theoretical curve is in the direction expected. The polarographic wave would be more drawn out, and hence the halfwave potential more negative, than if the solution were buffered a t the initial
PH.
Table I. Variation of E l l 2 as Function of p H for 1.01 X 104M Total Ammonium Ion a t 25” C. (For solutions below pH 9, i d measured a t
The assumed value of ( E l l z ) ,must be reasonably close to the reversible value and any overvoltage in the more acid solutions must be small. This must be so because the reduction in strongly basic solution was reversible and the calculated curve, using the average experimental value of ( E I , ~ ) ~ , fits the data quite well in the p H range of 12 to 13.
-2.20 volts; above pH 9, a t -2.35 volts us. S.C.E.) -Eiiz,
PH 4.2 5.9 5.9 6.8 7.7 7.8 9.1 9.4 10.6
ACKNOWLEDGMENT
11.9 12.7 12.9 13.0
The authors are indebted to Research Corp. for a Frederick Gardner Cottrell grant-in-aid which made this work possible. LITERATURE CITED
(1) Bates, R. G., Pinching, G. D., J . Am. Chem. SOC.72, 1393 (1950). (2) Berg, D., Patte-son, A., Zbid., 75, 5731 (1953).
Volts us. S.C.E. 2.049 2.063 2.065 2,044 2.051 2.045 2.100 2.126 2.150 2.216 2.249 2.261 2.273
id,
pa.
3.32 3.09 3.11 3.18 3.19 3.38 3.23 2.95 3.05 2.84 2.57 3.14 2.88
(3) DeFord, D. D., Hume, D. N., Zbid., 73,5321 (1951). (4) Deyrup, A. J., Ibid., 56,2594 (1934). (5) Heyrovsk?, J., Phil. Mag. 45, 303 (1923).
(6) Johnson, R. J., Ubbelohde, A. R., J. Chem. SOC.1731 1951. (7) Kalousek, M., Collection Czechoslov. Chem. Communs. 13, 105 (1948). (8) Kolthoff, I. M., Laitinen, H. A., “pH and Electrotitrations,” p. 162 Riley, New York, 1944. (9) Kolthoff, I. M., Lingane, J. J., “Polarography,” p. 75, Interscience, New York, 1952. (10) lbid., p. 85. (11) Kolthoff, I. M., Miller, C. S., J . Am. Chem. SOC.63, 1013 (1941). (12) LeBlanc, M., Z. physik. Chem. 5, 467 (1890). (13) MacInnes, D. A., “Principles of Electrochemistry,” p. 342, Reinhold, New York, 1939. (14) Moore, T. S., Winmill, T. F., J . Chem. SOC.91, 1373 (1907); 101, 1635 (1912). (15) Pech, J., Collection Czechoslov. Chem. Communs. 6, 126 (1934). (16) Podrouzek, W., Rec. trav. chim. 44, 591 (1925). (17) Vlrek, A. A,, Collection Czechoslov. Chem. Communs. 20,413 (1955). RECEIVEDfor review July 3, 1958. Accepted November 18, 1959. Taken in part from Ph.D. thesis submitted by Peter F. Linde to the graduate faculty of the State College of Washington, Pullman, Wash., 1953.
Viscosity Measurement The Kinetic Energy Correction and a N e w Viscometer M.
R.
CANNON
Chemical Engineering Deparfmenf, The Pennsylvania State Universify, University Park, Pa.
R. E. M A N N I N G and J. D. BELL’ Cannon Instrument Co., State College, Pa. Results of extensive studies of the magnitude of the kinetic energy correction in spectral viscometers, designed to magnify this correction, and in the various types of capillary viscometers that are now in world-wide use, are presented. A simple correlation was developed which takes into account the fact that the kinetic energy coefficient, m, varies with Reynolds number. A new improved viscometer is described which has a negligible kinetic energy correction over a viscosity range of 0.4 to 20,000 cs. It is excellent for precision work in kinematic dilution and intrinsic viscosity.
P
papers (3, 6, 6 ) have described various t,ypes of viscometers that are now in extensive use throughout the world. This paper discusses REVIOUS
Present address, General Co., Schenectady, N. Y.
