAL S T U D Y O F PRECISION CRYOSCOPY
1969
supply of the Chandler Laboratories was of sufficiently high quality for freezing-point work. (See p. 1979.) The ice used was commercial ice, sifted to remove small particles, and thoroughly washed with distilled water. Several previous on freezing-points have noted that clear samples of commercial ice were exceedingly pure, giving when melted a water having a conductivity commensurate with that of carefully distilled “conductivity water.” This observation was verified in the present work.
FIG.4 Freezing-Point Data
The solutions, after being made up, were always used within 18 hours. I t was observed that on several occasions when delays arose and precluded prompt use of solutions, that the concentrations appreciably diminished, due to slow adsorption on the vessel walls.
Experimental Results and Errors Obsewed Freezing-Point Depressions f o r Potassium C o b a l t i c ~ a n i d eand Potas,siiinz Ferricyanide The data obtained for these salts are presented in Tables I and 11 and Fig. 4. The conductivity measurements given in columns 3 and 4 show that if the proper precautions are observed, the change in concentration of the solution being measured from the time it is made up until the final equilibrium reading is made, cannot be a source of serious error. In a few of the earlier
1970
CAMPBELL ROBERTSON A N D VICTOR K . LA MER
measurements reported, however, a slight change in concentration, detectable by final conductivity measurements, took place, and in these cases i t was deemed proper to compute a corrected concentration from the conductivity-concentration relation for the salt in question.
Interpretation of the Results. The interest attached to these measurements turns upon the value of the limit of j/.mi, as m approaches zero, and on the path by which the limit is approached. The I’alue of the Limit. Since experiment can never give the limiting value, without extrapolation, appeal must be made to some theory of which the postulates are reasonable, and which is not inconsistent with experimental data a t finite concentrations. Both the original form of the Debye-Huckel theoryll and the mathematical extension given by Gronwall, LaMer and Sandved,I3 predict the same limiting value for jjm;; namely 2.76 for 1,3 valence types of salts. The various theoretical objections to the validity of this theory in finite concentrations, such as local smoothing, fluctuation terms, D in neighborhood of ions, and the assumption of spherical symmetry, which have been raised by such competent authority as R. H . FowlerI1*admittedly vanish as m approaches zero, so that this limiting value of j / m i should not be subject to an error greater than the uncertainty in D-3P, where D is the dielectric constant of water. The -4pproach to the Limit. The essential difference between the DebyeHuckel form of the theory and the purely mathematical extension given by Gronwall, LaMer and Sandved, resides in the form of the function produced by the introduction of the parameter “a” representing the ion size, which introduction is inescapable in finite concentrations. The original theory demanded that j/m.$ approach its limit concavely to the X axis, i.e. j/m& was always less than the limiting value, regardless of the value of “a” as long as it remains finite. On the other hand the equations of Gronpall, LaMer and Sandved, predict that whenever “a” is less than about 7-4for 1,3 types of salts, then j j m t will exhibit values greater than 2.76 and will approach this limit by passing through a maximum, the characteristics of the maximum being dependent upon the value of “a.” Signgcance of the Present Measurements. Extrapolation for the evaluation of limits is trustworthy only when experimental data and theory together justify a linear extrapolation. This is certainly not the case for aqueous solutions of electrolytes of valence type higher than ( I , - I ) , and all present day freezing-point measurements, including these reported, are insufficient to verify the ezact numerical value of the limit for j ;mi. Consideration of the data as plotted in Fig. 5 for the 3,’ saltsstudied, would lead us to place the limiting value of j .‘mi at a lower value than 2.76; perhaps 2.4. The allowable weight to be attached to the points, however, decreases decidedly a t the dilute end of the curve, and, if we consider only the more accurate points a t the four higher concentrations, it is evident
A CRITICAL STUDY OF PRECISION CRPOSCOPY
1971
,
0 0 0 0 0 0 0 0 0 0 0
.
.
.
0 0 0 0 0 0
U 2 u r a o m
..
.
.
.
.
.
.
.
* m m N
N 3
0m10
r - 0
g
0
d d m N m 3
N
N
oI
0":
I
. . .
