Capillary Radius and Surface Tensions Using Calculations Based on Tate's Law John D. Worley
St. Norbert College, De Pere, WI 54115 Surface chemistry is often relegated to the back of current textbooks of physical chemistry, which is both uufortunate and odd. This practice is unfortunate because material at the rear of a text is often thought to be less important to the well-trained chemist and student. If the material is passed over, the student may miss the chance to develop some of the basic concepts of surface chemistry. The practice is odd because surface chemistry is a basic part of industrial research and development. Research that uses the principles of surface chemistry is directed toward producing new and better adhesives, detergents, catalysts, foams, emulsions, and many other commodities found in our daily lives (1). The experiments found in physical chemistry laboratory manuals almost always use the duNuoy tensiometer to measure the surface tension of a liquid. However, many laboratories are not equipped with such a device or have only one. The experiment must then be carried out using some form of laboratory organization that may impede the full educational value. When the work is done one student at a time, too much time is required. Also teachers usually want students to do their own work. When the students work in pairs, one partner will often carry the brunt of the load. When the work is done as a class experiment, students do not get the hands-on experience desired. Our Experiment
This article describes an experiment that can be carried out without the use of such a tensiometer or any other particularly complex or expensive apparatus, except for a balance. This experiment can be used in a general chemistry lab or in a physical or biophysical chemistry lab. The results of the experiment will give the student an opportunity to work with the concept of surface tension and with the effect of surfactants on surface tension. Tate's Law The experimental method is based on a law first enunciated by W. Tate in 1864 (2).The law itself is straightforward and could be understood by a firsbyear student. It states that The mass of a drop formed at the tip of acapillary is directly proportional to the radius of the capillary tube.
However, the interpretation of the data is more challenging, and would be more appropriate for a physical chemistry lab. The part of the experiment that studies the surfadant is best applied in the general lab in a qualitative way. The quantitative application should probably be reserved for the physical chemistry laboratory Tate's law is the basis for the drop-weight method for determining surface tension.The proportionality constant, as shown below, contains the surface tension of the liquid being used. Tate's law may be used to determine the radius of an unknown capillary if the surface tenslon is known. Conversely, it may alsobe used to determine the surface tension if the radius of the dropping capillary is known. 678
Journal of Chemical Education
Objectives of the Experiment In this experiment, the radius of an unknown capillary is determined using the known surface tension of water. Then the surface tension of two organic liquids can be determined. Finally, the effect of a surfactant is determined. This effed ~-~~~~mav ~" be determined aualitativelv or auantitatively. If the determination is quantitative, the critical micelle concentration may also be determined. Apparatus The experiment requires an electronic balance that reads to the nearest 0.1 mg. It should have an opening for loading from the top. We use Sartorius and Ohaus balances for this experiment in our lab. This work was performed on a Sartorius 200-g balance. You will also need a Pasteur pipet, a few small beakers, four 6-in. test tubes if the critical micelle concentration is to be determined, and afew plastic weighingboats. Asmall rubber bulb for suctionine the liouid is also essential. We used a 9411. Pasteur pipei: The pipets should be inspected with a maenifvineelass before beeinnineso that the d ~ e t s used will Eot Laveany cracks or ZefectsL their tiPC Procedure The pipet is washed with soapy water and thoroughly rinsed with distilled water. Students will probably benefit from at least one practice run before taking their turn at the balance. Suction water into the pipet until it is about one-third filled. Some students may find the technique works better with more or less water. While holding the pipet tip to the bottom of the beaker with a gentle pressure, remove the bulb and place your finger over the opening to prevent the liquid from draining. Remove the pipet and wipe off the tip with a tissue. Allow drops to form slowly at the end of the pipet holding it in a vertical position. About one drop every 30 s seems to work well, though the time is not absolutely critical. However, the liquid should not run out so fast that the drop can not dangle from the end of the pipet. A plastic weighing boat is placed on the pan of the balance and tared. The student holds the pipet about l in. above the tray through the opening at the top of the balance. Two hands should be used to steady the pipet. The top of the balance case may be fully opened. If you rest your arms gently on the frame of the balance, it can give some helpful support. The first drop is released. A second student records the mass and tares the balance. Record the first numbers that appear on the readout, and tare immediately. The numbers will change over time because of evaporation, but as long as the liquid is relatively nonvolatile, there will be no problems with taring the balance. Between 10 and 20 drops are adequate to complete the experiment. The students then reverse roles. A different pipet may be used for each student.
