VOl. 5, No. 1
ANALYTICAL EDITION
70
of cells have been used, but the authors found the following type very advantageous: Satd. calomel-unknown electrode
I
Glass-buffer-satd. calomel known pH electrode
If E, is the potential recorded for the cell with the unknown and Eb is the potential with a buffer Of known substituted for the unknown, then AE = E,
Comparing it to a buffer of pH = 5.00, when Ea = 0.1483 and = 0.0502,
E,
- Eb = 0.0591 ApH at 25” C.
To illustrate, using fresh strawberry juice and comparing = 0.0292 and
it to a buffer whose pH is 3.00, when Ea E, = 0.0502, pH = 0.0502 - 0.0292 0.059
+
3,00 = 3,37
pH
=
0.0502 - 0.1483 0.059
+ 5.00
=
3.34
By this means measurements are rendered independent of the concentration of the buffer in the glass cell and also of the “asymmetry” potential of the electrode. LITERATUREI CITED (1) (2) (3) (4)
Elder, L. W., J.Am. Chem. Soc., 51, 3266 (1929). Elder and Wright, Proc. Nat. Acad. Sci., 14, 936 (1929). Stadie, J. Biol. Chem., 83,477 (1929). Stadie, O’Brien, and Lang, Ibid., 91, 243 (1931).
RECEIVED June 24, 1932.
A New Surface Tension Balance RICHARDJ, DEGRAY,Department of Chemistry, Lehigh University, Bethlehem, Pa,
T
HE measurement of surface tension is even older than the term ‘$xmface tension.” Thus Guthrie (6) and Tate (IO) mention that drop weights of liquids seem to bear some relation to the earlier observed phenomenon of capillary rise, but say nothing about the reasons therefor. Guthrie (6) postulates a picture of a film surrounding a bubble; this is advanced as a new idea, and possibly was the original thought on the subject. Since then, of course, work on surface tension by many methods and with many objects in view may be found. An excellent bibliography and comparison of methods then extant may be found in the paper by Dorsey ( 2 ) .
A H
OF TORSION BALANCE FIGURE1. DIAGRAM
Apparently detachment methods were discussed first in
1887 by Timberg ( l l ) ,who compared the results of the measurement of surface tension by capillary rise, dropweight, and pull-on-a-ring. The correlation was satisfactory to him, though he used the simplest of formulas to calculate the surface tension from the pull exerted:
rections to be applied to the measurements. Thus we find plates, rings, spheres, rectangles, cylinders, and even evaporating dishes being used, and the formulas becoming more and more involved (1, I d ) . Finally DuNouy (3) designed a surface tension balance similar in principle to Timberg’s ( I I ) , but by his wide application of this balance, particularly in the physiological field, he populariaed the method. His results were calculated by Timberg’s simple formula (Equation l), probably because of the intricacy of all correction formulas derived up to that time by the many investigators. I n this country detachment methods have been confined to the ring method, possibly because of the popularity of the DuNouy balance, but in Germany the bent-wire method has found favor. I n both methods the general scheme for the attainment of absolute results was to measure a liquid whose surface tension was known from some other, more accurate method, and adjust all other readings accordingly. The bent-wire method was raised to the eminence of an absolute method by the theoretical derivation of corrections by Lenard (9) and measurements by Dallwitz-Wegener and Zachmann (9) checking the theory against practice. A similar step in the development of the ring method did not come, however, until 1930, when accurate measurements of correction factors to be applied to this method by Harkins and Jordan (6) were combined with a derivation of the corrections for the curvature of the column of liquid raised by the ring by Freud and Freud (4). Since the theory yields the same factors as did comparison with values obtained by the capillary-rise or drop-weight methods, the ring method may now stand on its own values as an absolute method of surface tension, of inherent accuracy practically equal to that of other methods, and rapidity and convenience exceeding most others.
THEORY OF where y is the surface tension, W the force necessary to tear the ring from the surface, g the acceleration of gravity, and L the length of the wire forming the ring. No corrections were applied to this formula, which really assumes that the ring raises a cylinder of liquid, though discussions of the shape of liquids raised by solid bodies may be found as far back as that of Wilhelmy (13) in 1863. Further developments along this line are confined to variations in the shapes withdrawn from the liquid and cor-
THE
TORSION BALANCE
The work of Lenard et al. (9) was performed with a torsion balance. The work of Harkins and Jordan employed a modified chainomatic balance. Since the torsion balance had been so successful in the bent-wire method, it was hoped that it could be applied as well to the ring method, employing as many of Harkins and Jordan’s precautions as would be commensurate with practical applications, yet retaining the inherent advantages of great speed with high accuracy of this type of instrument. A direct-reading instru-
January 15, 1933
INDUSTRIAL AND ENGINEERING
ment was desired, if possible, so that the value of dynes per centimeter might be obtained without calculation. This means that compensating devices for correcting slight differences in the lengths of rings and in the change of acceleration of gravity must be devised. To do this, the basic theory of a torsion balance must be considered.
