Measurement of Vapor Pressures by Means of Matched Thermistors

304

A. P. BRADY, HARRIElTE HUFF, AND J. W. MCBAIN

MEASUREMENT OF VAPOR PRESSURES BY MEANS OF MATCHED THERMISTORS’ A. P . BRADY, HARRIETTE HUFF, AND J. W . McBAIN’ Stanford Research Institute, Stanford, California Received January I , 1060 INTRODUCTION

In the course of investigating the properties of aqueous solutions of colloidal electrolytes in this laboratory, it has become increasingly apparent that reasonably precise thermodynamic data on these solutions are of fundamental importance. At 0°C. the determination of the freezing-point lowering is a convenient and accurate method. Unfortunately, many compounds of interest both from a theoretical and from a practical standpoint are quite insoluble in water a t 0°C. Furthermore, the evidence for the rate of change of activity with temperature is conflicting, making uncertain the extrapolation of properties measured a t 0°C. to room and higher temperatures. For these reasons a determination of both the solvent and solute activities over a wide range of temperatures, desirable with any electrolyte solution, assumes special importance with colloidal electrolytes. The purpose of the present paper is to describe the theory and operation of an apparatus suitable for such determinations. Smith and Robinson (13) and O’Connor (10) have successfully measured vapor pressures of soap solutions at various temperatures by the isopiestic method. Although capable of great precision at higher concentrations, this method becomes extremely tedious at concentrations much below 0.1 m. Since it was desirable to make reasonably rapid measurements down to at least 0.01 m, it was decided to adapt the principle of the inherently less precise, but more sensitive, “thermoelectric osmometer” of Hill (6) and Baldes (1, 2). In this latter method one measures the steady-state temperature difference established when a drop of solution and one of solvent are exposed to solvent vapor and heat exchange is allowed only through the vapor phase. The principal modification in the apparatus described here lies in using “thermistors,” thermally sensitive resistors, as the temperature-sensing elements rather than the thermocouples and galvanometer used by other investigators ( 5 , 11, 12). The thermistors may be incorporated in an A.C. bridge, thus without loss of sensitivity eliminating the extremely troublesome effects of vibration, changes in galvanometer sensitivity, parasitic electromotive forces, etc., encountered in the D.C. method. Two further very real advantages of the A.C. method are (1) the ease and rapidity with which a reading can be made, thus enabling the operator quickly to detect and correct minor accidents such as a poor connection or the falling-off of a drop, and (2) the relative sturdiness of construction of the thermistors as compared to the thermocouples. The ther1 This work was carried out under a contract between the Office of Naval Research, United States Navy Department, and Stanford Research Institute. 2 Present address: National Chemical Laboratories, Poona, India.

MEASUREMENT OF VAPOR PRESSURES WITH TEERMI8TORS

305

mistora are completely glass enclosed; the coating has been found experimentally to be sufficiently thin over the sensitive elements to give adequate heat transfer. APPARATUS

A drawing of the apparatus approximately to scale and a schematic representation of the attendant electrical circuit are given in figure 1. The central glass tube can be lifted to bring the thermistors (Western Electric type V616, resistance about 5000 a, temperature coefficient about 3.7 per cent/%. at 30°C.) into the manipulation chamber for placing drops of solvent and solution on their ends. A metal cover prevents solution from dropping into the measuring chamber during this operation. The humidity in both the manipulation and the measuring chambers is maintained by wet blotting paper; the latter chamber in addition contains a layer of aluminum foil to serve as both a radiation shield and a tem-

!I AMPLIFIER

'ORs

L A 1 FOIL

OSCILLATOR

FIG.1. Vapor pressure apparatus

perature equalizer. The outer walls of the apparatus are of glass rather than metal in order to damp out fluctuations in bath temperature. The difference in resistance between the two elements was measured by connecting into a Wheatstone bridge as indicated in figure 1. A Dyke-Jones type bridge (Leeds & Northrup catalog No. 234117) was used in the present work because it was available, but much of this type of bridge is not utilized, since the variable resistance need have a total value of only about 20 ohms, and it was not convenient to use the Wagner ground. The parallel leads to the thermistor undoubtedly give an error in the absolute resistance measured, but the geometry is fixed, so the factor so introduced is incorporated into the calibration curve. The frequency used throughout was 500 c.p.s.,-sufficiently high for convenient amplification, but not high enough for body capacity to be troublesome. Detection of the balance point in some experiments was with a narrow band amplifier (built by Mr. C. G. McGee) and an oscilloscope;in others with a Hewlitt-Packard type 415A standing wave meter. The two detection methods had about the same sensitivity.