.
10 r - 3 e
m. i 0.
0 9 .
0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
0.
d r O 1 0 m N VI N d r - m 0 " "
.
.
.
N
d
b
m
"'9
h m m mVI-
3
__ ~
, 00 00 00 0 00 00 0
-
g.2; 0 9m v0 . "0 0 0 3
N
I972
CAMPBELL ROBERTSON A N D VICTOR K. LA MER
that the limit of 2 . 7 6 would be entirely compatible with these points. The final decision, however, as to the correctness of the theoretical limit, on the basis of freezing-point measurements, must be deferred until it is possible to obtain experimental data a t even lower concentrations. On the other hand, the data submitted for potassium cobalticyanide and potassium ferricyanide yield evidence in favor of the correctness of the extension of Gronwall, LaMer and Sandved, as regards the path of approach to the limit, since j/m+ passes through a maximum value instead of along a concave path.
FIG.5 Freezing-PointFunction for
( I , -3)
Salts
In evaluating the significance of the curve, as determined by these experimental points, the accidental errors may be considered as one group, and the constant and systematic errors in another. The deviation of the points among themselves is a sufficient criterion of the accidental errors to demonstrate that the shape of the curve is not fortuitous. The independent runs indicated by the differently shaped points were made a t different times, and indicate that the precision of reading and the errors in making up the solutions were sufficiently low to permit determination of the approximate form of the curve. Other considerations which must be entertained are the possibilities of systematic or constant errors being of such a nature and magnitude as to bring about the observed juxtaposition of the experimental points. It may be observed from the tabular summary of errors given elsewhere (p. 19;4), that most of the constant and systematic errors affect the result by a constant
A CRITICAL STUDY O F PRECISION CRYOSCOPY
I9i3
percentage, although some, such as the error in reading the galvanometer, would take the form of a constant fraction of a degree. I n order to check the possibilities of systematic errors, two series of values for j/m$ were calculated, using values lower by I X I O - ~ O C than those actually found, as a criterion of the effects of constant degree errors, and values of observed depression uniformly lower by o.5yc, as a criterion of the effect of constant percentage error. The curves obtained by the use of these values still show the "hump" shape, and are illustrated in Fig. 6.
FIG.6
,
Effect of errors on t h e l funrtion m)
I t is shown in the section on errors, that the maximum probable errors are distinctly less than these assumed values, and we may therefore conclude that these measurements prove the existence of the shape of curve, a t low dilutions, predicted by the extension of the Debye-Hiickel theory, as given by Gronwall, LaMer and Sandved. *
Summary of Errors In order that we may make a significant estimate of the meaning of the results for such sensitive measurements as these, it is necessary to systematically tabulate the possible sources of error, and satisfactorily account for each. *The "hump" appears a t higher concentrations in the corresponding curves for the activity coefficient of the solute than for that of the solvent. The data for the activity coefficient of CdSO, recently obtained by LaMer and Parks ( J . Am. Chem. Soc., 53, June (1931)) using the method of electromotive force and the similar results ohtained for ZnSOn Cowperthwaite and LaMer ( J . Am. Chem. Soc., 53, June (1931)) furnish conclusive proof of thevalidity of the Gronwall, LaMer and Sandved extension of the theory.
I974
CAMPBELL ROBERTGON AND VICTOR K. LA MER
The form adopted for this purpose is that of Table 111. The errors are grouped first in three principal categories: errors of the thermoelement and electrical measuring system; errors in concentration of solutions; error in the equilibrium state. Within each of these heads the individual errors are further classified into the usual categories of constant, accidental, and systematic. I n the table each error is named and a figure given as a description of its magnitude. In the pages following the separate errors are discussed and the source of each figure in the table is given.