CHEMISTRY
71
If the balance is set properly, with zero load (W = 0) the balance will be in equilibrium when the scale reading is O(S = 0 ) . Hence
o+z
or
fS
= =
at+O+AT AW
(3)
X
=
W,sof = A
(5)
(4)
and for direct reading,
A necessary condition for direct reading is that A must equal f. By algebraic manipulation of Equations 2 and 5 other expressions may be obtained on which the method of setting the balance for direct reading is based. The derivations of these equations are simple and will not be given. The methods and final equations follow: To find T: with no load, adjust the zero; add weight W and obtain reading SI;move suspension hook in or out and obtain reading 82; remove W and adjust the aero. Replace W and obtain reading 8 3 . The value of T may be found from
FIGURE 2. BALANCE AS BUII,T Two springs carry the load on the balance. One spring is fastened to the beam and to the index finger, so that as this moves over the scale more and more force is exerted upward on the right-hand side of the beam, thus balancing the force of gravity acting on the load which is suspended from the hook H . The second spring is attached to the beam and to an auxiliary handle by means of which constant loads or tares may be compensated for by the exertion of a constant force to balance them. Any change in g or L (Equation 1) affects y geometrically, so the zero adjuster cannot compensate for such conditions. The simplest method of adjustment is variation of the length of the balance arm A, accomplished by threading the hook on the arm, thus permitting the load to be moved out or in as the readings are to be higher or lower, respectively. The balance as designed and built is shown in Figure 2. It now remains to devise some method whereby the hook may be moved to its proper position so as to yield direct readings. The difficulty is that movement of the hook changes the torque due to the weight of the hook itself and the zero adjuster must be varied accordingly. No simple ratios apply, therefore, and more complicated equations must be sought. A skeletonized balance arm is shown in Figure 3. A load W and the weight T of the hook and any unbalancing t of the balance arm itself may be considered as making up the total downward pull on the arm. Whether the balance arm is unbalanced, and which side is heavier is immaterial: if the load t is considered as being concentrated at distance a from the fulcrum, moving the hook does not change this load, Movement of the hook does change the length of arm A and varies the torques exerted by weight W and tare T . This downward torque is balanced by the two springs. The torque exerted by the zero adjusting spring may be called Z and that of the second spring X . When the balance was built, a spring of certain modnlus was combined with a scale of certain size. Whatever the values, the torque X exerted by the spring is translated into scale reading S by a factor f, or X = fS. Equating upward and downward torques,
T is a constant inherent in a given balance, and once determined may be used whenever desired. Knowing T, the balance may be set as follows: with no load, adjust the zero; add weight W and obtain reading 81; calculate Sz from (W
+ T)(1 - & / W )
(7) Move the hook until reading 8 2 is obtained, remove the weight W , and adjust the zero. The balance is now ready to read directly in milligrams. 8 2
=
APPLICATION TO SURFACE TENSION Harkins and Jordan (6) modify Equation 1 by including a factor F which depends on several variables. Thus Equation l becomes
* y = -
Wg F 2L
If the balance is t o be equipped with a scale reading in dynes per centimeter, Equation 8 should give the relation between X
k
W
FIGURE3. SKELETONIZED BALANCE ARM
such a scale and the milligram scale. F bears no simple relation to W , and must be applied separately; no scale could be made easily to include F. Using g = 981 and L = 4.000 cm., y = =
1 dyne/cm.
98lW/S 8.1555 mg.
(9)
Thus if the balance is set to read directly in milligrams, tho relation between the surface tension and milligram scale should be as shown above. In order to mark the scales accurately, a dividing engine was used whose head did not carry parts of the circle permitting such a ratio, so the factor actually employed was 1 dynejcm.
=
8.146 mg.