306

A. P. BRADY, HARRIETTE HUFF, AND J. W. MCBAIN CALIBRATION

Table 1 and figure 2 give the change in resistance a t 30°C.with water on one thermistor and various concentrations of potassium chloride or potassium sulfate on the other. The quantity AT is the difference in the bridge reading between

TABLE 1 Difference i n resistance, AT, at SOOC. between water on both thermistors and various concentrations of potassium chloride or potassium sulfate on one thermistor YOLALITY

I

OSMOTIC CONCEHTPATION

m

m

22,

I

I

11

I

1

- - ,I

&/v6m

&

ohms

I

I

m OR J p m . MOLES/LITER

FIQ.2. Calibration curves for the thermistors a t 30°C.

water on both thermistors and water on one and solution on the thermistor with the higher resistance. The “zeroing” with water on both thermistors is necessary every few readings until the thermistors have been thoroughly conditioned, but since the whole operation requires but about 10 min. it does not constitute a serious source of delay. From figure 2 it can be seen that in a plot

MEASUREMENT OF VAPOR PRESSURES WITH THERMISTORS

307

of AT us molality, m, the two salts give different, slightly curved lines. If the respective molalities are converted to so-called osmotic concentrations, vgm, where v is the number of ions and y is the osmotic coefficient, the points for the two salts fall on the same straight line. Each resistance giyen in table 1 and figure 2 is the average of several determinations with a standard deviation of the mean of about 0.5 per cent. The value of dAr/d(vgm) in figure 2 is 44.0 =t0.13. Calibrations at O", 30°, and 5OoC., as \vel1 as a large number of determinations on other systems (a), indicate that a single determination may be made with a standard deviation of about 1 per cent or about 0.1 millimole osmotic concentration (& 0.0000G mm. of mercury at 3OoC.), whichever is the greater. TEMPERATCRE DIFFERENCE BETWEEN THE DROPS

Over a temperature range of a few degrees the temperature dependence of the resistance of these oxides can be expressed (3) as:

In the present case the two thermistors were of sufficiently similar charact,eristics so that BI = B?;however, TI was somewhat different from r2. The quantity B is actually somewhat temperature-dependent; for these thermistors it is 3580 at 0°C. and 3400 at 40°C. It is readily shown that if the temperature of the thermistors is varied simultaneously (i.e., by changes in the bath temperature) d In (rl - r 2 ) - -B dT T* whereas if the temperature of thermistor No. 1 alone is varied because of a difference in drop composition

It therefore is evident that to minimize t,he effects of fluctuatioris in bath temperatwe T I - rz should be as small as possible, and t o achieve the greatest sensitivit,y the variable drop should be on the thermistor with the greater resistance. The thcrmist.ors used here were mat,ched to about 10 per cent. Hence the fluctuations in air temperature inside the measuring chamber must bc less than 10 per cent of the error in temperature difference to be tolerated. This occasioned little difficulty in the present set-up, since the bath variation of less than 0.001"C. was considerably attenuated in the measuring chamber, whereas errors of about 5 X 10-5 "C. arose from ot,her sources. The maximum temperature difference that can be set up between the water and drop of solution is one that will completely cancel the lowering of the vapor pressure of the solvent by the solute. I t is of interest to compare this idealized

308

A. P. BRADY, HARRIETTE HUFF, AND J. W. MCBAIN

temperature-concentration curve, in which heat transfer by conduction from the drop to the ambient atmosphere may be neglected, with the observed temperature-concentration curve. For any temperature and concentration change of the drop (at constant total pressure) : d In p

d In p

If conductive heat transfer may be neglected, d In p = 0, and the mole fractions involved are low enough to substitute 1/55.56 for the coefficient of dvgm, and TABLE 2 Comparison of maximum and observed rates of change of resistance with osmolic concentration