( A ) The Thermoelement and Electrical Measuring System. The electrical system consists of means whereby the electromotive force in the thermoelement is balanced against a potential drop produced across a standard I-ohm copper coil, which electromotive force a t balance is taken as that of the thermoelement. (I.) Electromotive force measurement. The known electromotive force E l produced across the I-ohm copper coil is calculated from the electrical relations of five resistances and the potentiometer readings:
TABLE I11 Summary of Possible Errors Magnitude
Error
Group I . Errors in Thermoelement and Electrical System Constant (I). Error of standard cell. (Ia). Error in resistance coil values. (2). Error in calibration of thermoelement Systematic ( 3 ) . Inhomogeneity of thermoelement metal. (4). Stray electromotive forces ( 5 ) . Parasitic electromotive forces. ( 6 ) . Peltier or current heating in thermoelement. Accidental (7). Reading error of galvanometer.
0.01
yo
nil 5 x Io-T. 0.1
I
Group I I . Errors in Concentrations. Constant (7a). Calibration errors. (see under 16). (8). Attainment of equilibrium temperature. (9). Effect of dissolved salt and lower temperature on air solubility.
X
%
10%.
0.025%
0.02
70
A CRITICAL STUDY OF PRECISION CRYOSCOPY
I975
TABLEI11 (Continued) Magnitude
Error
Systematic (IO). Errors of rinsing procedure. ( I I). Possible insufficiency of pre-cooling. ( I 2). Melting by stirring air. ( 1 3 ) . Condensation from stirring air.
I
x
Io-".
0.02
y*
nil nil
Accidental ( 1 4 ) . Purity of salts. (IS). Purity of water (16). Error in making solutions. ( I 7). Contamination by dissolved glass. (IS). Effect of thermal leakage of freezing-point vessels.
0.1
5
x
%
10-6C.
0.05
0.1 0.01
% %
yo
Group I I I . Error tn Stabilaty and Homogenezty of Equilibrium Systemafac (19). Effect of imperfections in thermal isolation on the equilibrium temperature attained and recorded.
El is an exact function of Ea,the potentiometer reading, and a factor made up of the resistance values of the various coils employed. Below are tabulated the percentage accuracies of the factors of the above equation, together with notation as to the source of the precision figure. Factor
E3
Rz 8: RS (RI
R1 RP
+ + RJ
Accuracy 0 01%
0 01% 0 01%
0 01%
Source of Figure
Calibration of type K potentiometer and Bur. of Stand. calibration of std.cel1 Maker's calibration Calibration against Rz Calibration of the box R4 against the std. res. R3
+ +
Lead resistance enters into none of the factors above save (R1 Rz R4). I n this case the total lead resistance is 0.23 ohm, as measured after the apparatus was set up. Since Ra is never less than 1000ohms, neglecting the lead resistance entirely will introduce a maximum error of only 0 . 0 2 ~ ~ .The net accuracy, or extent to which the electromotive force is to be taken as the value in volts, then is, by least squares,
Error = .\/~(o.oI)* =0.022% The electrical measuring system thus is of such accuracy that its error may be entirely neglected in any consideration of the accuracy of the final result.
19i4
CAMPBELL ROBERTSON AND VICTOR K . LA MER
(2.) Calibration of the thermoelement. The magnitude cf the absolute temperatures used as fixed points for the calibrntion was known to o.reC., or less than o.04yo. In consideration of this and of the siriall second derivative of the calibration curve, it is safe to say that the error in the slope at 0°C. is less than o.oj%. ( 3 . ) Inhomogeneity of the thermoelement wires. '\5-hite3:has pointed o u t that for constantan mire this may be a serious prror. If the composition of the wire is not uniform along its entire length, a slight change in the positioii of the tenipcrature gradient along the length of the element will produce a difference in the observed electromotive force, even though the ends are iiinintained at constant temperatures. The error due to inhomogeneity in the copper-nickel element was checked by varying the position of the 2 5 degree thermal gradient used for the calibration. S o difference in the observed electromotive force was noted, as the position of the gradient, was shifted several centimeters. I t should be observed that with an element having as many as 48 junctions, the probability is high that small variations will cancel each other.* (4). Stray voltages. Since the electromotive force corresponding to a temperature difference of O.OOOOI~C. is only 9 x IO-^ volts, it is obvious that a very small leak from tlirb I I O volt power line, or from other sources, is capablt. of introducing serious error. Fortunately the method of equipotential shielding, referred to in the description of the apparatus and first brought out by White:3a ma) be depended upon to entirely remove this danger. ( 5 ) . Parasitic electromotive forces. These have been discussed on page 1968 and, as there noted, the only remedy is to keep the entire system at as uniform a temperature as possible, by avoiding temperature changes within the room, and by lagging where necessary. The size and direction of the parasities may be ascertained by immersing both legs of the thermoelement in pure ice arid wat>er. This was always done before and after measurements and the observed parasitic was never greater than j X IO-^ volts. ( 6 ) . Peltier and current heating in the thermoelement. The galvanometer draws a finite current from the thermoelement, but the errors thus produced are very small, as the following calculation shows: Peltier Heat = - EI, and if we assume a temperature difference of O . O I O , and the switch closed for I O sec., the heat developed is 4.3 X 1 0 - l ~cal. The current heat for the same temperature difference and time would = 12R = 2 X IO-' cal.