(10)
ANALYTICAL EDITION
72
The result of this variation and of many other errors will be discussed after the remaining calculations have been outlined. The remaining calculations are merely adapting t,he data of Harkins and Jordan (6) to the special conditions obtaining here. In their Table VIII, F is given as a function of R3/V and of R/r, where R is the (average) radius of the ring, r the (average) radius of the wire forming the ring, and V the volume of liquid raised by the ring. The length of the wire is taken as 4.000 cm. and the diameter of the wire as 0.0140 inch, or 0.01778 cm. Hence R = 4 / 2 ~= 0.63661 cm. r = 0.01778 cm. R/r = 35.805
In Harkins’ Table VI11 the values of F are given for values of R/r of 34 and 36: values of F corresponding to an R/r of 35.8 are obtained by interpolation. When the rupture of the film occurs, the weight of the liquid suspended from the ring and raised above the surface of the main body of liquid just exceeds the tension of the surfaces a t the ring. If this break occurs a t a scale reading of S dynes/cm. and the balance is set properly to read directly in milligrams, then by Equation 10 the weight of the liquid raised is W = 0.008146 Sgrams
D is the density of the liquid, the volume raised is
and if
V = 0.008146 S / D
CC.
Hence Ra/V = 31.672 D/S
(12)
To tabulate values of F for given values of S is impossible, since F depends on R3/V, and this in turn involves D, which is specific for each liquid and temperature. However, we could tabulate values of S / D against the corresponding values of F by calculating R3/V from Equation 12. This, with the value of R/r given in Equation 11, will permit the interpolation of the corresponding value of F from Table VI11 in the article by Harkins and Jordan (6). These values are given in Table I.
Vol. 5 , No. 1
S gives the true surface tension y as based on the empirical corrections of Harkins and Jordan (6),which in turn were justified by Freud and Freud (4)on purely theoretical grounds.
SOURCES OF ERROR Harkins and Jordan (6) claim an accuracy for the ring method, using their corrections, of 0.25 per cent. The accuracy of the interpolated values of the correction factors themselves is not claimed to be better than 0.4 per cent. Hence the error in the final value of y (Equation 13) will be the square root of the sum of this error in F, squared, plus the error in 8, squared. It remains to evaluate the error in S. 1. Error due to the wron relation between the two scales, as shown by Equations 9 anf 10: Equation 8 shows that S is directly proportional to g. Equation 10 really assumes a value of g of 982.2, so if we take 981 as the correct value, the error in g will produce an equal error in S , or
Error = (982.2 - 981.0)/(981)
=
0.12%
2. Error in R: R, or rather L, to which R is directly proportional, is guaranteed to kO.1 per cent. Any error in R changes R ~ TR3/V, , and L. Effect of error in R on R/r: A variation of 0.1 per cent in R produces an equal variation in Rlr. The effect of this change on the value of F interpolated from Harkins’ Table VI11 is at the maximum 0.0002, or 0.03 per cent. This is ne ligible, and in the following calculations a value of RTr of 35.8 will be used as before, and Table I remains applicable. Effect of error in R on R8/V and on L: No simple relation applies here, so let us assume actual figures and calculate the errors produced by variations in these figures. Assume water at 20’ C. gave a reading S = 74.6. The density is 0.9982. Using a value of R of 4.004 cm., we can calculatethe actual value of y which was measured: (Equation 10) (Harkins’ Table VIII) (Equation 8)
W = 8.146 X 74.6 = 607.7mg. V = 0.6077/0.9982= 06088 RS/V = 0.258S/0.6088= 0.426 F = 0.977 (0.977)= 72.74 Y = (0f’077)(981) 8.008
However, we would normally calculate the value of I:
y
from Table
FACTORS vs. S / D TABLEI. CORRECTION S / D O 100 90 80 70 60 50 40 30 20 10 0
1
2
3
4
5
1.020 1.021 I.. 023 1.025 1.005 1.006 1.008 1.009 0.987 0,9890.991 0,993 0.969 0.971 0.973 0.975 0.952 0.953 0.956 0.957 0.933 0.935 0.937 0.939 0.912 0.915 0.917 0.919 0.887 0.889 0.891 0.894 0.856 0.860 0.864 0,867 0.797 0.806 0.814 0.822 ,
..
..,
6
. ,. ... ... .. . . ,.
7
. ..
8
9
1.032 1.033 1.017 1.018 1.001 1.003 0.984 0.986 0.966 0.968 0.948 0.950 0.929 0.931 0.907 0.910 0.881 0.884 0.848 0.852
. ..
