# 1911 Encyclopædia Britannica/Thermoelectricity

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​THERMOELECTRICITY. 1. Fundamental Phenomena.—Alessandro Volta (1801) showed that although- a separation of the two electricities was produced by the contact of two different metals (Volta Effect), which could be detected by a sensitive electrometer, a continuous current of corresponding magnitude could not be produced in a purely metallic circuit without the interposition of a liquid, because the electromotive force at one junction was exactly balanced by an equal and opposite force at the other. T. J. Seebeck (1822), employing a galvanometer then recently invented, which was more suited for the detection of small electromotive forces, found that a current was produced if the junctions of the two metals were at different temperatures. He explained this effect by supposing that the Volta contact electromotive force varied with the temperature, so that the exact balance was destroyed by unequal heating. The intensity ​ of the current, C, for any given pair of metals, was found to vary directly as the difference of temperature, t-t', between the hot and cold junctions, and inversely as the resistance, R, of the circuit. We conclude by applying Ohm's law that the electromotive force, E, of the thermocouple may be approximately represented for small differences of temperature by the formula
| $E=CR=p(t-t')$ | (1) |
2. Thermoelectric Power, Series, Inversion.—The limiting value, dE/dt, of the coefficient, p, for an infinitesimal difference, dt, between the junctions is called the Thermoelectric Power of the couple. One metal (A) is said to be thermoelectrically positive to another (B), if positive electricity flows from A to B across the cold junction when the circuit is completed. The opposite convention is sometimes adopted, but the above is the most convenient in practice, as the circuit is generally broken at or near the cold junction for the insertion of the galvanometer. Seebeck found that the metals could be arranged in a Thermoelectric Series, in the order of their power when combined with any one metal, such that the power of any thermocouple p, composed of the metals A and B, was equal to the algebraic difference (p'−p″) of their powers when combined with the standard metal C. The order of the metals in this series was found to be different from that in the corresponding Volta series, and to be considerably affected by variations in purity, hardness and other physical conditions. ]. Cumming shortly afterwards discovered the phenomenon of Thermoeleclric I rwersion, or the change of the order of the metals in the thermoelectric series at different temperatures. Copper, for instance, is negative to iron at ordinary temperatures, but is positive to it at 300° C. or above. The of a copper iron thermocouple reaches a maximum when the temperature of the hot junction is raised to 270° C., at which temperature the thermoelectric power vanishes and the metals are said to be neutral to one another. Beyond this point the diminishes, vanishing and changing sign when the temperature of the hot junction is nearly as much above the neutral point as the temperature of the cold junction is below it. Similar phenomena occur in the case of many other couples, and it is found that the thermoelectric power p is not in general a constant, and that the simple linear formula (1) is applicable only for small differences of temperature. More accurately it may be stated that the thermoelectromotive force in any given circuit containing a series of different metals is a function of the temperatures of the junctions only, and is independent of the distribution of the temperature at any intermediate points, provided that each of the metals in the series is of uniform quality. This statement admits of the simple mathematical expression
| $E=\int _{t_{0}}^{t'}p'dt+\int _{t'}^{t''}p''dt+$&c. | (2) |
where p′, p, &c., are the thermoelectric powers of the metals, and to, t′, t″, &c., the temperatures of the junctions. There are some special cases of sufficient practical importance to be separately stated.
3. Homogeneous Circuit. Strain Hysteresis.—In a circuit consisting of a single metal, no current can be produced by variations of temperature, provided that the metal is not thereby strained or altered. This was particularly demonstrated by the experiments of H. G. Magnus. The effects produced by abrupt changes of temperature or section, or by pressing together pieces of the same metal at different temperatures, are probably to be explained as effects of strain. A number of interesting effects of this nature have been investigated by Thomson, F. P. Le Roux, P. G..Tait and others, but the theory has not as yet been fully developed. An interesting example is furnished by an experiment due to F. T. Trouton (Proc. R. S. Dub., 1886). A piece of iron or steel wire in the circuit of a galvanometer is heated in a flame to bright redness at any point. No effect is noticed so long as the flame is stationary, but if the flame be moved slowly in one direction a current is observed, which changes its direction with the direction of motion of the flame. The explanation of this phenomenon is that the metal is transformed at a red heat into another modification, as is proved by simultaneous changes in its magnetic and electrical properties. The change from one state to the other takes place at a higher temperature on heating than on cooling. The junctions of the magnetic and the non-magnetic steel are therefore at different temperatures if the flame is moved, and a current is produced just as if a piece of different metal with junctions at different temperatures had been introduced into the circuit. Other effects of “ hysteresis ” occur in alloys of iron, which have been studied by W. F. Barrett (Trans. R. S. Dub., January 1900).