'C. 0

30 50

I

i

per Lcnl

175 58.2 29.0

90.1

51

44.0

I1 86

25.0

L / R P for the coefficient of d T (where L is the heat of vaporization of the water), giving

(&aar RT'

= 55.56L

(3)

Combining equations 2 and 3 gives the maximum rate of change of resistance with concentration which would presumably be realized in an evacuated chamber, =

T~RB L X 55.56

(4)

whereas the actual value of dR/dvgm is, of course, the slope of a calibration curve such as in figure 2 . Table 2 gives a comparison of the maximum and observed rates of change of resistance with concentration at three temperatures. INFLUENCE OF DROP SIZE

The radius of the drop in the present apparatus can be varied about twofold. Within these limits no effect of drop size upon the resistance could be detected. Similar independence of drop size in the thermocouple method has been shown by Roepke (10) and noted by Baldes (2). This is to be expected from the following considerations. A coefficient for mass transfer can be defined by

m

=

kA(H

- Ho)

(5)

MEASUREMENT OF VAPOR PRESSURES WITH THERMISTORS

309

where m represents the mass of water transferred from the drop per unit time, k the film coefficient for mass transfer, A the area of the liquid surface, H the absolute humidity corresponding to the saturation a t the temperature of the liquid surface, and H othe absolute humidity of the ambient air. Similarly, a film coefficient for heat transfer can be defined by Q = hAt (6) where Q denotes the heat transferred from the liquid surface per unit time, h is the film coefficient for heat transfer, and t is the difference between the temperature of the liquid surface and that of the ambient gas. If heat exchange is only through the vapor phase, to a sufficient approximation Q = - m L in the steady state. Denoting the maximum temperature difference attainable between the drops (as given by equation 3) by T , and noting that both t and r are very small compared to the absolute temperature T,combination of equations 5 and 6 with the gas laws gives t 1 (7) hMAP + -

5

(E)

(k)

where M A is the mean molecular weight of air and P is the air pressure. The individual film coefficients, h and k , vary in a complicated manner with drop size and shape, but it has been shown both theoretically and experimentally ( 7 , 13) that the quotient, h / k , is independent of these quantities and is equal to about 0.22 for water vapor in air. Hence the measured quantity, r , should be independent of the size and shape of the drops. Baldes (1) has derived an expression that can be put in the form

where k' denotes the thermal conductivity of the air and D the diffusion constant of water vapor in air. In the derivation of equation 7a the rather unreal physical picture of transfer of mass and heat between a spherical drop and the walls of the vessel by diffusion alone is used, neglecting convection (radiation transfer can be shown to be negligible). Actually, however, the same derivation would apply if Baldes' (r - T O ) is interpreted as the effective film thickness for mass and heat transfer. Assumption of the same effective film thickness for these two processes is nearly, but not quite, correct (13), so equations 7 and 7a lead to nearly the same numerical value of t / r . Equation 7, however, does not assume a spherical drop. At 30°C. equation 7 leads to a value for 100t/r of 76.9 per cent, and equation 7a of 79.1 per cent, as compared to the observed value of 77 per cent in the last column of table 2. The agreement between the observed and theoretical efficiencies indicates that heat exchange is indeed almost all through the vapor phase. Loss by conduction along the thermistors would not only lower the efficiency, but its relative importance would depend upon drop size, and hence it is highly undesirable.

3 10

A. P. RRADY, HhRRIETTE HUFF, .4ND J. W. MCBAIN INFLUENCE OF SURFACE FILMS

The principal immediate application of the apparatus is the study of surfaceactive agents. Solutions of these materials with their surface films might from a priori reasoning be expected to give false results by the present method because of the possibility of the film's affecting the mass and heat transfer in an unequal manner. It has been shown by Roepke (12), however, that films of oleic acid and proteins do not affect the results by the thermocouple method, and Fineman and McBain (5) have shown that addition of a small amount of soap (enough to give a surface pressure of 35 dynes/cm., but not enough to give appreciable additional lowering of the vapor pressure) to potassium chloride solutions does not affect the temperature difference between the drops. On the other hand, it was felt desirable to compare directly the results of the present method on a soap I

4

I

I

I

I

j

I

I

I

+

0.4,

FREEZING

0

O'C,

0

30.C.

