913
V O L U M E 25, NO. 6, J U N E 1 9 5 3 The method is rapid and simple and affords stoichiometric results without necessitating the use of elaborate and costly equipment. It should prove of particular value to the forensic chemist concerned with the microchemical identification of dangerous drugs because it enables him to titrate directly those complex compounds which he is so often called upon to prepare and characterize. ACKNOWLEDGMENT
The authors are indebted to L. I. Pugsley for many constructive criticisms and helpful suggestions. LITERATURE CITED
.luerbach, &I. E., Drug Standards, 19,127 (1951). Beckett, A. H., Camp, R. M., and Martin, H. W,, J . Pharm. and Pharmacol., 4, 399 (1952).
Blumrich, K. G., and Bandel, G., Angew. Chem., 54,374 (1941). Conant, J. B., and Hall, N. F., J . Am. Chem. SOC.,49, 3047 (1927).
Conant, J. B., and Werner, T. H., Ibid., 52, 4436 (1930). Duquhois, P., Anal. Chim. Acta, 1, 50 (1947). Duquhois, P., J . pharm. chim., 26, 353 (1937). Ekeblad, P., J . Pharm. and Pharmaeol., 4 , 636 (1952). Frita, J. S., ANAL.CHEX,22, 578, 1028 (1950). Hall, N. F., and Werner, T. H., J . Am. Chem. Soc., 50, 2367 f 1928).
Harria, L. J., J . Bid. Chem., 84, 296 (1929); Biochem. J., 29, 2820 (1935).
Haslam, J., and Hearn, P. F., Analyst, 69, 141 (1944). Henry, Th. A,, “The Plant Alkaloids,” 3rd ed., pp. 511-23, London, J. & A. Churchill, Ltd., 1939.
(14) (15) (16) (17) (18)
Herd, R. L., J . Am. Pharm. Assoc., Sci. Ed., 31, 9 (1942). Higuchi, T., and Concha, J., Ibid., 40, 173 (1951). Higuchi, T., and Concha, J., Science, 113, 210 (1951). Kahane, E., Bull. SOC. chim. France, 18,92 (1951). Kleckner, L. J., and Osol, A., J . Am. Pharm. Assoc., Sci. Ed., 41,
573 (1952). (19) Markunas, P. C., and Riddick, J. A., AXAL. CHEX..23, 337 (1951); 24, 312 (1952). (20) Nadeau, G. F., and Branchen, L. E., J . Am. Chem. SOC.,57, 1383 (1935). (21) Palit, S.R., ANAL.CHEM.,18, 246 (1946). (22) Pifer, C. W., and Wollish, E. G., Ibid., 24, 300 (1952). (23) Pifer, C. W.,and Wollish, E. G., Ihid., p. 519. (24) Pifer, C. W., and Wollish, E. G., J . Am. Pharm. ASSGC., Sci. Ed., 40. 609 11952). (25) Pifer; C. W., Wollish, E. G., and Schmall, AI., A s ~ L .CHEM., 25, 310 (1953). (26) Schuyten, AI. C., Acad. roy Belg. classe sci., 32, 866 (1896) (27) Seaman, W., and Allen. E., ANAL.CREM.,23, 592 (1951). (28) Seaman, W., Hugonet, J. J., and Leibmann, W., Ibid., 21, 411 (1949). (29) Souchay, P., Bull. soc. chim. France, (5), 7,797,809,835 11940). (30) Toennies, G., and Callan, T. P., J . Bid. Chem., 125, 259 (1933). (31) Tomicek, O., Collection Czechoslov. Chem. Commzms., 13, 116 (1948). (32) Wagner, C. D., Brown, R. H., and Peters, E. D., J . Am. Chem. Soc., 69, 2609 (1947). (33) Wilson, H. K,,J . SOC.Chem. I n d . (London), 67, 237 (194% (34) Wittmann, G., Angew. Chem., A60, 330 (1948). RECEIVED for review October 30, 1962. Accepted March 13, 1963 Presented before the Division of Analytical Chemistry at the 122nd IIeeting of the AMERICANCEEMICAL SOCIETY, Atlantic City, K,J.
Refractometric Analysis of Flowing Solutions HARRY SVENSSON Laboratories of LKB-Produkter Fabrihaktiebolag; Stockholm, Sweden This work was initiated in the course of the design of two new types of recording refractometers in order to elucidate clearly the special problems inherent in the refractometry of flowing solutions. The conditions under which errors in the refractivity-volume curve are significant are given quantitatively where possible, and some information regarding the suitability of different cell constructions and different methods of recording is gained. Moreover, the refractive properties of two specific types of cells are analyzed and compared with each other. The analysis includes sensitivity, optical and volumetric resolving powers, light-transmitting capacity, and range of linear response. A cell described long ago by Hallwachs is found to have, under specified constructional and operational conditions, a linear range corresponding to the refractivity increment of 30% sucrose with an error less than 10-5. The results should interest not only chemists using refractometry as an analytical tool, but also instrument manufacturers and scientists who wish to build their own instruments.
for flowing solutions can serve as a means for keeping the concentration at a desired, constant value ( 1 1 ) . FACTORS AFFECTING USEFULNESS OF RECORDIhG REFRACTOMETERS
Optical Resolving Power. This concept may be defined a3 the reciprocal of the least difference in refractivity that can be detected v,-ith certainty by the instrument:
.