4. Law of Successive Temperatures.—The E.M.F. of a given couple between any temperatures t' and t” is the algebraic sum of the between t' and any other temperature I and the between t' and t". A useful result of this law is that it is sufficient to keep one junction always at some convenient standard temperature, such as o° C., and to tabulate only the values of the in the circuit corresponding to different temperatures of the other junction.
5. Law of Intermediate Metals.—A thermoelectric circuit may be cut at any point and a wire of some other metal introduced without altering the in the circuit, provided that the two junctions with the metal introduced are kept at the same temperature. This law is commonly applied in connecting a thermocouple to a galvanometer with coils of copper wire, the junctions of the copper wires with the other metals being placed side'by side in a vessel of water or otherwise kept at the same temperature. Another way of stating this law, which, .though apparently quite different, is really equivalent in effect, is the following. The of any couple, AB, for any given limits of temperature is the algebraic sum of the E.M.F.s between the same limits of temperature of the couples BC and CA formed with any other metal C. It is for this reason unnecessary to tabulate the E.M.F.s of all possible combinations of metals, since the of any couple can be at once deduced by addition from the values given by its components with a single standard metal. Different observers have chosen different metals as the standard of reference. Tait and J. A. Fleming select lead on account of the smallness of the Thomson effect in it, as observed by Le Roux. Noll adopts mercury because it is easily purified, and its physical condition in the liquid state is determinate; there is, however, a discontinuity involved in passing from the liquid to the solid state at a temperature of -40° C., and it cannot be used at all with some metals, such as lead, on account of the rapidity with which it dissolves them. Both lead and mercury have the disadvantage that they cannot be employed for temperatures much above 300° C. Of all metals, copper is the most generally convenient, as it is always employed in electrical connexions and is easily obtained in the annealed state of uniform purity. For high temperature work it is necessary to employ platinum, which would be an ideal standard for all purposes on account of its constancy and in fusibility, did not the thermoelectric properties of different specimens differ considerably.
6. Thermoelectric Formulae.-On the basis of the principles stated above, the most obvious method of tabulating the observations would be to give the values E, of the E.M.F. between 0° C. and t for each metal against the standard. This involves no assumptions as to the law of variation of E.M.F. with temperature, but is somewhat cumbrous. In the majority of cases it is found that the observations can be represented within the limits of experimental error by a fairly simple empirical formula, at least for moderate ranges of temperatures. The following formulae are some of those employed for this purpose by different observers:—
| E$_t$=bt+ct$^2$ | (Avenarius, 1863.) |
| E$_t$=at+bt$^2$+ct$^2$ | (General type.) |
| log E=a+b/T+c log T | (Becquerel, 1863.) |
| E$_{(t–t′)}$=c(t–t′)(2t°-(t+t′)) | (Tait, 1870) |
| E$_t$+E$_2$° = 10$^{a+bt}$+10$^{a′+b′t°}$ | (Barus, 1889.) |
| t=aE+bE$^2$+cE$^2$ | (Holborn and Wien, 1892.) |
| E(t–t′) =b(t–t′)$^{4/3}$ | (Paschen, 1893.) |
| E(t–t′) =a(t–t′)+b(t–t′) | (Steele, 1894.) |
| E$_{(T⋅T°)}$=mT$^n$−mT°$^n$, E$_t$=mt$^n$ | (Holman, I896.) |
| E$_t$ =bt+c log T/273, (c=Ts) . | (Stanfleld, 1898.) |
| E$_t$=−a+bt+ct$^2$ | (Holborn and Day 1899.) |
| E$_t$ =at+ct$^2$+s°(T log $_e$T−273 log$_e$273). |
       (Where s=s°+2cT, and c is small. See sec. 15.) ​
For moderate ranges of temperature the binomial formula of M. P. Avenarius is generally sufficient, and has been employed by many observers. It is figured by Avenarius (Pogg. Ann., 119, p. 406) as a semi-circle, but it is really a parabola with its axis parallel to the axis of E, and its vertex at the point t=−b/2c, which gives the neutral temperature. We have also the relations dE/dt=b+2ct and d $^2$E/dt$^2$=2c. The first relation gives the thermoelectric power p at any temperature, and is probably the most convenient method of stating results in all cases in which this formula is applicable. A discussion of some of the exponential formulae is given by S. W. Holman (Phil. Mag., 41, p. 465, June 1896).