THERMISTORS THERMISTORS

POINTS

1

I

02

0.0

i

I 01

I 0.2

I

I

03

04

I 05

I 06

I 07

I OB

I

fl FIG.3 . Osmotic Coefficient us. square root of molality for potassium laurate at 0" and 30°C.

by a thermodynamic method. T o this end the thermistors were calibrated a t 0 ° C . with potassium chloride and experiments run with potassium laurate, whose freezing-point depression has been well investigated (4,9). Figure 3 gives a graph of osmotic coefficient t i s . molality for potassium laurate a t 0°C. by both the thermistor and freezing-point methods, and at 30°C. by the thermistor methods. I t can be seen that the two methods agree quite well where they can be compared. The rather marked drop in critical concentration for micelles of this colloidal electrolyte in changing from 0°C. to 30°C. completely confirms conclusions derived from conductivity measurements (4a) and illustrates the danger of temperature extrapolation of the properties of these solutions. SUMMARY

The theory and operation of a modification of the Hills-Baldes vapor pressure apparatus are described. In the present apparatus, the thermocouples and gal-

VAPOR PRESSURES OF SOLUTIONS OF DETERGENTS

31 1

vanometer are replaced by two thermistors incorporated in an alternating current bridge. This leads to greater rapidity and ease of operation, with a sensitivity at least equivalent to the best published thermocouple results. Examples of runs on potassium chloride, potassium sulfate, and the colloidal electrolyte, potassium laurat'e, are given. REFERENCES (1) BALDES,E. J . : Biodynarnica No. 46 (1939). (2) UALDES, E. J.,AND JOHSSOS,A . F.: Biodynarnica No. 47 (1939). (3) DECKER,J. A,, GREEN,C. B., A N D PEARSON, G. L.: Elec. Eng. Trans. 66, 713 (1946). (4) BRADY, A. P . : Thesis, Stanford University, 1944. (4a) BRADY, A. P., AND HUFF,HARRIETPE: J . Colloid Sci. 3, 511 (1948). M. N., AND MCBAIN,J. W.: J . Phys. & Colloid Chem. 62,881 (1948). (5) FINEMAN, (6) HILL,A . V.: Proc. Roy. SOC.(London) Al27, 9 (1930). (7) HILPERT,R.: Forsch. Gebiete Ingenieurw. B. 2, Forschungsheft 366, 1-22 (1932). (8) HUFF, H . M., MCBAIN,J.W., AND BRADY,A. P.: J . Phys. & Colloid Chem. 66, 311 (1951). (9) MCBAIX,J . W., A N D J~OLDUAS, 0. E . A , : J. Phys. Chern. 47, 94 (1943). (10) O'CONNOR, J. J . : Thesis, Stanford University, 1948. (11) ROEPKE,R. R.: J. Phys. Chem. 46, 359 (1912). H. N L . ,: J. Biol. Chem. 133, 103 (1940). (12) ROEPKE,R . R., AND M ~ S O T. K . : Adsorplion and Eztraclion, p . 53 el seq. hlcGrnw-Hill Book Company (13) SHERWOOD, Inc., New York (1937). (14) SMITH,E. R . B., AND RoRrvsoH, R. A , : Trans. Faraday SOC.38, 70 (1942).

T H E VAPOR PRESSURES OF XQUEOUS SOLUTIOSS OF SOlIE DETERGESTSI HARRIETTE HUFF, J. W. hlcBAIS,* A N D A . P. BRADY Stanjord Research Institute, Stanjord, California Received January 23, 1950 ISTRODVCTION

Many interesting properties of solutions of soaps and detergents, such as their lowering of surface and interfacial tension and their solubilizing and emulsifying ability. are closely connecatcd with the thermodynamics of the solutions. With a few exceptions (4, 9, 10, 11) nll applicable thermodynamic information has come from measurements of the freezing-point depression. Too often, therefore, an attempt at correlation of thermodynamics with an observed property of a detergent is at best an intelligent guess, either because the substance is too insoluble at 0°C. for investigation, or because a long temperature extrapolation must be made. The prescnt work was undertaken to fill this gap partially, by This work was carried out under a contract between the OEce of Sav:rl Rrse:ircli, United States Navy Department, and Stanford Research Institute. * Present address: Nationnl Chemical Laboratories, Poon:i, Indin.