T. Cyr, J. Prudhomme, and M. Zador Universite de Montreol Montreol 101, Canada
A Stabilized Linear Direct Reading Conductance Apparatus The solvolysis of t-butyl chloride
A simple ac conductivity apparatus for experiments in chemical reaction kinetics is dcscrihed. The instrument is sufficiently reliable to be used by first year undergraduate students and can be assembled within a few hours. I t can be used whenever the main purpose is the recording of a chanee in conductance against time such as in a titration or i n reaction kinetics where the reaction rates are too high to he followed hv a bridge type instrument. T h e major shortcomings of mbst ac conductivity meters is that they are either too slow, as in the case of Wheatstone bridge circuits, or that they are not linear and accurate. Included among the latter are a series of commercial circuits using a rather unstable oscillator followed by a resistive voltage divider and an AC voltmeter such as shown in Figure 1. The amplitude of the oscillator output signal Vo is very important since the ac voltage at A in Figure 1 is simply VoR2/(Rl Rz). Any instabilities in Vo are directly reflected as inaccuracies in the detected signal. If R1 is chosen as the conductivity cell resistance and Rz is chosen to be a few ohms in value, then the detected signal is approximately linear when R1 is very much greater than Rz. However, such a circuit does not lend itself to scale expansion as a result of its inherent instabilities. Moreover, since most detectors are of the type indicated a t B in Figure 1, the resistor Rz must be changed constantly to keep the detector within its linear range. With the advent of the Heath analog computers, operational amplifier circuits such as that shown in Figure 2 were found to sienificantlv"imorove . the linearitv. (.I .. 2). but not however the accuracy of the conductance experiments. The faults of the circuit shown in Firmre 2 are numerous and require some elaboration. The o&llator uses what is known as the "Twin-Tee" filter and although this gives fairly good frequency stability to the output signal VO, the amolitude of the sienal VO is not at all controlled except to some extent by tKe satuiation of amplifier 1. The signal Vn with its instabilities are then am~lifiedhv amplifier 2 shown to the center of Figure 2. he gain of amplifier 2 can he shown to he a function of RI, normally taken as the cell resistance, Rz, selected to he within a factor on one hundred of that of R1, and the amplifier's open loop gain, -K. The ac signal at A in Figure 2 is approximately -VoKRz/(Rl + R2 + KRI) which can he made nearly equal to -VoRz/Rl if K is sufficiently large in magnitude. Operational amplifiers with K greater than 100,M)O are not uncommon and if they are chosen also for a sufficiently high input resistance, then the output signal of amplifier 2 is
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Figure 1. Circuit diagram of a typical commercial conductivity meter.
0
nemdS.
+
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where a, = l/R1, the cell conductivity in mhos. Operational amplifiers including those of the Heath analog computer have an output impedance, Rout, which is often several hundred ohms in value. Normally this is not important in operational amplifier circuitry, hut for the circuit shown in Figure 2, this impedance must be included in R1 572
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JFigure 2. Circuit diagram for conductivity meter as shown by Ewing and Re~iley( I ) .
in eqn. (1). The output voltage amplitude
is no longer directly proportional to ac = l/R1 as desired except to a certain degree of approximation when R1 is of the order of several hundreds of thousands of ohms. Another nonlinearity becomes evident when one considers the diode detector, D, in Figure 2. One feature of semiconductor diodes is that their forward current impedance varies as a function of the forward current. As a result of this, the recorder ac and dc input impedances will become very important when VI has a low voltage value corresponding to R1 large and uc small. The effect of the recorder impedance will become evident as a knee or curve in the detected signal versus a, graph when large cell resistances are obtained. It is for these reasons that most experimenters have rejected the usual operational amplifier circuits. The limitations of the circuits discussed above can however he avoided. The major fault of the circuit shown in Figure 2 is made evident by eqn. (2). If however the amplifier circuit is rearranged as shown in Figure 3, the effect of the output resistance of the oscillator is avoided. The output signal Vl is given by
which in the limit of large amplifier open loop gain, K, becomes
This signal is much easier to handle than that given in eqn.