R, = l / A n
(1)
The requirements to be met by the optical resolving power can be found by considering the refractivities of the components (in distillation) or the specific refractive increments of the components (in chromatography and other cases where a solvent is transporting the components), and by deciding how small an admixture of one component in another or in the solvent is to be detected. If in a chromatographic fractionation the least specific refractive increment is 0.001 (for a 1% concentration increase) and if that component is to be detected down to a coilcentration of O.Ol%, then a resolving poiyer of lo5will be neceamry. If the optical reqolving power is written as follon.4: ds
R
ECORDING refractometers have recently found application in chromatographic analysis and a number of instruments for
that purpose hare been dexribed (3,4,7 , 9 , H , 14-16). That is, however, not the only field where such instruments can be used. Fractional distillation is another example where a recording refractometer can serve as an aid in the subdivision of the effluent into fractions. Moreover, in the production of chemicals on a large firale by continuously operating methods, a refractometer
where X is the numerical value of the directly recorded qualitywhether this is an angular deflection, a linear di~plncement,an electric current, a number of interference fringei, etc.-then it appears as the ratio between the Tensitivity :
S = dX/dn
(3)
ANALYTICAL CHEMISTRY
914 and the uncertainty, AX, of the instrument. .4t a constant resolving power, the individual values of the sensitivity and the uncertainty are of little importance. I t can be agreed, however, that an arrangement characterized by high values of sensitivity and uncertainty is more convenient in use and requires less trained personnel than one giving low values to both variables. Direct reading by the unaided eye on a large paper graph must be considered as more convenient than reading in a microscope on a micro film. The uncertainty of the reading in an instrument is composed of two parts, the optically conditioned uncertainty and the uncertainty due to secondary influences. The nature of the secondary uncertainty depends upon the basic principles of the instrument. If it is equipped with an electronic system and with an electromagnetic pen recorder, we have to do with secondary uncertainties in the instability of the electronic y t e m , in the influence of wetching and of weather conditions on the coordinate scales of the paper, and in the thickness of the line drawn by the pen. In a photographically recording instrument, the secondary uncertaiiity is defined by the resolving poFer of the plate and by the reading accuracy of the microscope. In general, the optical resolving power can be expressed mathriiiaticall> in terms of the wave length of the light, the cell thickness, etc., but such expressions are valid only under the conclition that the secondary uncertainties are smaller than the optirnlly conditioned uncertainty. Othernise the experirnentally ttetermined optical resolving power must necessarily remain lower than the theoretical one. This may be illustrated by the follor\ ing example. In photographic recording of interference ftinges, it has been found possible to localize a fringe to within 1 of the distance between neighboring fringes, which thus repre-ents the optically conditioned uncertainty. The secondary uncertainty is in this case defined by the reproducibility of the leading microscope, which can be said to be 3 microns. The theoretical resolving power can thus be reached only a t a sensitivi t s great enough to bring every two neighboring fringes farther apart than 150 microns -4lthough mathematical expressions for optical resolving powers in general do not contain the sensitivity as a factor, a high sensitivity is nevertheless important. -4s a matter of fact, the euperimentally determined resolving power rises with the sensitivity until the point is reached where the optically conditioned uncertainty is as great as the secondary uncertainty. Volumetric Resolving Power. I t is ne11 known from the theory of optical instr6ments in general that an increase in size means a possible increase in resolving poiver until other than optical influences (vibration, temperature fluctuation) or trivial optical influences (inhomogeneity of optical glasses) set a limit. When flowing solutions are concerned, this way of increasing the optical resolving power would involve measurements on greater homogenized volume fractions of the flowing solution. However, this method will necessarily smooth out the refractivity-volume curve to be recorded. I n other words, the increased optical resolving power is bought a t the cost of decreased volumetric resolving power. The latter quantity may now be strictly defined as the number of independent measurements that can be performed on 1 ml. of the flowing solution, or as the reciprocal of the cell volume:
Rv
=
l/4V
(4)
Here V is the running volume coordinate and A V the cell volume. Maximum Correctly Recorded Slope of Refractivity. Before this topic is treated, the different mutual relationships that are possible between the cell and the pipeline must be considered. If the cross section of the cell perpendicular to the direction of flow is not greater than that of the pipeline, mixing in the cell need not be considered, since the mixing which exists is only the unavoidable mixing due to the drag effect a t the walls and occurs everywhere in the pipeline. In such a case the cell can be de-
signed so as to give a reading which is representative for the average refractivity within the cell volume, Every refractivity curve of low curvature will then be correctly recorded, but no upper limit for its slope can be demonstrated. If the cross section of the cell is greater than that of the pipeline, the mixing in the cell must be taken into account, This mixing is then either uncontrolled and incomplete, in which c&se no quantitative treatment of the effect of the volumetric resolving power on the recorded curve can be carried out, or it is controlled and complete. The latter case can be treated mathematically, and it is then necessary to distinguish between two methods of recording: the discontinuous method and the continuous method. If the instrument is operating discontinuously, it takes a ieading on a completely homogenized volume fraction AV (which is identical with the cell volume), then this volume is replaced, without mixing, by an equal volume of fresh flowing solution. The new volume fraction is also completely homogenized, a neiv reading is taken, and so on. If the instrument is operating continuously, complete mixing within the cell is ascertained a t every moment while the liquid is constantly flowing. In this technique, the initial cell contents are removed only to 63 % ( 1 - l / e ) after passage of one cell volume, to 86% ( 1 - l / e * ) after passage of two volumeq, to 95% ( 1 - l / e 3 ) after passage of three volumes, etc. This circumstance gives rise to an error in the recorded curve which is already a function of the first derivative of the refractivity, n’ ( V ) . In order to get an idea of the order of magnitude of this error, we have to consider the effect of the replacement of an infinitesimally small volume element dV by an equal volume of fresh flowing solution of different refractivity. If a t a certain moment the refractivity within the cell is n ( V ) An, where the first term represents the correct and desired value, which would prevail in the middle of the cell if it had the same cross section as the pipeline, and the second term represents the error to be derived, then the new refractivity after exchange of the volume dVis:
+
n(V
+ dV) + An f d(An) = (AV
- dV)n( V ) + An
+dV n(V
+ AV/2)
AV
(5)
Here n(V+dV) is the new correct and desired value, and d(An) is the differential of the error to be derived. Equation 5 is the differential equation of the error, and by developing n( V dV) and n ( V + A V / 2 ) into their first approximations, n( V ) n’( V ) dV and n( V ) n’(V)AV/2,respectively, it can be reduced to:
+ +
+
d(An)
+ r1VAn dV + 31 n’(V) d V = 0
(6)
This equation can be solved by standard methods, and the general solution is:
where (&)o is the error when V = Vo. The integral above can be developed into a power series of AV or can be evaluated graphically, but it is sufficient for our purpose to calculate the error for a constant n’(V) after the passage of an appreciable volume V - Vo. Equation 7 then reduces to:
Consequently, in continuous recording under constant stirring within the cell, there appears an error in the recorded curve which is equal to the product of the slope of the curve and the cell
V O L U M E 25, NO. 6, J U N E 1 9 5 3 volume. This error is, however, without significance as long as it is smaller than the reciprocal of the optical resolving power. Hence the following expression is gained for the maximum correctly recorded first derivative of the refractivity:
V n’(V) = R R O
If an instrument has R , = 106 and Rv = 10, therefore, a refractivity curve with a greater slope than 10-4 refractivity units per milliliter is in error. I n regions of low curvature, this erior can be corrected by the use of Equation 8. Maximum Correctly Recorded Curvature of Refractivity Curve. This concept is meaningless in the continuous method of recording under constant homogenization of the cell contents. since an error is present already at the curvature 0. The eriors to be derived here, therefole, peitain only t o the discontinuous method explained above and t o the case 4 here the cross section of the cell is not greatei than that of the pipe line. The iefractivity given by the instrument is then exactly expreazed by the integral :
V
+ AV/2
where n( V ) is again the correct and deqired value of the refractivity in the middle of the cell. If the integral function of n ( V ) is denoted by Ar(\ V),the error can be written in the form:
If now the two S function. are developed into pon-ers of AT7 with neglect of higher powers than 3, we get:
Again, if this error is smaller than the reciprocal of the optical refiolving pori-er, it plays no role whatsoever, but the smoothing-out effect of the cell volume on the recorded curve begins to become noticeable a t this value of the error. Consequently, the maximum curvature of the refractivity curve that is coriectly reproduced by the instrument is given by the equation:
The great inferioritj- of the continuous method with constant homogenization of the cell contents in coniparison with t’he two methods just treated is quite apparent. An automatically operating discontinuous method of recording is difficult to arrange technically. The most preferable arrangement, therefore, seems to be a continuously operating one with a cell of cross section not greater than that of the pipeline, and with no honiogenization within the cell. I n order to secure the necessary optioal resolving power and sensitivity of the cell, it is then in general necessary either to pass the light in the direction of flow in a rectilinear cell, or to pass it perpendicular to the direction of flon- in a cell with a serpentine channel. Maximum Rate of Liquid Flow. -1 recording refractometer does not function properly if the rate of liquid flow through it is increased too much. The liquid must acquire a definite temperature before it enters the cell, and t,he device for thermostating and the accuracy required here (which of course depends on the optical resolving power) will therefore have an influence on the permissible rate of flow.. I n a discontinuously recording instrument, the rate of flow is, in addition, limited by the maximum number of readings that can be taken per time unit, this limitation being the more serious the smaller the cell is. The maximum rate of f l o is ~ of course a function of the tem-
915 perature difference between the cell and the flowing liquid where it enters the apparatus. It should be measured experimentally by pressing a liquid of constant refractivity and temperature, different from that in the cell, through the instrument. Starting with a very low rate of flow, the instrument will give the correcmt refractivity value. On progressively increasing the rate, the critical value at which the instrument begins to show a noticeahle error can be observed. Maximum Recordable Slope of the Refractivity Curve. 111 most continuously operating methods, there is a critical value of the refractive index derivative beyond which the recording fails altogether. The reason for this, and the value of n’(V)a t which it occurs, depends on the design of the instrument, and it is not possible t o give a mathematical expression for it. For proper functioning many refractometers require that the refractive index n-ithin the cell be constant within the optical resolving power so that the upper limit of n‘(V) depends on the effectiveness of the homogenization in the cell made by the flowing solution itself or by a special stirring device. That is t,he case with all refractometers where it is necessary t o have an optical image of the light source (instruments based on the critical angle for total reflection; differential prism refractometers; Rayleigh interference refractometers), whereas instruments giving an optical image of the cell (other kinds of interference refractometers) are free from this restriction. The upper limit to the first derivative may also be defined by the inertia of a pen recorder, or by the sensitivity of the film in a photographically recording technique. The maximum recordable slope of the refractivity curve is an important apparatus constant. I t can easily be determined esperimentally. It may depend on the sign of the slope. It may also be the dwivative lvith respect to time, n’(t),that has a critical limit, and not n’( V). Range of Linear Response. I t is desirable that the directly recorded quantity X be a linear function of the refractivity. From the point of view of convenience this is important as one otherwise has to tranPlate the reading into refractivity by way of an evaluation curve or table. Among currently used methods for meawring refractivity, only the interferometric ones can be made strictly linear. Even in interferometry, however, the linearity is very often destroyed by the use of nonlinear optical compensators. Other methods, such as those based on angular deflections in prismatic cells, are linear only within certain refractivity ranges, the magnitudes of which change from one cell to another. -4primary nonlinear optical response can, however, be transformed into a linear ultimate response by the incorporation into the instrument of special mechanical devices, or more simply, by the use of specially ruled paper in the recorder. The range of linear response can be defined in terms of the optical resolving power-i.e., as the range within which the deviation from linearity is too small to be revealed by the instrument. I n such a case, the relative accuracy of the reading will increase throughout the range. I t can also be defined in terms of a predetermined relative accuracy which it is considered unnecessary to surpass. Complete Measuring Range. If the response is not strictly linear, the complete measuring range is in general greater than that within which linearity prevails. I t is natural that, the greater the resolving power and the sensitivity, the smaller the complete measuring range if the instrument, is not excessively large. However. it is often possible t o combine a high optical resolution Ivith a large range by having eschangeable cells with different properties in this respect, or by other devices. According to the principle of the instrument, the measuring range can be given in trrms of absolute refractivity units, or it has to be given in terms of refractivity differences. I n the lat,ter type of instruments, the zero response can he placed a t any desired value of the absolute refractivity scale. Adaptability to Different Colors. Colored substances occur every now and then in chemical practice, and consequently the
.