0.787
Since the dynes/cm. scale was formulated on the basis of Equation 8 without the F (which then is Equation 1) the scale reading S is
s = -wg 2L
and from Equation 8 y - ’ Zwg z
F
Hence y =
SF
(13)
Therefore, by dividing the reading S by density of liquid D , the value of F may be read in Table I. Then F times
3. Error in r: The size of the wire is specified as 0.0140 0.0001 inch. Assuming r to be 0.0141 inch, or 0.017907 om., we find R/r to be 35.551 instead of 35.8. From Harkins’ Table VI11 we find that this change a maximum variation in - nroduces . Fof 0.25 per cent. 4. Error in densit : For benzene at 20’ C. a change of 6 in the third place in t i e value of the density is required to vary the value of F 1 in the third place. Hence an error in density of 0.07 per cent affects y to 0.01 per cent. 5. Error in temperature: The effect of change in density with temperature is so slight that the final effect on y is negligible, as stated above. For water the value of y varies 0.14 per cent er degree Centigrade a t 20’ C. Without apparatus such as &scribed by Harkins and Jordan (6) the ring method is notoriously weak in the matter of temperature control. For work of ordinary accuracy, however, variations of less than 1’ C. are negli ible. The simple expedient of opening and closing windows%as sufficed to maintain the room in which this instrument is used to within 1’ C. with very little trouble. 6. Compensation of other errors: This balance was designed with adjustable features in order that it might be set t o give correct readings. Under the discussion of error 2b a sample set of data was used and y showed to be in error by 0.30 er cent, if the error in R be i nored. In setting the balance, {owever, the reading 74.6 woufd be considered unsatisfactory and the hook would be moved out until the break occurred a t 74.4 instead, for a t this value
INDUSTRIAL AND ENGINEERING CHEMISTRY
January 15, 1933
s
D
== 74.4
SI$
= 0.9982
;4g;
y =
SF = 72.76
That is, the balance now gives correct values for water and any other liquid in that range of surface tension. This was accomplished by lowering the readings 74.4/74.6, or 0.9973 times the old values: this invalidates Equation 10. Measuring a liquid of true surface tension of 29.00 dynes/cm. and density 0.879, if the balance were still set for direct reading in milligrams, Equation 10 applies and the reading at which the break occurred may be calculated to be 32.2. Had the balance not been properly set for water, the value of y found would be = 32.2/0.879 = 36.6 = 0.903 SF = 29.08
-
y =
Error
0.24%
However, in the adjustment the readings were changed by the factor 0.9973, so the break would now occur at a reading of
tension of water a t the depired temperature, then the hook may be moved out or in until the rupture occurs a t the reading desired, and the balance will be set for direct reading in dynes per centimeter for any liquid. For this adjustment of the hook, T should be found in dynes per centimeter from Equation 6 and then XZcalculated from Equation 7, using as W the desired reading in dynes per centimeter. Table I1 gives the apparent values of the surface tension of water at various temperatures. If the rupture occurs a t the value given in Table 11, the reading when corrected mill be the true surface tension of water (7). TABLE11. APPARENT VALUESFOR THE SURFACE TENSIONOF WATER c. Dynes/cm. c. D“ v n e n,/ r-m 0
74.6 74.5 74 . _.474.3
18 19
20 ~.
21
S = 32.2 X 0.9973 = 32.1
= 36.5 = 0.903 y = SF = 28.99 Error = 0.07%
‘I?
That is, setting the balance to read correctly for water corrects the readings in the neighborhood of the majority of organic liquids to less than 0.1 per cent error. By similar calculations it may be shown that moving the hook to eliminate the error due t o variation in r (which was 0.25 per cent) in the neighborhood of 70 dynes/cm. also corrects readings in the neighborhood of 20 dynes/cm. to within 0.01 per cent. 7. Miscellaneous errors: Harkins and Jordan (6) mention other sources of error such as tilting the ring out of the plane of the surface, lack of cleanliness, insufficient area of surface, etc. Such errors involve technic rather than theory and must be left to the intelligence of the operator. Klopsteg (8) points out that a considerable error may be introduced if the balance is not in its equilibrium position at the moment of rupture. He also discusses the importance of correcting for the weight of droplets of the liquid adhering t o the ring after the rupture. The more recent work of Harkins and Jordan (6), however, considers this correction to be erroneous. Concerning the area of surface, DuNouy (3) states that the minimum diameter of the watch glass used for a containing vessel is 5 cm. when a ring of 4-cm. circumference is used. The balance as designed accommodates a 6-cm. watch glass, or, if the door is open, an 8-cm. glass may be used. If the liquid tested wets glass, DuNouy states that 3.5 cm. diameter of containing vessel is sufficient. The use of a 5-cm. glass resting on an 8-cm. glass allows overflowing of the surface of the liquid and tends to insure cleanliness of surface.