Electric
the meaning of the kinetic energy correction and describes a new viscometer. The viscosity equation that is applicable to capillary viscometers is usually written as follows:
The quantity B/t is called the kinetic energy correction. I n a well designed viscometer B/t is usually a small per cent of the Ct term. The origin of the above equation is obscure and will be derived. DERIVATION OF VISCOSITY EQUATION
If an energy balance is made about a capillary viscometer of any t,ype (Ubbelohde, Zeitfuchs, Cannon-Fenske, etc.) for unit mass of fluid the following equation is obtained:
XI
-
Xz = F
F,
+ Fo + F.
(2)
This equation says that the driving
fluid head (XI - X?), which is the vertical distance between the two menisci, is balanced by the total friction encountered. The total friction consists of several components. By far the greatest quantity of friction is developed in the capillary. In a properly designed viscometer the flow will always be viscous and this portion, F,, can be calculated by I’oiseuille’s law :
T h e quantity, F,, is friction due to turbulence caused by the contraction of the stream as it leaves the,bulb to enter the capillary. In modern texts this quantity has bccn correlated as a function of the kinetic energy that the flowing fluid has when inside the capillary. The quantity, F,, is friction caused by the turbulence due to the expansion of VOL. 32, NO 3, MARCH 1969
355
Table 1.
Reynolds Number Range and Magnitude of Maximum Kinetic Energy Correction (Viscosity range of 0.3 to 1.5 cs. for viscometers now in world-wide use)
Viscometer Type Cannon-Fenske routine Cannon-Fenske opaque Ubbelohde Ubbelohde Fits Simons Zeitfuchs Zeitfuchs cross arm SIL Atlantic Cannon-Manning semimicro for transparent liquids Cannon-Manning semimicro for opaque liquids Cannon-Ubbelo hde Cannon-Ubbelo hde dilution Cannon-Ubbelohde semimicro Cannon-Ubbelohde semimicro dilution B S U-tube B S miniature B S reverse flow
Size 25 25
C Constant,
Viscosity = 1.5 Cs.
100 100
370 440 53 410 330
2.9 3.1 0.06 2.8 2.8
Efflux time, sec. 750 750 500 150 500 500 500 500 500
Re. No. 5.6 5.6 12 380 15
K.E. corr., % ’ 0.007 0.003 0.02 0.18 0.02
...
Cs./Sec. 0.002 0.002 0.003 0.01 0.003 0.003 0.003 0.003 0.003
25
0.002
3.2
150
38
0.05
750
1.5
0.0004
25 25 25
0.002 0.002 0.002
3.2 56 56
150 150 150
38 140 140
0.05 0.83 0.83
750 750 750
1.5 5.6 5.6
0.0004 0.006 0.006
25
0.002
3.2
150
38
0.05
750
1.5
0.0004
25 A M1
0.002 0.003 0.001 0.003
3.2 120 35 58
150 100 300 100
38 420 35 330
0.05 4.0
750 500 1500 500
1.5 17 1.4 13
0.0004 0.03 0.0008 0.02
oc 1 1
... 1
...
1
the stream as it leaves the capillary. It also has been correlated as a function of the kinetic energy of the fluid inside the capillary. Bwause both F , and F , have been correlated as a function of kinetic energy, they can be added such that
F,
E, Cs.-Sec.2 56 23 86 61 88 93 1.8 83 85
Viscosity = 0.3 Cs. Efflux time, K.E. sec. Re. No. corr., % 150 140 0.83 150 140 0.34 100 300 2.8
Ua + F. = m9
€
1
0
z
2
1.9
IL
-
-
-
-
0
-
041-
-
W
a
VISCOMETER
a
$ a
xu
0,005 0.02 0.02
I I I I I I
- I -
N-ll (SUSPENDED LEVEL1
I VISCOMETER 8-2
U
Equations 3 and 4 can now be substitutfd into Equation 1, yielding Equation 9.
0.02
2.0 17 13
-
P
(4)
0.1
18
0 7 -
Y
5
100 100
...
-
c
w
i00
0 2
-
A
VISCOMETER 8-4 ALL CAPlLCARlES HAVE T R U M P E T SHAPED ENTRANCES AND E X I T S
W
$
0.1
Equation 5 can be solved for o : p , kinematic viscosity. to get Equation 6.
40
70
100
400
2 00 .REYNOLDS
Figure 1 .