*Since this work was completed, a paper has come to our attention entitled "Thermoelectricity of Nickel Wire" by T. Tsutsui: Tokyo Scientific Papers, Institute of Physical Chemical Research, 11, 93 (1929), in which it is stated that nickel wire may give serious errors of inhomogeneitv, if the wire is bent very often. As above noted, the first copprrnickel element tested a i a whole did not display this trouble, but a second one subsequently made in this laboratory was not so satisfactory, and it is possihle that nickel is in general an unreliable element for couples.
A CRITICAL STUDY OF PRECISIOS CRYOSCOPY
19i7
Since both temperature and time given are much larger than ever used in this work, it is obvious that the error is nil. ( 7 ) . Galvanometer sensitivity. The sensitivity of the galvanometer used as a balancing instrument is such that, when connected with the @-junction copper-nickel thermoelement, I min. on the illuminated galvanometer reading scale is equal to o . o o o o ~ T . The error in thc observation of the position is certainly less than 0 . 2 5 mm. and the degree of precision of the reading is equal, conservatively, to I X IO-~Y'.
Errors in Concentration. (8). Attainment of equilibrium. When the solution is introduced, it is at o°C., and the lowering of the temperature of the contents of the vessel to the freezing point for that concentration will bring about a certain amount of melting, and thus a dilution. h simple calculation, however, will show that at concentrations such as those worked with, this error is negligible. Consider a solution of 0.001 molal potassium ferrocyanide, for which the freezing-point depression is about 0.013 degrees. The flasks contain zoo g of ice, of specific heat 0.j, 2 j o grams of water, specific heat, 1.0, and the water equivalent of the flask is about 2 j grams. The total mater equivalent is 3 7 j grams, which for 0.013 degrees change requires 4.87 calories. This will melt 0.062 grams of ice, which, for the 2 5 0 grams of solution introduces an error of 0 . 0 2 jyo. The error is less when the solution is more dilute. (9). Change in air solubility. Fox24made an extended study both of the solubility of air in pure water at temperatures from 0°C.to zg°C., and of the change in solubility due to the presence of small amounts of sodium chloride. From his results the following may be calculated : Solubility of air in pure water a t oo = 1.28 X Io-jm 1 Change in solubility due t o small quantities of dissolved sodiuni chloride = j . 2 2 X IO+ mols!/moI KaC1 Change in solubility with temperature = 2.35 X IO-^ mols,'degree The change in solubility of air with salt concentration is linear, within the limits of error, for these small concentrations, and since the error due to the lower temperature of the solution is likewise linear within the limit of residual error, it is a simple matter of division to obtain the result that for S a C l both errors are of constant percentage, and are respectively - 0 , 0 2 6 ~ ~and +o .oo4yOof the observed freezing-point depression. (IO). Errors of rinsing. Since the solutions of known concentration are brought into equilibrium contact with the ice by a series of rinses, the question arises as to whether this process is effective and accurate. Horvorka and Rodebush,' who also used the rinsing method, report that they rinsed three times, using each time only enough solution to fill their vessel about one quarter. Furthermore, this rinse solution was introduced from n fixed delivery tube a t about the center
I978
CAMPBELL ROBERTSON AND VICTOR IC. LA MER
of their vessel a t the top, and the rinse, as it ran through, would be almost certain to follow a few channels. The procedure in this work was to completely fill the freezing-point vessel with the solution of known concentration three times, each time allowing the circulators to run for several minutes. In this way the error due to adhering liquid films on the ice surface is eliminated. To test this point, several measurements of freezing-point depression were made using the third rinse as the equilibrium solution, and in every case the difference between this reading and that of the fourth final filling was less than 1.5 X IO-^ degree. ( I I). Error in pre-cooling. The error which would result, if the solution as run in was not exactly a t o°C., was obviated by the overnight standing in the ice-water thermostat. Furthermore, for some weeks of trials a thermoelement was kept in one of the ten bottles in the bath and the difference in