8
APPARATUS FOR STUDENT CONDUCTANCE MEASUREMENTS PAUL BENDER, WENDELL 1. BIERMANN, and ALVIN 6. WINGER University of Wisconsin, Madison, Wisconsin
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steady current flowing in a d.-c. circuit depends upon the magnitude of the applied potential and the resistance of the circuit. In the case of alternating current, however, the equilibrium current is limited by the circuit impedance, which depends not only on the circuit resistance hut also on the reactance due to the capacitance and inductance in the circuit. The reactance, and hence the impedance, is a function of the f r o quency of the current; the phase relationships between the currents and voltages in the various branches of an a.-c. circuit depend also upon the impedance of the components involved.
In the measurement of the resistance of a metallic conductor the direct current Wheatstone hridge of Figure la is regularly employed. The two points, A and B, across which the galvanometer used as detector is connected, are brought to the same potential (measured relative to some common reference point) by proper adjustment of the calibrated resistances RI, &, and R,. The familiar relation R - R* - R,; 5 bridge ratio = - R, RI then permits the calculation of the unknown resistance R,. In introductory treatments in undergraduate physical chemistry it is often suggested that this method can be extended satisfactorily to the measurement of the conductances of electrolytic solutions simply by replacement of the d.-c. battery and galvanometer by an a.-c. voltage source and detector. This is not possible, for in alternating current operation it is necessary to take into account the capacitative and inductive couplings in the bridge circuit which have no effect on direct current operation. At high frequencies this becomes a formidable task, hut at the relatively low frequencies a t which ordinary conductance measurements are made (500 to 2000 cycles), only the capacitative effects must he considered if noninductively wound standard resistances are used in the bridge. Flow of alternating current will take place capacitatively between any two conductors separated by a dielectric and between which an alternating potential is maintained. It follows that a complete treatment of capacitative effects in conductance measurements by the hridge method must consider inter-capacitances between the various bridge elements and between the hridge elements and ground. This problem has been treated in detail by Hague,' and with particular reference to electrolytic conductance measurements by Shedlovsky2 and Jones and J o ~ e p h s . ~The following discussion will be based upon the circuit of Figure l b , in which are shown the capacitative couplings of primary importance in a, practical hridge assembly for student conductance measurements. The principal capacitance encountered in the bridge arms proper is found in the conductance cell and its
-
1 Hncm, B., "Alternating Current Bridge Methods," 5th ed., Sir Isaac Pitman and Sons, Ltd., London, 1945. a SHEDWVSKY, T., J. Am. Chem. Soc., 52,1793 (1930). JONES,G., AND R. C. JOSEPHS, ibid., 50,1049 (1928).
213
APRIL. 1950
associated leads. The cell arm then acts like a parallel combination of a resistance, R,, and a capacitance, C.. When measurements are made on solutions of high specific conduotance, the reactance due to this parallel capacitance is of negligible importance except in measurements of extreme accuracy. With conductance water or other liquids of low conductance in the cell, however, this is no longer true, and the bridge arm shows its true nature as an impedance rather than a pure resistance. Experimentally this behavior is reflected in the impossibility of achieving a sharp bridge balance. In d.-c. bridge operation it is necessary that the potentials of the points across which the detector is connected be matched a t balance in magnitude only; in a x . operation they must be equivalent not only in magnitude but also in phase. In the bridge circuit of Figure l b the required phase balance cannot he effected because of the capacitative reactance in the cell arm. This interference with accurate resistance measurements may be eliminated by introducing a compensab ing capacitative reactance into one of the other bridge arms, as C. in Figure lc. It may be shown that in this case the necessary phase balance as well as magnitude balance may be obtained by proper adjustment of R,, Re, R,, and C,; a t this balance point R. - = -R1 = -C , R.
RI
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I
C,
It is seen that it is not necessary to know the actual values of C. or C. in order to determine the cell resistance; for a particular cell the resistance can be correlated in the usual fashion with the specific conductance of the cell electrolyte through the cell constant. Unfortunately, the balance so effected still will not be indicated sharply by the circuit of Figure lc, because of capacitative couplings between the circuit elements and ground; of these the most important, between detector and ground and oscillator and ground, respectively, are shown explicitly as Cdand C.. Even though the points A and B may be properly matched in potential by the method given above, a current will still flow through the detector and back to the oscillator through the ground capacitances if this potential is not ground potential. This phenomenon is often referred to as the "head effect" because this parasitic current travels to ground via the head of the observer if a headphone is used as the detector. The detector will respond to this current just as well as to that produced by the difference in potential of points A and B when the bridge is unbalanced. As a result, the precision with which bridge balance can be established is adversely affected, since the background in the detector output due to the head effect inevitably obscures the small change in the detector output produced by a slight bridge unbalance. In order to eliminate the head effect it is obviously necessary to bring the points A and B to ground potential a t balance; this is best accomplished by means of the Wagner ground device shown in Figure Id. The detector is first connected between A and B, and the
Figure 2.