700
1000
2000
NUMBER
Variation of kinetic energy coefficient, m, with Reynolds number Data from ( 2 )
Equation 6 is the viscosity equation and is usually written as follows: (7) KINETIC ENERGY CORRECTION
I n Equation 7 it is evident that
I n the past, it was assumed that C and B are constants. This assumption was correct in regard to C and incorrect in regard to B. Experimentally it is possible to prove whether or not B and C are constants. Instruments have been designed such that B/t is a negligible percentage (below 0.05%) of Ct. Hundreds of experiments indicate that C is a true constant for a capillary viscometer,
356 *
ANALYTICAL CHEMISTRY
B can be a constant only if m is a constant. From a theoretical standpoint m would be expected to increase with Reynolds number because all other forms of fluid friction do. A literature search revealed that few accurate data on m were in existence, because most observations n’ere made on viscometers which were designed for measuring viscosity and consequently B/t was always a very small percentage of Ct, which meant that variations in B and m could not be detected. Accordingly, special viscometers were designed to magnify the kinetic energy correction such that i t was 30% a t a Reynolds number of 500 and an efflux time of 500 seconds so that the correction was 150 seconds and, consequently, it could be measured accurately. At a
Reynolds number of 100, the correction was 1.5% of 1340 seconds, so that good accuracy in measurement was obtained over a wide range of Reynolds numbers. Figure 1 summarizes results on three viscometers that are typical of the results obtained on all viscometers tested. Note that m is not a constant but approaches zero as the Reynolds number approaches zero. The most important part of the curve is that between a Reynolds number of 80 and 500, because all liquid viscosity measurements in present-day viscometers that require a significant kinetic energy correction fall within this range. The analytical relationship for trumpet shaped capillary ends in this range is m = 0.037 (Re)o.&
(8)
The analytical relationship for square-
Table II.
A-
Figure 2. Improved Ubbelohde viscorneter available in sizes to cover range of 0.2 to 20,000 centistokes All dimensions in mm. A. Charge bulb E. Efflux bulb C. Meniscus bulb 1. Large tube 2. Upper capillary 3. Vent Ti. Timing mark Tn. Timing mark
cut capillary ends gives a higher value for m, but they are not recommended for use because they cause a higher kinetic energy correction and have no advantages. If the kinetic energy correction is below 374 the kinematic viscosity term in the Reynolds number can be replaced by Ct, without introducing a net error in viscosity above 0.09%. With the above simplification one obtains the following : m
=
0.037
m
(x) (4) VDp
-
0.074 zI')-(
V rDtCt
=
If kinematic viscosity and the viscometer constant, C, are expressed in units of centistokes, then E in units of centistoke-second2 of Equation 9 is
Calculation of Viscosity by Equation 1 1 in the Range Where Kinetic Energy Correction Is Significant
Viscosity Size of K.E. Error in True Efflux from Time, Equation 11, Viscosity, Correction, Viscosity, Viscometer Type0 See. cs. cs. 72 % 0.1 Ubbelohde 127.3 1,261 1.260 0.3 100.9 0.997 0.0 0.997 0.6 ~. . 0.0 0.951 0.7 96.4 0,951 0.1 0.9 88 0 0 867 0.866 0.1 1.3 0 768 78 4 0.769 0.0 0.673 0.673 69 0 1.9 0.0 0.4694 5.2 49.7 0.4694 74.7 1.259 0.4 0.1 Cannon-Ubbelohde 74 7 1 259 1 260 0.1 59 3 0 996 0.8 Size 100 0.997 0.1 No. .491 56 8 0 952 0.9 0.951 1.2 0.0 C = 0.01692 cs./sec. 0.866 51 9 0 866 0.3 1.7 E = 27.3 cs.-sec.2 46 3 0 4'71 0.768 0.3 2.4 40 9 0 675 0.673 0.0 6.7 29.6 0,4695 29 6 0 4695 0.4694 0.1 26.3 0,4054 26 3 0 4054 9.8 0.4051 12.4 0.0 2244 .55 00.3688 3688 0.3687 0.1 0.2 Cannon-Ubbelohde 158.1 1.261 1.260 0.0 0.3 Size 75 125.2 0.997 0.997 0.1 0.3 No. A40 119.4 0.950 0.951 0.1 108.8 0.865 0.5 C = 0.007987 cs./sec. 0.866 E = 46.3 cs.-sec.p 96.8 0.768 0.0 0.6 0,768 0.1 85 2 0.674 1 .o 0.673 0.1 2.7 60.3 0.4688 0.4694 0.0 4.1 52.8 0.4051 0.4051 0.2 5.3 48.7 0,3695 0.3687 a For these viscometers C was determined by viscosity standards in the range where kinetic energy corrections were negligible and E was determined by one viscosity liquid listed in Table 111, For the first and second instruments n-hexane a t 68" F. was used; for the third instrument n-hexane a t 100" F. was used for determination of E .