916
ANALYTICAL CHEMISTRY
adaptability t o different wave lengths should also be considered when judging the general usefulness of an instrument. If a mercury discharge lamp is the only light source available, it is hard t o use the instrument for redcolored solutions. An interferometer operating with white light does not function properly if the solution has a color of any tint, especially not if the cell thickness in the direction of the light is great. The interferometer channel c ill then act as an efficient filter m o n o c h r o m a t o r which makes t h e o b s e r v a t i o n of the central white fringe all the more difficult.
@ +-/
+ Figure 1. Refraction in a Differential-Prismatic Cell
*42 =
REFRACTIVE PROPERTIES OF THE DIFFERESTIAL-PRISMATIC CELL
Differential-prismatic cells have been used repeatedly in refractometers for stationary and flowing liquids ( 1 , 2, 4, 5, 8-14, 16), but nowhere a full account of their refractive properties seems to have been given. Basic Equations and Sensitivity. The construction of the differential-prismatic cell is shown in Figure 1, where the notation to be used in this article is also given. The light is supposed t o enter the solvent chamber and then to pass the solution chamber. The angles a,a’,p, and p’ are positive when the light pencils rise toward the right and negative in the opposite case. The angle v is defined as positive when the partition wall goes from the top left to the lower right corner and negative for the other diagonal of the rectangle. This notation, although not quite logical, has the advantage of giving a positive sign for a’ - cy in the final equation. The follo~ingbasic equations are easily derived from the figure: sin a = n sin 6 sin a’ = n(l sin(u
- 6)
= (1
+ 6) sin 8’
+ 6 ) sin(v - 6’)
-a
=
n6 tan v
a = 2 tan v S
(19)
When a parallel beam of light of this aperture is concentrated by a telescope objective into an optical image of a very narrow slit, the breadth of the central diffraction fringe becomes:
wheref2 is the focal length of the objective and X the wave length of the light. Let us now suppose that the receiver is capable of localizing this optical image to within a certain fraction, e, of the breadth given above. The least angular deflection of the light beam th:s+ the receiver is capable of detecting is then given by the equatioii A a ’ = tan A m
,eL -f2
=
2eXS a
(14)
This expression is identical with the optically conditioned uncertainty A X defined in Equation 2 and in the text below. The optical resolving power is consequently obtained by dividing the sensitivity S by this uncertainty:
(15)
(231
(16)
The solution of these equations is, as a first approximation for small values of a and 6: 01‘
---
(17)
The sensitivity of the cell will be defined as the derivative da’ld(n6) and is thus found to be:
Consequently the sensitivity can be chosen a t will, but a given cell has a fixed sensitivity. However, if all cell walls are optically plane and transparent, each cell can be used for two different sensitivities by turning it through 90” in relation to the optic axis. Optical Resolving Power. The optically conditioned uncertainty in determining the angle of deflection depends on the diffraction of light a t the effective aperture of the apparatus. Let us assume that this aperture is identical with the solution compartment of the cell, because otherwise the apparatus is badly constructed. (The cell construction in Figure 1 is not good enough from this point of view, since both compartments define the effective aperture; the solvent chamber should be made greater than the solution chamber.) The exit aperture is then a cos a‘/tan v, where a is the internal cell dimension in the direction of the optic axis. For optical resolving power, however, only the order of magnitude is important, and the cos a’ factor can be omitted. Hence we have:
Volumetric Resolving Power. If h is the dimension of the cell perpendicular t o the paper, the volume of the cell is found to be: a2h Av = 2 s The volumetric resolving power is the reciprocal of this volume. Light-Transmitting Capacity. Equations 23 and 24 show that the product of optical and volumetric resolving powers can be written : R,Rv =
S ~
rXah
If e and a are fixed according to the desired optical resolving power, it is found that the only way of getting, simultaneously, a high volumetric resolving power is to make S large and h small. Both methods tend to decrease the cell aperture, and the ultimate limiting factor is therefore the available light intensity. I t can be said about optical instruments in general that if there is an abundant light intensity a t the receiver, there remains a possibility of increasing either the optical resolving power or some other important property. I n such cases the instrument is either badly constructed or improperly used. The available light intensity is always one of the limiting factors. It is therefore neceasary in this case to consider the light-transmitting capacity of the cell under discussion before its general efficiency can be judged. If a single-slit source is used, the tolerable width of this slit is related to the diffraction in its optical image thus: the greater the
V O L U M E 25, NO. 6, J U N E 1 9 5 3 diffraction, the wider the slit can be without appreciably lowering the optical resolution. For the sake of simplicity, it will be stated that the slit may be allowed to be so wide as to make its optical image, calculated on a geometric-optical basis, equal in size to the breadth of the central diffraction fringe resulting from an infinitely narrow slit, although this width has already somewhat lowered the resolving power. If the focal lengths of the collimating and telescope lenses are fi and fi, respectively, the permissible width, T . of the light source slit is therefore given by the equation:
917 grid used by the present author ( 1 3 ) in a combination of the differential prism with Rayleigh’s interferometer is perpendicular to the grid used by Muller and Frachtman, but nevertheless this arrangement should give about the same light intensity. Range of Linear Response. Brice and Halwer (1) and Kegeles and Sober (9) have recently discussed the deviation from linearity of the differential-prismatic cell. According to thow papers, the exit angle is given by the second-degree approximation: 01’
The light flux outside a collimator is independent of whether the filament of the lamp itself or an optical image thereof is in the focal plane of the collimating lens, provided the filament in the former case and the slit in the latter are of the same dimrnsions. K e can consequently calculate the light flux on the assumption that we have a straight cylindrical filament of thicknesq r and length s in the focal plane of the collimating lens. Let the intrinsic brilliancy of the lamp be B candles per square centimeter, then the total light flus from the lamp will be TrsB and, a t the collimating lens, the illumination has dropped to rsB/A.f:. If this e\pression is multiplied by the entrance pupil of the cell, which is ah/S, one obtains the desired light-transmitting capacity, since reflexion and absorption losses are not being considered:
Efficiency. This concept will be defined as the product of optical resolving power, volumetric resolving poxer, and lighttransmitting capacity. Thus by the aid of Equations 25 and 27:
=
nS6
- n-2 S3P
if the entrance angle is assumed to be 0. The present author (IS) found the same expression for the fringe density in the optical image of the prism even without the assumption of LY = 0. The 62 term in the above equation determines the estension of the linear range. If that term is just equal in size to the optically conditioned uncertainty in a’,given by Equation 22, the border of the linear range as defined in terms of optical resolving power has been reached. Thus a refractivity difference satisfying the inequality
is within the linear range. This range is not very large. At a resolving power of 106 and a sensitivity of 1, we find for water solutions ( n = n8 = 0.00516. The linear range defined in terms of a certain relative accuracy 1 ’p (where p means the tolerable relative error), on the other hand, is given by the inequality: 2
ns
T h i s equation is, together with those for the component factors, very useful in the design of recording refractometers. It shows, among other things, that the efficiency is proportional to the sensitivity, although this quality is generally recognized as a rather unimportant factor. I t also shows that the focal length of the collimating lens is an important factor, which is not generally realized. Further, whereas the optical resolving power is proportional to the cell thickness, the efficiency is inversely proportional. Consequently, when it is a question of increasing the optical resolution, it is a much better way to increase the sensitivity of the receiver (to depress the value of E ) if this is possible than to increase the cell thickness. Finally it should be noted that the efficiency is independent of the wave length of the light. The follou.ing limitations in the validity of Equation 28 should b e kept in mind. The construction of the cell must be such that the whole volume is actually used optically. A “dead volume” will of course diminish the efficiency. Reflection losses have been disiegarded. They can be expected to become important a t very high sensitivities, when the light enters and leaves the cell partition wall under nearly grazing angles. There is, therefore, probablv an optimum sensitivity, which can, however, be judged to be rather high. The length of the slit should be interpreted as the actually useful length; if the cell height is small and its thickne- great, the cell will act as a stop for the slit length. Finally, the length of the slit and the focal length of the collimating lens are not independent of each other. To make their ratio large a well-corrected lens is necessary. Equation 28 was derived for a single slit as the light source. Kow Muller and Frachtman (26)have built a recording refractometer where a grid is the light source and the receiver is a photocell behind a similar grid. This technique is interesting since the light intensity is thereby multiplied by a factor equal to the number of lines in the grid, and the optical or the volumetric resolving power, or both, can be correspondingly increased. The light source
(29)
5 ps2
For S = 1, the responqe is linear to within 1% up to n8 = 0.02. Differential-Prismatic Cell with Four Different Sensitivities and Ranges of Linear Response. A perspective drawing of this cell on an enlarged scale is shown in Figure 2. It consists of tn.0 single cells cemented together, both having the same rectangular dimensions but different angles of the partition wall. The two smaller triangular U chambers are for the flowing solution Figure 2. Differand are accessible through the attached ential Prismatic capillary tubes a t the top and bottom Cell for Flowing of the cell. The txro bigger chamberq, Solutions for the solvent, are accessible through Perspective drawing a hole with a stopper a t the top of the cell. The sensitivities of the two component differential prisms are, for one direction of the incident light, 3 and 3/2, respectively. If the cell is turned through 90” in relation to the optic axis, one gets the tm-o additional sensitivities of ‘/a and s/a, respectively. The ranges of linear response are, in the order of increasing sensitivity n6 < 0.01500, 0.00612,0.00222, and 0.00096. The volume of the two compartments for the solution is 28 to 29 microliters.
-
REFRACTIVE PROPERTIES OF HALLWACHS’ CELL
In the year 1893, Hallwachs (6) described refractometric measurements in a cell of the type shown in Figure 3. He designed this in order to measure refractive index differences which were too great to be conveniently measured by the interference refractometer he had used earlier. Hallwachs used an extended light source and measured the exit angle corresponding to the border line between a bright and a dark field in the telescope, this angle
ANALYTICAL CHEMISTRY
918 corresponding on the entrance side of the cell to light pencils entering the partition wall under grazing incidence. He gave a detailed theory of the method for the case of @ = 0 in Figure 3. The author has now studied the general case of an arbitrary angle of incidence and found that Hallwachs' cell possesses some remarkable refractive properties which makes it very suitable for use in refractometers under more varied conditions than those used by IIallxachs. The following treatment follows closely that alrrady givrn for the double-prismatic cell.
rises toward infinity when this angle approaches 0 " and 90". The minimum sensitivity for water solutions is 2.67. The fact that the sensitivity can be continuously varied by turning t,he cell round an axis perpendicular to the paper in Figure 3 is a t the same time favorable and unfavorable. I t is favorable because the same cell can be used for solutions of n-idely different concentrations. But it is unfavorable because the angle of deflection is highly dependent on the orientation of the cell (except a t the mininium sensitivity), which necessitates an expensive cell holder and an accurate angle-reading device. Optical Resolving Power. Hall\vachs' original cell construction in Figure 3 is unsuitable for flowing solutions because the solution compartment of refractivity n( 1 6) encloses an appreciable dead volume-Le., a volume that can never be u$ed optically. The modified construction Pho\m in Figure 4, with the angle of the diagonal n-all equal to arc sin l / n , keeps the dead volume a t a minimum n-ithout appreciably lowering the usefulness of the cell for different semitivities (entrance angles between 0" and 90"). This cell construction \vi11 therefore be assumed to prevail in the following treatment in order to do full justice to thr cell. The optical resolving power of the cell is calculated in the same way as for the diffei.entiRl-pii.~iiiatic cell. The effective exit aperture of the cell is:
+
nt
1
Figure 3.