temperature frequently noted. It was never more than 0.01 degree and averaged more nearly 0.004~C. By referring to paragraph (8) of this section it may be seen that this source of error is negligible. (12). Melting by stirring air. The stream of air may melt ice and thus change the concentration. About 0.j liter of air per minute was run through each vessel, for a total time previous to the final readings of about 3 5 minutes. To test the possible error from this source, a Dewar vessel containing water at o°C. was inserted into 'the thermostat, and a stream of the same air used for stirring was bubbled through it for two hours. Since this flask contained water only, if the air had been warmer than oo, the temperature of the contents should have reflected it. No change greater than 0.01degree, however, could be detected in the two hours. This would not indicate that the air, if a t any temperature above o°C., was not enough so as to melt a significant quantity of ice and appreciably change the concentration. (13). Condensation from stirring air. A slight amount of water will be condensed from the stirring air, since this is saturated a t o°C., and the solution, being a t a slightly lower temperature, has a lower vapor pressure. This effect, however, due to the small total amounts of air used, and the extremely small difference in vapor pressures a t oo, is of an order of magnitude beyond consideration. (14). Purity of the salts. The purification and testing of the salts is described on page 1968. This error is independent of concentration, and may be regarded as negligible, after the three recrystallizations carried out. (IS). Punty of the water. The table below gives a series of conductance measurements on the distilled water supply extending over several months, including vacation periods and periods of laboratory session when the distilled water was being constantly used.
A GRITICAL STUDY OF PRECISION CRYOSCOPY
1979
Conductivity Values-Chandler Laboratories Distilled Water Supply-z5"C. May 30, 1930 to June 6 , 1930
Dec. 28, 1929 t o March 24, 1930 I.02 I
(r.0. X ro6)
.65
1.44 1.21
1.30
1.14 1.08
1.04
0.80
1.16
Supply accidentally contaminated and very bad from March 24 to May 30.
0.91
0.76 0.75 1.12
I . 14
1.06 .88
I . 13 1.17
I .07
The mean value is about 1.10, the highest is 1.65 and the lowest is 0.75, Kendall,*' in reviewing the question of purity of water for conductance measurements, summarizes an extended body of data and arrives at the following conclusion. Water, which may be described as "good standard conductivity water," containing dissolved carbon dioxide in equilibrium concentration with that in the atmosphere, has a specific conductance of 0.80 X IO-^ reciprocal ohms a t o"C., practically all of which conductance is due to this carbon dioxide It is thus evident that the distilled water used in the Chandler Laboratories is very nearly the equivalent of standard conductivity water. The importance of the small additional contamination for this freezingpoint work may be further defined from the following data: a large sample of water wm drawn and its conductance measured. Then small measured amounts of a 0.03 M KCl solution were added, and the conductances again determined. After subtracting from each reading the original conductance of the water, the additional conductances imparted by the small concentrations of KC1 were plotted. The plot of specific conductance against moles per liter from 0.00004 to 0.0006 molar gave a straight line which extrapolated accurately to the origin, the slope being 6.8 mols per reciprocal ohm. From this relation, it is possible to estimate with moderate precision the effect of contamination, as measured by conductance, on the freezing-point depression, the relation holding if the assumption is made that the unknown contaminating substance does not have abnormal ionic mobilities, differing greatly from those of KCI. If we assume that of the conductance of the Chandler water, 0.80 x 1 0 4 reciprocal ohms, is due to carbon dioxide, then the additional conductance of, say, 0.30 X IO-^ should correspond to 2.1 X IO-^ mols of dissolved substance, and should affect the freezing-point depression by .756 X IO-5%. And it is, of course, only a difference in concentration between the liquid in the two freezing-point vessels which will give rise to an error in the results. Consequently the consistent use of the same water supply for making up solutions and for filling the reference flask largely cancels out the above small effects.