Student Conductance Bridge Assembly
bridge is balanced as well as possible in the usual fashion. The detector is then connected between A and P, which is equivalent to connecting i t across a new Wheatstone bridge network the four arms of which are RI, Rz, R., and Ra When this auxiliary bridge is brought to balance by adjustment of the potentiometer contact P, A will be at ground potential because P is grounded directly. The detector is then switched back to B, and the main bridge adjusted to a new balance which now can be obtained more accurately because of the reduction of the head-effect interference by means of the Wagner ground. This rebalancing procedure, however, mill necessarily changk the potential of A with respect to ground, so the head effect cannot be completely eliminated in one step. This method of successive approximation is therefore continued until the desired precision of bridge balance is obtained. The Wagner ground may he used with bridges having any ratio. It not only makes the elimination of the head effect possible but it may he shown to eliminate the effects of all ground capacitances in the bridge network. At high frequencies adjustable capacitances in parallel with R. and Ra are required for proper operation but these are not needed in ordinary conductance work. A bridge assembly for student conductance measurements which can be recommended for undergraduate physical chemistry laboratory work is shown in Figures 2 and 3. The basic bridge employed is a commercial student-type Wheatstone bridge; both the Leeds and Northrup and Rubicon models are in use. The
JOURNAL OF CHEMICAL EDUCATION
214
Figure 3.
Internal Assembly
wiring diagram for the unit is given in Figure 4; in addition to the features of Figure Id, the oscillator and detector are transformer-coupled to the bridge, and an amplifier is included in the detector section. While the transformers specified lack the special shielding desirable in bridge-coupling transformers, they have been found quite adequate, readily available, and relatively inexpensive. The oscillator is connected to the bridge terminals marked "Galvanometer" and the detector to the "Battery" terminals to simplify the use of the Wagner ground. If the opposite arrangement were employed, the balance condition for the auxiliary bridge would become Rs - R, R. R* and the Wagner ground setting would be strongly dependent on the cell resistance even if the bridge ratio were held constant. The indicated Wagner ground connection is much more convenient, since it requires a minimum of adjustment. The five pairs of resistors are selected to provide Wagner ground balances near the midpoint of the potentiometer (used as the fine adjustment) for the bridge ratio values of 100, 10, 1, 0.1, 0.01 required in student conductance work. The Wagner ground potentiometer should be a heavy duty type since ordinary radio potentiometers do not stand up in this application. The value of C. required to compensate for the capac-
itance in the cell arm depends upon the value of the bridge ratio; as previously noted C. = C, (&/R,). An ordinary Roseveare type conductance cell of approximately unit cell constant, with two-foot leads, may require as much as several thousand micramicrofarad balancing capacitance when filled with conductance water. A small variable condenser in parallel with switch-selected fixed mica condensers provides a fine adjustment with an adequate capacitance range. The amplifier, which is of conventional design, can be seen in the upper corner of Figure 3. The use of shielded wire as indicated in Figure 4 is essential if efficientperformance is to be obtained; decoupling filters are required to prevent interaction between different amplifiers operated from a common power supply, which should be voltage regulated. The bridge input voltage was made variable instead of the amplifier gain, after experience with both arrangements; the maximum input voltage is fixed a t the master oscillator. While a number of oscillator circuits are available for conductance work, the resistancecapacitance coupled audio o~cillator~.~ offers particular advantages, including excellent wave form and ease of adjustment. A single oscillator may be used to control a number of output amplifiers feeding different bridge assemblies. The required isolation of the oscillator from the amplifiers is effected by a cathode follower stage. Shielded cable is used to connect the 1000cycle source to Jones plug receptacles on the power distribution panel for the bridge units. I t is necessary to provide shielding around these a.-c. receptacles. It might be argued that from the teaching standpoint it would be preferable to supply the Wagner
' TERMAN, F. E., R. R. BASS, W. R. HEWLETT, AND F. C. PYOC. I. R. E., 27,649 (1939). CAHILL, REICA,H. J., "Theory and Applications of E l e o t r o n Tubes," 2d ed., McGraw-Hill Book Company, Inc., New York, 1944.
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APRIL, 1950 ground, balancing capacitances and amplifier as separate units auxiliary to the bridge. This arrangement was tried initially but was found unsuccessful in actual laboratory use, principally because of the many leads required and their associated stray 1000-cycle fields. It appears impossible to overemphasize the necessity for adequate shielding in these units, especially when several are to be used in the same location. The accuracy which can he obtained with this equipment is surprisingly good, particularly since t,he basic bridge units are built for direct-current and sixty-cycle use. In measurements on standard fixed resistors (noninductive) of from 50 to 500,000 ohms an accuracy of about 0.1 per cent and a precision considerably higher has been obtained. In the more pertinent case of actual conductance measurements, the performance attainable is equally satisfactory, as is shown in Figure 5. Student results are necessarily less accurate, but good workers regularly obtain AO values for KC1 which are within 0.3 per cent of the accepted value. Although the conductance bridge circuit is decidedly more complicated than the simple d.-c. Wheatstone bridge considered in introductorv ~hvsicscourses, we have found that the
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