Table 2. Correction Values rlV('"1
f
0.98171 0.96341 0.94512 0.92682 0.90853
0.02 0.04 0.06 0.08 0.10
f
0.89024 0.87195 0.85365 0.83536 0.83285
rlP) 0.12 0.14 0.16 0.18 0.1827'
rlP31
f
0.81707 0.77133 0.72560 0.72368 0.71856
0.20 0.25 0.30 0.3021" 0.3077'"
'Water " nButyl Alcohol Toluene where f is the wmection factor (4). The corrected radius for the capillary was 6.40 x 104 m. Measurements with a microcaliper showed the outside diameter to be 1.3 mm. An iron wire of l-mm diameter could be fit snugly inside the tip. The calculated radius appears reasonable. The average masses for n-butyl alcohol and toluene (17 drops each) were 0.0077 g and 0.0078 g, respectively. The surface tensions for each of these compounds were calculated using eq 2 to be 26 mN/m and 26 mN1m. The Handbook of Chemistry and Physics give the values a s 24.4 mN/m and 28.27 d m , respectively (5).The errors are in the 8% range. The alcohol appears to have been overcorrected, a n d t h e toluene appears to have been underwmected. If no corrections are applied, the errors are about 20% for each liquid. Results Using Surfactant The effect of a surfactant on surface tension can be determined visually or quantitatively by this method. The student can see that the size ofthe drop that detaches is much smaller when the surfactant is present. Table 3 gives the average drop masses for five solutions containing SDS a t the indicated molarities. Table 3. Effect of SDS on Drop Size and Surface Tension
Avg Mass
Std Dev
M x lo3
0.0130 0.0119 O.OH 0 0.0104 0.0121
0.0008 0.0006 0.0004 0.0004 0.0010
2.0 4.0 6.0 8.0 10.0
Surface Tension 31.7 29.0 26.8 25.4 29.5
The figure contains a graph of surface tension (calculated using eq 2) versus molarity. The graph shows a minimum a t 0.008 M. This is the point a t which SDS is present a t its critical micelle concentration (me). Micelle formation occurs only above the cmc. At concentrations above the cmc, the surface tension shows no further change or may show an increase due to the release of surfactant molecules from the surface to the interior for micelle formation. The
680
Journal of Chemical Education
S~rlacetension expressed in mN/m versus the concentrat on of somolanty
aim oodecyl su fate expressed in
student may wish to note that the cmc wuld be predicted from the average drop mass alone. One literature value for the cmc for SDS is 8.0 x 104 M (6). An additional capillary tube with a much larger diameter was examined. The averagedrop size using water was 0.0552 g. The radius as calculated using eq 1without correction was 1.19x lo3 m or 1.19 mm. The inside diameter as measured by a rule was 2.5 mm. The measured and calculated values appear reasonably close. Conclusions Tate's law can be used to determine the radius of a n unknown capillam with reasonable accuracy. Surface tensions can be determined within reasonable tolerances due to the simplicity of the method. The eKwt of a surfactant on surface tension can be determined or demonstrated visuallv. The critical micelle concentration c w also be determined. The apparatus required is simple and inexpensive, except for the balance. The experiment requires operator skilis for good results. Acknowledgment The author thanks the American Society for Engineering Education for four Summer Faculty Fellowships and the Bio/Molecular Engineering Group at the Naval Research lab for making them all interesting and for rekindling an interest in surface phenomena. Literature Cited 1.Oaipow, L. I. Surfma Chemistry; Reinhold;New Yo*, 1962. 2. Tate,T.Phil. Mog. 1884.27.176. 3. Harkins, W. D.; Bmm, F E. &Am. Chem. Soc. 1919,41,499. 4. A d s m n , A. W Physlml Chemisfry ofSurbzzs. 5th ed.;Wiley; New York,1990; pp 21-23.
6 . c ~ c ~ a m f b mofchemistry k MdPhpieJ; Mthed.; Weast,R.C..Ed.; CRCRess:Baea Raton, FL,1983-1984; pp F3M6. 6. Williams, R. J.;Phillips, J. N.;Mysels, K J. h m .Famdny Soe. lSS6.51.728.