To summarize the errors; by adjusting the hook the variations in R and r may be compensated for to 0.1 per cent. In the example used it will be noted that the error in the scale relation was included tacitly; this, too, is compensated for and included in the 0.1 per cent. The error due to density was 0.01 per cent, and that due to temperature 0.14 per cent. It is obvious that temperature cont8rol is the greatest need. Applying the Theory of Errors, the combined effect of these errors on the value of S may be calculated by the square root of the sum of the squares and found to be 0.2 per cent. Since y
=
XF
and the error in F was given as 0.4 per cent, the error in y may be calculated similarly to be less than 0.5 per cent.
SETTING THE BALANCE If the values of R, r, and g are correct, the balance may be set to read directly in milligrams, and the surface tension readings will also be found to be correct. If any of these factors is in error, the hook must be moved to compensate for that error. Water is the usual standard for surface tension work. If it is known what reading should be obtained so that, when properly corrected (by Table I), it equals the true surface
73
22 23 24 25
74.1 74.0 73.9 73.7
For work confined to a narrow range of values of surface tension, a simpler adjustment may be made. Thus, if the hook is moved until the rupture occurs a t a reading equal directly to the true surface tension, F is automatically included in the reading and the balance may be used to read directly with no correction whatsoever. Thus, if set for water a t 21’ C., the value of F included in the setting is 0.977, and the balance may be used from y equals 71 to 77 dynes/cm. with a maximum error of 0.5 per cent. A balance set to read directly on water, however, would be in error by 14.3 per cent when measuring the surface tension of benzene.
EXPERIMENTAL RESULTS I n order to check the equations derived, the balance was set to read directly in milligrams and then used to determine the surface tensions of water, benzene, glycerol, and carbon tetrachloride. The setting of the balance to read in milligrams showed Equations 6 and 7 to be correct. The results are given in Table 111. TABLE111. EXPERIMENTAL RESULTSOF BALANCE READINQS LIQUID Hz0 CsH6 CsHsOa CCln
ACTUAL READINQ 74.4 32.0 67.2 31.3
(Temperature, 20° C.) CORRECTBID DENSITY READINQ 1. C. T. 0.9982 72.77 7 2 . 7 5 =t0 . 0 5 0.8788 28.87 28.88 f 0 . 0 3 634 f 3.0 1.221 63.38 1.594 26.74 26.77 f 0 . 1
ACKNOWLEDGMENT Much credit must be given the Roller-Smith Company, Bethlehem, Pa., which so successfully translated the theoretical requirements into actual fabrication of the desired instrument. LITERATURE CITED (1) Cantor, M., Wied. Ann., 47, 399-423 (1892). (2) Dorsey, N. E., Bur. Standards, Sci. Paper 21, No. 540 (1926). (3) DuNouy, P. L., J. Gen. Physiol., 1, 521 (1919); “Surface Equilibria of Biological and Organic Colloids,” p. 19, Chemical Catalog, 1926. (4) Freud and Freud, J. Am. Chem. Soc., 52, 1772-82 (1930). (5) Guthrie, F., Proc. Roy. SOC.(London), A13, 444-57 (1864). (6) Harkins and Jordan, J. Am. Chem. Soc., 52, 1751-72 (1930). (7) International Critical Tables, Vol. IV, p. 447; Vol. 111, p. 25, McGraw-Hill, 1926. (8) Klopsteg, P. E., Science, 60, 319-20 (1924); 63, 599-600 (1926). (9) Lenard, Dallwitz-Wegener, and Zaohmann, Ann. Phusik, 74, 381-404 (1924). (IO) Tate, T., Phil. Mag.. (4) 27, 176-80 (1864). (11) Timberg, G., Wied. Ann., 30, 545-61 (1887). (12) Verschaffelt, J., Communications Phys. Lab. Univ.Leiden Suppl., No. 42e (1918). (13) Wilhelmy, L., Pogg. Ann., 119, 176-217 (1863). R B I C B I I VApril ~ D 5, 1932.
.