7. Experimental Results.—In the following comparative table of the results of different observers the values are referred to lead. Before the time of Tait's researches such data were of little interest or value, on account of insufficient care in securing the purity of the materials tested; but increased facilities in this respect, combined with great improvements in electrical measurements, have put the question on a different footing. The comparison of independent results shows in many cases a remarkable concordance, and the data are becoming of great value for the testing of various theories of the relations between heat and electricity.
| Metal. | Tait (0° to 300°) | Tait (0° to 300°) | Steele (0° to 100°). | Steele (0° to 100°). | Noll (0° to 200°). | Noll (0° to 200°). | Dewar and Fleming (+100° to −200°). | Dewar and Fleming (+100° to −200°). |
| | p. | 2c. | p. | 2c. | p. | 2c. | p. | 2c. |
| Aluminium | −0.56 | +.0039 | −0.42 | +.0021 | −0.41 | +.00174 | −0.394 | +.00398 |
| Antimony | . . | . . | +42.83 | +.1450* | . . | . . | +3.210 | +.02817 |
| Bismuth | . . | . . | . . | . . | . . | . . | −76.870 | −.08480 |
| Cadmium | +4.75 | +.0429 | +4.79 | +.0389 | +4.71 | +.0339 | +4.792 | +.02365 |
| Carbon | . . | . . | . . | . . | . . | . . | +12.795 | +.03251 |
| Copper | +1.81* | +.0095 | +3.37 | +.0122 | +3.22 | +.0080 | +3.156 | +.00683 |
| Cobalt | . . | . . | . . | . . | −19.252 | −.0734 | . . | . . |
| Gold | +3.30 | +.0102 | +3.19 | +.0131 | +3.10 | +.0063 | +1.161 | +.00315 |
| Iron | +14.74 | −.0487 | . . | . . | +11.835 | −.0306 | +14.522 | −.01330 |
| Steel (piano) | +9.75 | −.0328 | . . | . . | . . | . . | +9.600 | −.01092 |
| Steel (Mn 12%) | . . | . . | . . | . . | . . | . . | −5.73 | −.00445 |
| Magnesium | +1.75* | −.0095 | . . | . . | −0.113 | +.0019 | −0.126 | +.00353 |
| Mercury | . . | . . | . . | . . | −4.03 | −.0086 | . . | . . |
| Nickel | −24.23* | −.0512 | . . | . . | −20.58 | −.0302 | −18.87 | −.05639 |
| Palladium | −8.04 | −.0359 | . . | . . | . . | . . | −9.100 | −.04714 |
| Platinum | −1.15* | −.0110 | . . | . . | −4.09 | −.0211 | −4.347 | −.03708 |
| Silver | +2.86 | +.0150 | +3.07 | +.0115 | +2.68 | +.0076 | +3.317 | +.00714 |
| Thallium | . . | . . | +1.76 | −.0077 | . . | . . | . . | . . |
| Tin | −0.16 | +.0055 | −0.091 | +.0004 | −0.067 | +.0019 | +0.057 | +.00021 |
| Zinc | +3.51 | +.0240 | −1.77* | +.0195 | +3.318 | +.0172 | +3.233 | +.01040 |
Explanation of Table.—The figures marked with an asterisk (*) represent discrepancies which are probably caused by impurities in the specimens. At the time of Tait’s work in 1873 it was difficult, if not impossible, in many cases to secure pure materials. The work of the other three observers dates from 1894–95. The value of the thermoelectric power dE/dt at 50° C. is taken as the mean value between 0° and 100° C., over which range it can be most accurately determined. The values of d $^2$E/dt$^2$ agree as well as can be expected, considering the difference of the ranges of temperature and the great variety in the methods of observation adopted; they are calculated assuming the parabolic formula, which is certainly in many cases inadequate. Noll’s values apply to the temperature of +100° C., Dewar and Fleming’s to that of –100° C., approximately.