a
AB = a tan p' cos
-
a'
(41)
and thr hrradth of the central diffraction fringe in the optical image of a slit becomes:
Refraction i n Hallwachs' Cell
Basic Equations. K i t h the notation used in Figure 3, the follov ing equations are immediately evident : sin a = n sin p
+ 6 ) cos p' = n(1 + 6) sin p'
fl(1
=
/1
(32) cos p
(33)
sin a'
(34)
(43)
(35)
The optical resolving power is the ratio between the sensitivity and this uncertainty. Hence, after some wbstitutions with the aid of the Equations 32 to 34:
Squaiing and adding all thrre equations give?: sin2 LY'= sin2 a
The uncertainty in the &termination of the eur angle a' then become':
+ n26(2 + 6 )
n hich equation can also be put into the form:
cos 2a' = cos 2 a - 2n26(2
+ 6)
These equations have been derived without any approximations nhatsoever. For small refractive indev differences, they can be 6). simplified by omitting the term 6 in the parenthesis ( 2 Further, if the angles a and a' are small enough for neglecting their fourth powers beside the 6 term, Equation 35 can he i e d u c ~ d to:
+
and, consequently, for grazing incidence at the partition w i l l : a' = n
4%
If, further, 01 is put equal to 90" - w and a'
(38) =
YO" - w ' . the cor-
responding limiting equation for grazing i n d e n c e at the outer wall is obtained: w' =
n
4x6
(44)
(36)
(39)
Equations 38 and 39 indicate an extremely high sensitivity in the range of small refractivity differences. On-ing to thr square-root dependence, a considerable measuring range is nevertheless available. Sensitivity. R e define the sensitivity of the cell as the derivative:
I t should be observed that a ros 8' is the longest way that any light pencil travels through the solution to be meawred. I n casep where the effective aperture is not defined by the cell, the resolving power ip directly pioportional to the length differenve be, a tween the longest and shortest I path. of light pencils through the solution. We thus find that the resolving power is a t ita maximum a t grazing incidence on the outer wall ( LY = 90") and a t its minimum under the conditions used by Hallwachs. The difference between the t w o end values of the resolving power is, however, small; for water solutions, they differ only by 50%. V o l u m e t r i c Resolving Power. From the definition given above, the follon ing equation is easily derived:
Rr =
I t is consequently at a minimum when the exit angle is 45" and
Figure 4. Hallwachs' Cell, Modified C o n s t r u c t i o n for F l o w ing Solutions
2
dnT1 "7
a*n
(45) . .
Light- T r a n s m i t t i n g Capacity. The use of grazing incidence from
919
V O L U M E 2 5 , NO. 6, J U N E 1 9 5 3 a narrow light-source slit in a collimator means that the entrance pupil of the cell is vanishingly small, which gives rise t,o a very faint illumination at the receiver. The same applies to grazing incidence at, the partition wall. In addition, the reflexion losses are very much greater near grazing incidence than a t normal and moderate angles of incidence. One is therefore inclined t,o state wit,hout any further mathematical considerations that, with due consideration of bot,h optical resolving power and light transmission, the cell has its best opemting conditions at entrance angles well away from grazing incidence a t any of the cell wills. Matt.ers are, however, not quite as simple as that. A closer inspection of Equation 37 for small angles reveals that the esit angle is not influenced by the entrance angle as long as the term a2 is negligibly small compared with the term 2nQ. Consequently an extended light’ source can be used for grazing incidence, and the permissible extension of the source rises with the refractivity difference to be measured. The conditions are similar for grazing incidence a t the outer wall. Hallwachs’ cell has the quality of concentrating an incident beam of light, containing a broad range of entrance angles, into a beam of light characterized by one single and well-defined exit angle. This is why Hallwachs never succeeded in finding a sharp boundary between the bright and dark fields in his telescope. For small refractivity differences the permissible extension of the light source becomes too small to give a sufficient light-transmitting capacity. This conclusion agrees with the fact reported by Hallwachs that the light intensity did not allow accuratemeasurements to be made belowa refractive indes differenceof 0.00033. If reasonable figures w e put into Equation 44, one obtains resolving powers much gi.eater than the reciprocal of this value. The conclusion that the cell is more effective away from grazing incidence thus seems to be justified, after all. The above qualitative considerations on the light-transmitting capacity of the cell have shown that a precise mathematical treatment thereof would have to be rather complicated. One would have to take into account that the maximum permissible width of the light source is a function of the entrance angle and of thr refractivity difference to be measured. The Fresnel equations, from which the reflection losses would have to be derived, are also very difficult to handle. However, the light-transmitting capacity can easily be derived if the treatment is limited to entrance angles for which the reflection losses can be neglected-Le., for angles remote from grazing incidence. Under such conditions, the term containing 6 in Equations 35 and 36 is much smaller than the other two terms. Consequently, the light source slit has to be sufficiently small to make its optical image, calculated on a geometric-optic basis, not greater than the central diffraction fringe in this image. If the focal length of the collimating lens isf,, the width T of the light source slit is therefore given by the equation: r =
2 .fix a tan 0‘cos
a:’
(46)
E =
(47)
Here primed and unprimed angles have been made equal, since only orders of magnitude are of importance when light intensities are concerned. Equation 47 is identical with the corresponding equation for the differential-prismatic cell. Efficiency. Multiplication of 44, 45, and 47 now gives:
(48)