The advantage of Equation 11 over the commonly used Equation 7 is that E is a constant over the important Reynolds number range, whereas B is a variable. It is advisable to determine E experimentally rather than to calculate it from instrument dimensions, because its value is influenced b y the degree of tapering of the capillary ends. IMPORTANCE OF KINETIC ENERGY CORRECTION
The kinetic energy correction is important only in the viscosity range below 1.5 cs. and, as Table I shows, some instruments have a negligible correction for viscosities even lower than 0.3 cs. While most laboratories select instruments which require a n efflux time in excess of 100 seconds to reduce timing errors, there are cases where lower flow times are used and the kinetic energy correction is high. Table I1 illustrates the degree of precision obtainable by Equation 11 for such cases. The instruments listed in Table I are available in many different capillary bores to cover a liquid viscosity range of 0.3 to 30,000 cs. &lost of them are listed b y the American Society of Testing Materials (1) and the British Institute of Petroleum ( 7 ) . Table I11 lists viscosity data obtained in master viscometers for pure hydro-
Table 111. Viscosities of Some Pure Hydrocarbons a t Three Temperatures
Vixosity, Centistokes CannonTemD.. Cannon- Ubbelohde " F: ' Master semimicro Average n-Decane 6% 1 21302 1. -... 2 ~ n 0 1 2602 0.9963 0.9970 100 0.9976 0.8657 0.8655 122 0.8654 _I
Methylcyclohexane 122
0 9508 0 7676 0 6730
0 9518 0 i677 0 6735
0 9.513 0 7676 0.6732
68 100 122
n-Hexane 0 4692 0 4696 0 4038 0 4053 0 3685 0 3689
0 4694 0 4051 0 3687
68 100
carbons that can be used for determining E experimentally after C has been determined b y a viscosity standard with a viscosity high enough to neglect the E l f 2 term. Such viscosity standards are commercially available from the Cannon Instrument Co., State College, Pa., through a contractural agreement with the American Society for Testing Materials. N E W VISCOMETER
Figure 2 illustrates a recently patented (4) glass capillary viscometer of the Ubbelohde type with improvements. The dimensions are such that the kinetic energy correction is VOL. 32, NO. 3, MARCH 1960
357
negligible for viscosities as low as 0.4 cs. For this instrument,. over a range of 0.4 t o 20,000 cs., Equation 11 reduces to Equation 12. Y
= q / p = Ct
(12)
A charge of only 1.0 ml. is required. It can be used as a dilution viscometer if desired and it is available in 12 different sizes covering a viscosity range of 0.2 to 20,000 cs. This type of viscometer has the same liquid driving head a t all temperatures and consequently the viscometer factor (or C constant in Equation 12) is the same a t all temperatures.
It is used by simply placing the instrument in the bath, pouring in a sample of approximately 1 ml. (the precision of viscosity measurement is independent of the amount charged), waiting 5 or 6 minutes for bath temperature to be attained, and then measuring the efflux time in seconds for the efflux bulb to discharge through the capillary. Viscosity in centistokes is then obtained by multiplying the efflux time in seconds by
the viscometer constant which is the C factor of Equation 12. If absolute viscosity is desired, i t is obtained by multiplying centistokes by density in grams per cubic centimeter, thus obtaining centipoises. More detailed instructions applicable to all capillary-type viscometers are available (1, 7 ) . NOMENCLATURE
B = variable in viscosity equation, stokesecond C = viscometer constant, stokes per second D = capillary diameter, cm. E = kinetic energy factor, stoke-second2 Fy = friction in capillary P, = friction a t capillary entrance F. = friction at capillary exit 9 = acceleration due to gravity, em. per second2 L = capillary length, cm. m = kinetic energy correction coefficient r = capillary radius, cm. t = efflux time, seconds U = velocity of fluid in capillary, cm. per second V = efflux volume, cm. XI - X2 = distance between menisci in viscometer
=
tl
absolute viscosity, poises kinematic viscosity, stokes
’
y = - =
P
=
p
density, grams per cc. LITERATURE CITED
(1) Am. SOC. for Testing
Materials, Philadelphia, Pa., “ASTM Shndards on Petroleum Products and Lubricants.” D 445-53T. (2) Bell, J. D., thesis, “Variationof Kinetic Energy Correction Coefficient with Reynolds Number Capillary Viscometers,” in partial fulfillment for the degree of master of science, Department of Chemical Engineering, Pennsylvania State University, January 1947. (3) Cannon. M. R.. IND.ENG. CHEM.. ANAL.E;, 16, 708’(1944).(4) Cannon, M. R., U. S. Patent 2,805,570 (Sept. 10, 1957). (5) Cannon, M. R., Fenske, M. R., IND. ENQ.CHEM..ANAL.ED. 10.’ 297 (1938). . . ( 6 ) Ibid., 13, 299 (1941). (7) Institute of Petroleum, London, W. 1, England, “Standard Methods for Test; mg Petroleum and Its Products, \ - I
IP71/58.