Figure 3. Black circuit diagram of modified conductivity meter (1) since we can electrically subtract the constant term Vo from our recorder deflection by some sort of zero offset circuit. The practical circuit shown in Figure 4 includes a standard Wien-bridge oscillator circuit which can he found in most elementary electronics courses. The resistors and capacitors shown a t the far left of the diagram are the frequency sensitive elements and the oscillator frequency can he calculated as the inverse of their time constant, f = 1/ (2zRC) = 1/(2z X 15000 X 0.1 X 1 0 - 9 = 106 Herz for the switch S1 in the position shown. The other position of S1 corresponds to an oscillator frequency of 3 KHz. The two frequencies are necessary to permit us to eliminate the effects of cell canacitance which can cause some error if it is exceptionally large.' The notation 6K8. 2K7. 3K3, and 4ft7 shown in Fiaure 4 is used to eliminate confusion over the decimal point location and these values should read 6800, 2700, 3300 and 4.7 ohms, respectively. The output signal of amplifier 1 is rectified by the diodes at B. Amplifier 3 serves to buffer the output signal Vo from the varying impedances of these diodes. The rectified signal, which is directly proportional to the amplitude of Vo, is compared with a reference voltage a t A by amplifier 2 and the amplified difference signal is fed to the light hulh. The resistance of the light hulh increases as more voltage is applied across it, and by placing it in the negative feedback path to amplifier 1 we have managed to stabilize the oscillator amplitude. The oscillator circuit presented a non-distorted sine-wave stable in amplitude to five parts in ten thousand. The Fairchild 741 operational amplifiers were chosen for their simplicity and cost ($0.38 each). The zero offset circuits shown by the 10K potentiometers ahove amplifiers 4 and 5 were not necessary in this part of the circuit. Amplifier 4, which corresponds to the amplifier shown in Figure 3, was a Fairchild 740 type operational amplifier which is nearly identical to the 741 amplifiers except for its exceptionally high input impedance, 1012 ohms. The
extra expense is well justified ($7) since often cell resistances of ten megohms or more may be encountered during an experiment and the circuit requires that the amplifier input impedance always must he very much greater than the cell resistance. The 10K zero offset trimming potentiometer is adjusted to eliminate any dc bias current from the conductivity cell. The 100 uF capacitor near J1 was later added to further prevent such dc currents which naturally can lead to electrodeposition. A calibration position C on SZis also shown. This extra position was useful during the laboratpry experiment where calibration marks at 0 pmhos and 1000 pmhos were desired a t the recorder output. The sample cell is not shown hut is connected between the socket J,.and the earth side of the 10K ohm CAL resistor. The detector diode D has been biased to eliminate the nonlinearity discussed after eqn. (2) above. Such a detector circuit is generally known as a biased "voltage clamp." The amplifier 5 serves to isolate the rest of the circuit from the recorder. The 10K zero offset resistor on amplifier 5 is a zero offset control adjusted to suhtract the constant term from eqn. (4). The circuit components are inexpensive. They are mounted directly on Verohoard and are housed in a 4 X 5 X 8 aluminum chassis. In operation the switch Sz is put in the uppermost position and 10K CAL is adjusted to give a fuii scale redorder deflection corresponding to 1000 mhos. Note that TRIM is a trimming resistor adjusted to give the same signal amplitude for the lOOHz and the 3KHz signals. For most experiments Sz is placed in position 4 giving Rz = 20 Kohms and permitting a cell conductivity between 2 and 1000 m h o s to be displayed directly on a 1 V (1000 pmhos) recorder or digital voltmeter. For a cell having two 1 cm diameter platinum discs placed at 1 cm distance, .the cell capacitance usually can he neglected as a source of error. The connection J1 is the standard UHF socket used for most commercial conductivity cells. The nrincinal advantaees of this conductivitv meter " over those previously published are that it presents a nondistorted sine wave of constant amplitude, within 0.05%, the output signal is directly proportional to the conductance with a fairlv . hiah - decree of accuracv, within 0.1%. and that it has excellent stability over veriiong periods of 'The cell capacitance, C, must be included in the value of R1 by replacing R1 in eqns. (2) and (4) by RI X (1 + 2rfRlC)-"2. Thus for f = 100 Hz and for a cell capacitance of 50 picofarads as often found, the square root term becomes (1 + R1 X 3.14 X 10-8)1/2 and unless R1 becomes comparable to thirty rnegohms, the cell capacitance may be neglected.
Figure 4. Circuit diagram of conductivity meter. A Lambda LZ stabilized dual 15 V power supply is used. The feedback resiitors on S2 are selected to better than 0.1% accuracy. All capacitances Shawn are in microfarads. Volume 50, Number 8, August 1973
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time. These features may he compared most favorably to those circuits already published (3-5) as well as those commercially available.2 In addition, the total component cost is significantly less than $100. The conductance apparatus is used by our undergraduate students to study the kinetics of solvolysis of terthutyl chloride. This reaction is well suited to the undergraduate laboratory as shown by our experience of several years using a bridge type instrument described by Chesick and Patterson (6).The reaction
where R = H or CzHa, is of first order with respect to tert-butyl chloride. The reaction is followed by the measure of the conductance due to the hydrochloric acid formed. The recorded signal, St, at time t is given by
where a and b are constants. It can be easily shown that the integrated rate-law is given by In (S. - So) - In (S. 2
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SJ = kt
Such as the Radiometer Model CilM 2 C.
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where So, St and S, represent the signal a t 0, t and infinite time, and k is the rate-constant. The Guggenheim method previously used (6) can be discarded because with this apparatus fairly short reaction times (as small as about 1 min) can be chosen and so S, is observable. Furthermore, during one laboratory period, four to five experiments in a wide temperature range (30-60°C) can he chosen to yield a good Arrhenius plot. The corresponding activation energies are generally between 19-21 Kcal, slightly lower than the literature value of 23 Kcal(5). Different solvent mixtures of water and ethanol can be used. Between 40% to 60% V/V of water the rate changes by a factor of ten (5). This is a good demonstration of the sensitivity to solvent polarity of a reaction containing a highly polar transition-state. Acknowledgments should he expressed to Messrs J. Caron, R. Hartshorn and J. Seguin for their enthusiasm.
Literature Cited (1) Reilley,C.N.. J . CHEM. EDUC.,39,A853,A93311962). (2) Ewing, 0.W., a n d B r a ~ d c n , T . H . , A w I Cham., . 35, 182611963). (3) Groenberg,D.B.. J.CHEM. EDUC.,39.140(1962). 14) Sfoek,J.T., J.CHEM. EDUC., 44.573 (1967). (5) OI~cn,E.D.,Martin,R. J.,sndAhnell, J. E.. J.CHEM.EDUC.,47,542l1910). 161 Che~iek,J . P . . a n d P s t t e m n , A . . J.CHEM.EDUC.. 3'7.21211960). (7) Grunwe1d.E.. and Winstein, S., J A m e r Chom. Soc., 70.846(1948).