This equation is very similar to Equation 28, and most of what was said in that connection applies here too. The factor cos 8’ is to be expected since a large angle, p’, reduces the dead volume. Otherwise there is no hint here suggesting the use of a high sensitivity. Range of Linear Response. Equation 36 can be w i t t e n in the form :
201‘ = arc cos (cos 2 a - .in%
- 2~1~62)
(49)
The simplest way of analyzing the linearity is to develop this espression into powers of 6. To begin with, we stop with the second power, and get: 2n2
a’-a=-
4112
tan 2u sin 2a
) a2
(50)
The linear range defined in terms of optical resolving power is determined by the condition that the a2 term in the above equation be smaller than the uncertainty in the reading of the angle a’. The latter quantity is given by Equation 43, but we prefer here to put it equal to the ratio between the sensitivity S and the resolving power R,. Since the sensitivity within the linear range is identical with the factor of n6 in the above equation (which is the first approximation of Equation 40), we find that this range ie given by the inequality:
2n tan 2 a sin 2 a ( n S ) 2 R,(tan 20 sin 2 a - 4122) The range within which the relative error is less than other hand, is:
(51) p,
on the
For a certain entrance angle, the quadratic term in Equation 50 vanishes. This optimum angle is given by the equations:
(53)
sin 2a = 1 -
1 32 n
1 + ___ -, 2048 n s
p 4
‘ * * ’
(54)
I t is close to 41°, and there the functions 51 and 52 rise toward infinity. That is, however, not the case with the range of linear response. For this angle we have to consider the third-degree term in the expansion; it has been found to be:
Two subterms in this expression disappear for the angle 53, and the following equation remains: 01’
The calculation is then carried out in the same way as for the differential prism. A light source of thickness r, length s, and intrinsic brilliancy B has a total brilliancy of ?rrsB, and a t the collimating lens the intensity is rsB/4fT. If this expression ismultiplied by the area of the cell’s entrance pupil, ah tan p cos a , one arrives at the desired light-transmitting capacity of the cell:
sB 4 n 2 - 1 2eafi cos p’
-a
= __ 2n26 (1
sin 2 a
+ -)
8,1462
3 sin2 2 a
If the second term above is put equal to the uncertainty in the reading of the angle, SIR,, one obtains the following range of linear response, defined in terms of optical resolving power: (57)
With R, = lo6and n = 4 / 3 , this gives n 6 within which the relative error is less than
5 p
0.0128.
The range
is:
Finally it should be added that the sine of the double angle of deflection is a linear function of the rrfractirity difference within
ANALYTICAL CHEMISTRY
920
a still greater range. This is easily realized by introduction of t h e value of (a‘ -a) according to Equation 56 into the equation: (59) The third-degree term then disappears. -4 mechanical construction which allows direct measurement or recording of this sine function is easily designed. It is therefore worth while to deduce even this extended linear range. I t can be done quite easily in the following way. Starting from the equation with an unknown coefficient U , sin 2a’ cos 2 a = cos 2a‘ sin 2a:
+ sin2n2t2a: + D64 ~
(60)
t h e values of cos 2a‘ and sin 2a’ are introduced from Equation 36, and the equation is squared to get rid of the square root term, Already during this operation, all terms except those of the fourth degree in 6 can be discarded as being without interest, There are only three a4 terms, and they give the value of D:
cross section of a serpentine channel, which the light traverses perpendicularly to the direction of flow, might solve this problem in the most satisfactory manner. Two special cell types have been investigated with regard to some important refractive properties. The considerations on optical and volumetric resolving powers and light-transmitting capacity have not revealed any significant differences between the two types. If attention is paid to all these qualities, as has been done by introducing the concept of efficiency, the choice between them is quite arbitrary. I n other respects there are, however, striking and important differences. The sensitivity and range of linear response of a double-prismatic cell are determined by the angle of the partition wall; a given cell has a fixed sensitivity (or a t best, by turning the cell through go”, two fixed sensitivities). Hallwachs’ cell has a minimum sensitivity of 2.67, and it can then be varied continuously upward by a continuous variation of the entrance angle. On the other hand, this property involves higher demands on the cell holder.
where a is, of course, the optimum entrance angle given by Equations 53 and 54. Thus, for this entrance angle,
51 sin 2(a’ - a ) =
2n26 sin 2 a
2n464 - sin 4a ~
a n d in the same way as before the following range of linear response is found:
or,with cos 2a put equal to its approximate value of 1/’1n2: 1
I go With a resolving power of 106, the linear rsnge estends to n6 = 0.0473. DISCUSSION
The introductory discussion of the factors of importance for refractometers for flowing solutions mainly serves the purpose of elucidating the special difficulties and limitations that are present in this case in contrast to the case of stationary solutions of con&ant refractivity. An attempt has been made to show what accuracy it is on the whole possible to reach, and under which circumstances the recorded refractivity-volume curve can be expected to be in error. The equations presented should give some hints regarding the best compromise between the different requirements on the instruments in each special case. The predominating difficulty is to satisfy simultaneously the demands on a sufficient optical resolving power, volumetric resolving power, and light-transmitting capacity. This problem is essentially the problem of a suitable cell construction, although other features of the apparatus also play a role. The analysis has shown that, if the cross section of the cell is greater than that of the pipeline and if the refractometer requires a constant refractivity within the cell, a discontinuous method of recording gives better result than a continuous method. A cell cross section not greater than that of the pipeline is, however, greatly to be preferred because the mixing is then kept a t a minimum and a special homogenization device is superfluous. I n microanalysis, where the pipeline must be very narrow, it is difficult with such cells to reach the required optical resolving power and light intensity. With the optic axis in the direction of flow the light intensity suffers, and with the optic axis perpendicular to this direction, the re~olvingpower. It has been pointed out that a cell with a small