1980
CAMPBELL ROBERTSON AND VICTOR K . LA MER
(16). Errors in making up the solutions. It is not difficult to make up solutions with an error of less than o.oj%. Calibrated weights and flasks were used throughout. ( 1 7 ) . Possible solution of glass (Pyrex). The magnitude of this error was ascertained by measuring the increase of conductance of distilled water after standing in the bottles for several days. The mean of several trials showed an increase of concentration of o.oooo18 mol/l., estimated as KCl. To further diminish the errors due to contact of the solution with the bottles, a uniform technique of cleaning solution, steaming and standing in contact with distilled water for a t least z 4 hours, was employed for the cleaning of all glass with which the solutions came in contact. (18 . Thermal leakage. The thermal leakage modulus of the containing vessels was about 0.7 calorie per minute, per degree thermal head. This will introduce an error proportional to the thermal head. A calculation of its magnitude shows, however, that it is not large. Consider a solution having a freezing-point The solution is in the vessel for, say, 35 minutes. depression of o . o I O C . The total heat leak is 0.7 X 0.01 X 35 = . 2 4 5 calories. This will melt 0.003 grams of ice, which, in a total solution volume of 2 5 0 cc., means an error of only o.oorz%.
Errors due to Deciution from the Equilibrium State. (19). Shift in equilibrium. Error from this source may arise from a variety of causes. If the vessels contain insufficient ice, if stirring is not fully adequate, if the pressure on the contents of the vessels varies unequally, the equilibrium will be seriously disturbed. It has been pointed out in an earlier section that it is impossible physically to have an accurate solid-liquid equilibrium state in a vessel of finite size, due to the effect of hydrostatic pressure. Therefore, in each vessel, we have something more or less approaching an average temperature of that which tends to be attained in the different parts of the vessel. It is a mobile system and is sensitive to external influences. The only way by which the extent of a shift of equilibrium or its stability, may be judged, is by the constancy of the temperature readings attained. The constancy with this apparatus was very good. In one typical run the temperature remained constant to i 4 x I O - ~ O C . for 40 minutes. This is the most reliable criterion of equilibrium, as in the case of inter-comparisons of check runs, all the errors of the entire system enter, and it is difficult to ascribe causes with certainty.
The Design of Improved Freezing-Point Apparatus The design of apparatus brought out by AdamsYin 191j, and refined in various ways by subsequent workers, is limited to a precision of about I X IO-~OC. It would be of value for the study of solutions to be able to
A CRITICBL STUDY O F PRECISION CRYOSCOPY
1981
obtain measurements precise to I X I O - ~ O C . , and the subsequent discussion will attempt to demonstrate that this is not only 'pasible but would not entail great expense.
Shortcomings of the Present Apparatus. The original design of Adams as exemplified in Fig. I has persisted thus far, changed only in minor details. The crux of the problem is to obtain sufficient sensitivit,y of t,he thermoelement while retaining a high degree of thermal isolation of the system. High thermoelement sensitivity dictates two vessels very close together, to give a short thermoelement, while good thermal isolation requires that the vessels have much distance and insulation between them. I n the present design the necessary compromise results in a construction equally distributed between the conflicting demands. Thermoelement Sensitivity. It has been demonstrated in a previous section that,, given a certain allowable total thermal conduction of the thermoelement wire itself, the size of wires and number of junctions follow a definite optimum rule. The ultimate limiting factor is the allowable total heat conduction of the thermoelement wires. In the present design, where the thermoelement wires are bundled together and encased, the extraneous considerations which enter, such as the thermal conduction of the case, and the thermal conduction through the bundle of junctions to the junctions a t the center, make it impossible to calculate solely on the basis of conduction of the wire. The conduction of the thermoelement wires alone, even for a fairly large element, is very small. Therefore it would be possible, if the design of the two containing vessels were suitably changed, to use a thermoelement which would be very short and have perhaps two or three hundred junctions, without introducing a serious thermal leakage between the vessels. Thermal Isolation of the Equilibrium Mixtures. The ordinary glass Dewar vessel does not provide the best thermal insulation, because it is necessarily uninsulated a t the top. The connecting tubes and stirrers of a type necessary with Dewar vessels also contribute heat leakage paths of relatively large capacity.
