gradient in volts per centimetre, the conductivity will be in mhos (per cm. cube). [Example. Two parallel copper electrodes 40 X 30 cm. are immersed 6 cms. apart in a rectangular trough of 10 per cent. CuSO solution of 0032 conductivity, and a current of 60 amperes is passed between them. What voltage is required? Cross-section of trough, 40 X 30 = 1200 sq. cm. Current density 60 ÷ 1200 = 0'05 amp. per square centimetre. current density Conductivity 0'032 = .. potential gradient = potential gradient This for a distance of 6 cms. amounts to 9°36 volts, which is therefore the total fall of potential in passing from the one plate to the other.] The reciprocal of the conductivity is called the resistivity or specific resistance, and is measured in ohms (per cm. cube). In a trough or tube of uniform cross-section, containing two parallel electrodes, the current density is uniform, viz., the total current divided by the cross-section; and the potential gradient is uniform, viz., the difference of potential between the electrodes divided by the distance between them; hence we may write The last quantity is called the conductance of the trough or tube, and even if the vessel is not regular in shape this name may still be retained for the ratio current potential difference [Thus in the preceding examples the conductance of the trough is o'032 X The reciprocal of the conductance is called the resistance, and is measured in ohms; hence we arrive at the usual expression of Ohm's law resistance = potential difference (or voltage, or electromotive force) current [The resistance of the trough is 60156 ohms.] 6'4 In order to determine the conductivity of a solution the simplest method would be to place it in such a trough of uniform cross-section, pass a current through it, and use an ammeter to determine the strength of current and a voltmeter for the voltage between the electrodes. This method would not be satisfactory, however, on account of "polarisation" at the electrodes, i.e. change in the nature of the electrodes, especially evolution of gas, due to the current. There would not be much polarisation at copper electrodes in copper sulphate solution, and so the above method would be practicable; but, e.g., with platinum plates in dilute sulphuric acid the polarization would be so large as to cause quite erroneous results. To avoid this error a separate pair of electrodes through which current is not passed may be used for the potential measurement, and the measurement made by an electrostatic voltmeter (electrometer), i.e. one which takes no current. The connections are shown in Fig. 12: current is taken in and out by the large electrodes HJ; the current density is found by dividing the reading of the ammeter A placed in series with the electrolytic trough, by the cross-section of the latter. Thin wires H'J' are immersed in the liquid and lead to the electrostatic voltmeter V; the reading of the latter divided by the distance between the wires gives the potential gradient. This method is sometimes used for liquids of excessively low conductivity. In ordinary cases an alternating current method is always used. Current is led alternately one way and the other through the liquid, many times a second, so that the polarisation produced on a plate serving as cathode is immediately afterwards destroyed by its use as anode. The arrangement of apparatus is essentially similar to that employed in the case of solid conductors, viz. Wheatstone's bridge. We shall therefore proceed to describe in detail the method of carrying out conductivity measurements in this way.1 Wheatstone's bridge consists of an arrangement of conductors shown schematically in Fig. 13. Current from a B OG d FIG. 13. battery B is led to the point a, where it divides into the two branches abc and adc, reuniting at c to return to the other pole of the battery. Two intermediate points b and d of the branched conductors are con nected through a galvanometer G: then it may be shown that no current flows through G, provided the resistances ab, bc, ad, dc, are chosen so as to satisfy the relation ab × cd = bc × ad. Hence, if three of the resistances are known and the fourth is to be measured, the three can be adjusted until no current flows through the galvanometer, and the equation will then give the value of the unknown resistance. A battery and galvanometer are available in such an arrangement when a metallic resistance is to be measured; but for an electrolyte the battery must be replaced by a source of alternating current, the secondary of a small induction coil being the most convenient; and as with an alternating current an ordinary galvanometer will give no indication, some appropriate indicator must be used instead: an electro-dynamometer or a telephone, commonly. Induction coils suitable for the purpose are now sold by the German scientific instrument-makers. They should be small enough to be worked by a single dry cell, or by an accumulator with a metallic resistance in circuit to reduce the 1 The following details are nearly all taken from the admirable monograph of Kohlrausch and Holborn, Leitvermögen der Elektrolyte (Teubner, Leipzig, 1898). current to about half an ampere. The secondary should have only a moderate number-say 1000-of turns, as high electromotive force introduces disturbing effects. The interruptor may be a light steel spring covered by platinum at the place where sparking occurs, working against a platinum point; or a platinum point dipping into mercury: in this case the mercury must be protected against oxidation by a layer of distilled water, and the mercury connected to the positive pole. If a telephone is used as indicator, the frequency of the spring must be such as to give the best effect on it; i.e. the spring must give out a clean singing tone of the pitch of a treble voice, 250 to 1000 vibrations per second. With an electrodynamometer this is not necessary, but the frequency must in any case be fairly high in order to eliminate polarisation sufficiently. + Alternating current is now often supplied for domestic purposes; but if that is to be made use of, only a fraction of the voltage supplied must be employed on the bridge. The best plan is to feed 'a lamp (L) (Fig. 14), and a wire-resistance (R) of a few ohms, in series from the mains, and take off current from the ends of R to the bridge. But as the frequency of alterna LR FIG. 14. tion is usually only 100, the domestic supply gives a dull and unsatisfactory tone in the telephone, and is best used with an electro-dynamometer. As observing instrument the ordinary commercial telephone is the most useful. The resistances of the bridge are to be adjusted so that no sound is produced in it, or if that condition is unattainable, at least a well-defined minimum of sound. The telephone should be pressed closely against one ear, while the other is stopped up. The telephone should have only a moderate resistance-say 10 ohms—and by choosing a good instrument and working under favourable conditions, it is possible to measure to Toooo part, or even less; observation is easier if the tone is not too loud. The chief defect of the telephone is that when the resistances in the bridge are out of balance, the telephone does not, like a galvanometer, show which way they should be altered to obtain balance. In this respect an electrodynamometer is superior. It consists essentially of a coil suspended at the centre of a fixed coil, the planes of the two being at right angles. When current is led through both, the movable coil tends to set itself parallel to the fixed; the rotation is resisted by the suspending wire or wires, and the amount of turning measured with a telescope-mirror, and scale. For resistance measurements the fixed coil of the electrodynamometer should .be inserted in the main circuit (i.e. of the induction coil or that taken off the alternating mains), while the movable coil is put in the branch which otherwise carries the galvanometer or telephone. With this arrangement, when the unknown resistance is underbalanced by the three known, the movable coil will turn in one sense; if overbalanced, in the other; its use being therefore similar to that of the galvanometer in measuring metallic resistances. With regard to the composition of the arms of the bridge, the chief determining factor, when a telephone is used, is facility of adjustment. Plug resistance-boxes, such as are employed with a battery and galvanometer, would involve so much manipulation as to make it very tedious to find the minimum sound; hence the usual plan is to have a box for |