O
I
I
I
I
I
I
I
f
2
3
4
6
6
7
Figure 5. Comparison between Ranges of Linear Response of Hallwachs’ Cell (Upper Curves) and Diffewntial-Prismatic Cell (Lower Curve) as Functions of Sensitivity at a Constant Resolving Power
It is typical for both cells that a high sensitivity is connected with a small range of linear response, and vice versa. I n order to compare the cells more closely in that respect, the diagram in Figure 5 has been constructed. The ranges of linear response, defined in terms of optical resolving power, have here been plotted as functions of the sensitivity for both cells. The curves refer to a constant resolving power of 106, which for eHallwachs’ cuvet means that its cell thickness, a,varies along the sensitivity asis. At the higher sensitivities, the two cells behave similarly, but in the range of minimum sensitivity Hallwachs’ cell possesses a degree of linearity far beyond that of the differential-prismatic cell. At its optimum entrance angle, it has a linear range about 25 times greater than that of a differential-prism of the same senaitivity . ACKNOWLEDGMENT
This investigation is part of a research program for the development of improved methods of optical analysis of stationary and flowing liquids, which program is generously supported by tKe Swedish Technical Research Council. Additional financial aid has been given by LKB-Produkter Fabrikaaktiebolag, Stockholm, which is also gratefully acknowledged. The author
921
V O L U M E 25, NO. 6, J U N E 1 9 5 3 ie indebted to Ole Lamm of Stockholni for having dran-n his attention to Hallwachs’ work (6). LITERATURE CITED
Brice, B. A , , andHalwer, AI., J . Opt. Soc. Anter., 41, 1033 (1951). Cecil, R., and Ogston, A. G., J . Sci. Instr., 28, 263 (1951). Claesson, S., Arkiv. Rem. M i n e d . Geol.. 23 A, S o . 1 (1946). Claesson, S., “The Svedberg 1884 30,’8 1944,” p. 82, Uppsala, -4lmquist & Wiksell, 1944. (5) Debye. P. P., J . A p p l . Phys., 17,392 (1946). (6) Hallwachs, W., Ann. Phys. Chem., X e u e Folge 50,577 (1S93). :7j Jones, H. E., Ashman, L. E., and Stahly, E. E., -1s.4~.CHEM.,
(1) 12) (3) 14)
21,1470 (1949).
(8) Iiegeles, G., J . -4m.Chem. SOC.,69, 1302 (1947). (9) Kegeles, G., and Sober, H. A., ANAL.CHEM.,24,654 (1952). (10) Longsworth, L. G., IND. ENG. CHEW, ANAL.ED., 18, 219 (1946). (11) lliller, E. C., Cran-ford, F. !A7., and Simmons, B. J., -4x.4~. CHEY.,24, 1087 (1952). (12) Muller, R. H., and Frachtman, H. E., Intern. Congr. Pure and A p p l . Chem., p. 33 (1951). (13) Svensson. H.. Acta Chem. Scand.. 6.720 (1952). (14) Thomas, G. R., O’Konski, C. T.,‘and Hurd, C : D., AXIL. CHEY., 22, 1221 (1950). (15) Tiselius, A , , and Claesson, S., A ~ k i vKern. Mineral. Geol., 15 B, S o . 18 (1942). (16) Zaukelies, D., and Frost, A. A., ANAL.CHEW,21,743 (1949). RECEIVED for review l u g u s t 28, 1952. .4ccepted March 11, 1953.
Design and Use of an Electronic Pressure Controller F. J. DIGNEY AND STEPHEN YERAZUKIS Rensselaer Polytechnic Institute, Troy, .V. Y . Apparatus was designed which will maintain a constant pressure within small laboratory equipment such as an equilibrium still or distillation column. The pressure can be set to any desired value from vacuum to slightly above atmospheric pressure. Fluctuations from the control point are of the order of hO.1 mm. of mercury. Once the pressure is set, it will remain constant indefinitely. This equipment will be useful for maintaining constant pressure in small laboratory equipment automatically for hours, weeks, or months. It is especially indicated where very precise pressure control is required for long periods.
THE
successful pursuit of many laboratory problems, such as the determination of vapor-liquid equilibrium, may frequently depend upon a delicate control of pressure. rllthough numerous commercial controllers are available their expense or performance, particularly in so far as the need for continual observation and adjustment are concerned, often leave much to be tleeired. This pressure controller is not only easily and econoniically constructed but conveniently used, requiring a minimum of operator attention.
5.l 2 Y Y STOPCOCK
Sur
TUBING-
3 TO CMITROLLED
4 ELECTROCYTE RESERVE 6 ELECTRODE 5 REFERENCE PRESSURE B
APPAR4TUS
The controller consists of a simple manometer, one side of is connected to a reference pressure; the other is open to the system to be controlled. Any differential between these pressures will unbalance the manometer fluid. This effect is employed to actuate the corrective mechanism to restore the system to the desired control point. The barostat, Figure 1, which is of a size convenient for coniplete submergence in a small constant temperature bath includes the manometer and reference pressure bulb. The temperature as well as the volume and quantity of the gas sealed in the reference pressure bulb by the manometer fluid can be precisely controlled so that any desired reference pressure can be generated and maintained for an indefinite length of time. Since the refel ence pressure against P hich the system is balanced is dependent only on controlled variables, it is evident that atmospheric conditions cannot affect the operation of this pressure controller. The three electrodes provide the necessary contacts. -4s long as the pressure in the controlled system is equal to the reference pressure, the manometer is balanced and neither of the electrodes contacts the liquid, an electrolyte. On the other hand, if the elstem is off the control point, the manometer will be unbalanced with the consequence that one of the electrodes will contact the electrolyte, thus closing an electrical circuit and transmitting a signal to the main controller, Figure 2. This controller in turn operates a relay-powered tire valve in either the pressure or
7
R hich
I / -
I5
c
CI
23 CMI
,qI
/
I
LITER FL4SK
35 CM
A’2rr
lUeNG
4
2 r u CAPlLL4RI
/ L 3 8 CM
4 L
Figure 1.
The Barostat