RECEIVEDfor review August 14, 1959. Accepted November 23, 1959.
Polarographic Determination of Styrene Monomer in Polyester Resins WILLIAM M. AYRES and GERALD C. WHITNACK Chemistry Division, Research Department, U. S. Naval Ordnance Tesf Sfafion, China fake, Calif.
b A quick, precise, polarographic method for the determination of styrene monomer in polyester resins is presented. A solution of the resin in ethyl alcohol is polarographed in a tetrabutylammonium chloride supporting electrolyte (7570 ethyl alcohol). Dissolved oxygen need not b e removed prior to recording the polarogram. Data indicate that the analysis has a precision of about 5 parts per thousand, and the presence of phthalate and fumarate or maleate esters does not interfere.
B
polyester resins are used as inhibitors for solid propellant grains, a rapid and precise method of analysis was needed for styrene monomer in such mixtures. Current methods require either a vacuum distillation prior to analysis or a n accurate knowledge of all interfering constituents present, so that corrections may be applied (1, 2, 5 ) . Although the spectrophotometric method of Hirt (2) appears rapid and precise, the presence of phthalate and maleate or fumarate esters requires a two-component analysis for styrene ECAUSE
358
ANALYTICAL CHEMISTRY
and phthalate and then a correction for the absorption of the fumarate or maleate component of the polyester. The reduction of styrene a t the dropping mercury electrode has been reported by Laitinen and Wawzonek (4) to occur a t a half-wave potential of -2.35 volts referred to the saturated calomel electrode (S.C.E.). This potential is considerably more negative than the reduction potentials of phthalates (6) and maleates or fumarates (S), and should enable the measurement of the diffusion current of styrene in the presence of these esters. APPARATUS A N D MATERIALS
A Sargent Model XXI Recording Polarograph was used throughout these studies to obtain and record the polarographic data. I n the initial phases a Cary Model XI recording spectrophotometer was used for the spectrophotometric data. A semimicro vacuum distillation apparatus was used to purify the commercial styrene and to check the styrene content of the polyester resins. The polyester resins used were: Selectron 5119 (Pittsburgh Plate Glass Co.. Pittsburgh, Pa.), Vibrin 121
(Naugatuck Chemical Co., Naugatuck, Conn.), and Marco 28C and 28V (Celanese Corp. of America, New York, N. Y.). Solvents were: 95% ethyl alcohol (LAC Chemicals, Culver City, Calif.), absolute methanol (Mallinckrodt, analytical reagent), and chloroform, B and A Reagent (Allied Chemical and Dye Corp., New York, N. Y.). T h e supporting electrolyte used in the polarographic analysis was tetrabutylammonium chloride. A special polarographic grade of this material was purchased from Southwestern Analytical Chemicals, Austin, Tex. Five milliliters of a n aqueous 0.1M tetrabutylammonium chloride solution was added to 15 ml. of 95% ethyl rtlcohol containing the styrene. Thus, the styrene was polarographed in a solution that was approximately 75% ethyl alcohol. Redistilled mercury (c.P. grade) was used as the anode in all beakers for polarographic analysis. A capillary with a drop time of about 4 seconds per drop a t -2.50 volts (vs. mercury pool) in the supporting electrolyte was used in all work. The total pressure applied to the dropping mercury was 91.5 cm. On open circuit and a t zero applied potential the values for m and 1 were 5.37 mg. and 8.25 seconds per drop, respectively.