In using the above table to find the value of E or dE/dt at any temperature or between any limits, denoting by p the value of dE/dt at 50° C., and by 26 the constant value of the second coefficient, we have the following equations:—
| $dE/dt=p+2c(t-50)$, at any temperature $t$, Cent. | (3) |
| $E(_{t-t')=(t-t')(p+c(t+t'-100))$ | (4) |
for the E.M.F. between any temperature $t$ and $t'$.
8. Methods of Observation.—In Tait’s observations the E.M.F. was measured by the deflection of a mirror galvanometer, and the temperature by means of a mercury thermometer or an auxiliary thermocouple. He states that the deviations from the formula were “quite within the limits of error introduced by the alteration of the resistance of the circuit with rise of temperature, the deviations of the mercury thermometers from the absolute scale, and the non-correction of the indications of the thermometer for the long column of mercury not immersed in the hot oil round the junctions.” The latter correction may amount to about 10° C. at 350° Later observers have generally employed a balance method (some modification of the potentiometer or Poggendorf balance) for measuring the E.M.F. The range of Steele’s observations was too small to show any certain deviation from the formula, but he notes capricious changes attributed to change of condition of the wires. Noll employed mercury thermometers, but as he worked over a small range with vapour baths, it is probable that he did not experience any trouble from immersion corrections. He does not record any systematic deviations from the formula. Dewar and Fleming, working at very low temperatures, were compelled to use the platinum thermometer, and expressed their results in terms of the platinum scale. Their observations were probably free from immersion errors, but they record some deviations from the formula which they consider to be beyond the possible limits of error of their work. The writer has reduced their results to the scale of the gas thermometer, assuming the boiling-point of oxygen to be −182.5° C.
9. Peltier Effect.—The discovery by J. C. A. Peltier (1834) that heat is absorbed at the junction of two metals by passing a current through it in the same direction as the current produced by heating it, was recognized by joule as affording a clue to the source of the energy of the current by the application of the principles of thermodynamics. Unlike the frictional generation of heat due to the resistance of the conductor, which Joule (1841) proved to be proportional to the square of the current, the Peltier effect is reversible with the current, and being directly proportional to the first power of the current, changes sign when the current is reversed. The effect is most easily shown by connecting a voltaic cell to a thermophile for a short interval, then quickly (by means of a suitable key, such as a Pohl commutator with the cross connectors removed) disconnecting the pile from the cell and connecting it to a galvanometer, which will indicate a current in the reverse direction through the pile, and approximately proportional to the original current in intensity, provided that the other conditions of the experiment are constant. It was by an experiment of this kind that Quintus Icilius (1853) verified the proportionality of the heat absorbed or generated to the first power of the current. It had been observed by Peltier and A. E. Becquerel that the intensity of the effect depended on the thermoelectric power of the junction and was independent of its form or dimensions. The order of the metals in respect of the Peltier effect was found to be the same as the thermoelectric series. But on account of the difficulty of the measurements involved, the verification of the accurate relation between the Peltier effect and thermoelectric power was left to more recent times. If C is the intensity of the current through a simple thermocouple, the junctions of which are at temperatures t and t′, a quantity of heat, P×C, is absorbed by the passage of the current per second at the hot junction, t, and a quantity, P′×C, is evolved at the cold junction, t′ The coefficients, P and P′, are called coefficients of the Peltier effect, and may be stated in calories or joules per ampere-second. The Peltier coefficient may also be expressed in volts or micro volts, and may be regarded as the measure of an E.M.F. located ​ at the junction, and transforming heat into electrical energy or vice versa. If R is the whole resistance of the circuit, and E the of the couple, and if the flow of the current does not produce any other thermal effects in the circuit besides the Joule and Peltier effects, we should find by applying the principle of the conservation of energy, i.e. by equating the balance of the heat absorbed by the Peltier effects to the heat generated in the circuit by the Joule effect,
| $(P-P')C=C^{2}R=EC,$ whence $E=P-P'$ | (5) |
If we might also regard the couple as a reversible thermodynamic engine for converting heat into work, and might neglect irreversible effects, such as conduction, which are independent of the current, we should expect to find the ratio of the heat absorbed at the hot junction to the heat evolved at the cold junction, namely, P/P ′, to be the same as the ratio T/T ′ of the absolute temperatures of the junctions. This would lead to the conclusion given by R. J. E. Clausius (1853) that the Peltier effect varied directly as the absolute temperature, and that the of the couple should be directly proportional to the difference of temperature between the junctions.
* Fig. 1.—Diagram of Apparatus for
* Demonstrating the Thomson Effect.