A Xew Design. It is evident from a study of the Dewar vessel type of apparatus, that the most detrimental feature is the vessel itself, and in the following design a radical change from this type of construction is proposed. The new design is shown in center section in Fig. 7. I t embodies the following features: ( I ) The solutions are contained in two solid silver cylinders (A), of about 6 cm. inside diameter, 30 cm. long and about. 8 mm. wall thickness, having covers fastened with clamps. The cylinders are supported parallel to each other, about 1 2 mm. apart, by a framework of micarta, (B), which com-
1982
CAMPBELL ROBERTSON AND VICTOR IC. LA M E R
bines to a remarkable degree low thermal conductivity and high strength. T h e leakage modulus is made extremely small by constructing the supporting framework to make the heat path between the cylinders and from the outer environment very long. This is easily achieved by using a suitable construction of micarta tubing and rod. The outsides of the two cylinders are threaded with a pitch of about (2) I O per cm, (C), and the thermoelement consists of bare wires wrapped around the two cylinders, the threads serving to keep the junctions evenly spaced.
/
E
FIG.7 A Proposed New Design for Precision Freezing-Point Apparatus
Before wrapping on the thermoelement, the cylinders are given a thin coat of bakelite varnish, which is the only insulation necessary. The thermal insulation of the 8 mm. silver wall is considerably less than the insulation of a glass thermoelement case. (3) The assembly, consisting of the two cylinders, the thermoelement wrapped around them and the supporting micarta, is isolated from the outside environment by being placed entirely within a heavy glass cylinder, (D), with a removable ground-on cover. The large glass container is filled with rice flour, and evacuated to about 0.003 mm. This means of thermal insulation has been found to be far more effective than a vacuum alone. It has been used in research50 and tested by Professor S. L. Quimby of the Department of Physics of Columbia University, and found more efficient than any other combination of evacuated spaces.
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I983
(4) Access to the vessels is supplied by long thin platinum or bakelite tubes, (E), extending through the end of the outer glass container. The heat conduction here introduced may be made infinitesimal. ( j ) Stirring is effected by means of a rotating cage of platinum wire, working in pivots within the cylinders, and driven from outside the vessels by a thin bakelite rod, (G). ( 6 ) In operation, both silver cylinders would be packed with ice, the covers clamped on, and the assembly placed in the outer container. Water for the reference vessel, and solution for the other, would be introduced through the tubes. With the isolation here attainable, the thorough agitation of the cage stirrers, and perhaps a 250-junction thermoelement, it should be possible to obtain data precise to 1% at about j X IO-^ molal.
Summary and Conclusions The many sources of error inherent in the determination of freezing(I). point depressions in highly dilute aqueous solutions have been critically reviewed and numerically evaluated. The past and present means for minimizing these errors are discussed, and conclusions drawn as to their effectiveness. has been assem(2). A freezing-point apparatus precise to I X IO-T. bled, and the magnitudes of the specific errors of its operation considered in detail. ( 3 ) . The apparatus has been used to measure the freezing-point depressions of potassium cobalticyanide and potassium ferricyanide in the concentration range o.ooj M to 0.0003 hl. (4). The experimental data confirm the extension of the theory of Debye and Huckel as given by Gronwall, LaMer, and Sandved, as regards the form of the osmotic dev ation function in this region of concentration. ( 5 ) . As a result of this study, the design of a radically different form of freezing-point apparatus is submitted, which should overcome many of the difficulties and errors of the present type.
VII.
Bibliography
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LA MER
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