10. Thomson Effect.—Thomson (Lord Kelvin) had already pointed out (Proc. R.S. Edin., 1851) that this conclusion was inconsistent with the known facts of thermoelectric inversion. (1) The was not a linear function of the temperature difference. (2) If the Peltier effect was proportional to the thermoelectric power and changed sign with it, as all experiments appeared to indicate, there would be no absorption of heat in the circuit due to the Peltier effect, and therefore no thermal source to account for the energy of the current, in the case) in which the hot junction was at or above the neutral temperature. He therefore predicted that there must be a reversible absorption of heat in some other part of the circuit due to the flow of the current through the unequally heated conductors. He succeeded a few years afterwards in verifying this remarkable prediction by the experimental demonstration that a current of positive electricity flowing from hot to cold in iron produced an absorption of heat, as though it possessed negative specific heat in the metal iron. He also succeeded in showing that a current from hot to cold evolved heat in copper, hut the effect was smaller and more difficult to observe than in iron.
The Thomson effect may be readily demonstrated as a lecture experiment by the following method (fig. 1). A piece of wire (No. 28) about 4 cm. long is soldered at either end A, B to thick wires (No. 12), and is heated 100° to 150° C. by a steady current from a storage cell adjusted by a suitable rheostat. The experimental wire AB is connected in parallel with about 2 metres of thicker wire (No. 22), which is not appreciably heated. A low resistance galvanometer is connected by a very fine wire (2 to 3 mils) to the centre C of the experimental wire AB, and also to the middle point D of the parallel wire so as to form a Wheatstone bridge. The balance is adjusted by shunting either AD or BD with a box, S, containing 20 to 100 ohms. All the wires in the quadrilateral must be of the same metal as AB, to avoid accidental thermoelectric effects which would obscure the result. If the current flows from A to B there will be heat absorbed in AC and evolved in CB by the Thomson effect, if the specific heat of electricity in AB is positive as in copper. When the current is reversed, the temperature of AC will be raised and that of CB lowered by the reversal of the effect. This will disturb the resistance balance by an amount which can be measured by the deflection of the galvanometer, or by the change of the shunt-box, S, required to restore the balance. Owing to the small size of the experimental wire, the method is very quick and sensitive, and the apparatus can be set up in a few minutes when once the experimental quadrilaterals have been made. It works very well with platinum, iron and copper. It was applied with elaborate modifications by the writer in 1886 to determine the value of the Thomson effect in platinum in absolute measure, and has recently been applied with further improvements by R. O. King to measure the effect in copper.
11. Thomson’s Theory.—Taking account of the Thomson effect, the thermodynamical theory of the couple was satisfactorily completed by Thomson (Trans. R. S. Edin., 1854). If the quantity of heat absorbed and converted into electrical energy, when unit quantity of electricity (one ampere-second) flows from cold to hot through a difference of temperature, dt, be represented by sdt, the coefficient s is called the specific heat of electricity in the metal, or simply the coefficient of the Thomson effect. Like the Peltier coefficient, it may be measured in joules or calories per ampere-second per degree, or more conveniently and simply in micro volts per degree.
Consider an elementary couple of two metals A and B for which s has the values s′ and s″ respectively, with junctions at the temperature. T and T+dT (absolute), at which the coefficients of the Peltier effect are P and P+dP. Equating the quantity of heat absorbed to the quantity of electrical energy generated, we have by the first law of thermodynamics the relation
| $dE/dT=dP/dT+(s'-s'')$ | (6) |
If we apply the second law, regarding the couple as a reversible engine, and considering only the reversible effects, we obtain
| $(s'-s'')/T=-d(P/T)/dT$ | (7) |
Eliminating (s′−s″) we find for the Peltier effect
| $P=TdE/dT=Tp$ | (8) |
Whence we obtain for the difference of the specific heats
| $(s'-s'')=Td^{2}E/dT^{2}=-Tdp/dT$ | (9) |
From these relations we observe that the Peltier effect P, and the difference of the Thomson effects (s′−s″), for any two metals are easily deduced from the tabulated values of dE/dt and d $^2$E/dt$^2$ respectively. The signs in the above equations are chosen on the assumption that positive electricity flows from cold to hot in the metal s'. The signs of the Peltier and Thomson effects will be the same as the signs of the coefficients given in Table I., if we suppose the metal s′ to be lead, and assume that the value of s′ may be taken as zero at all temperatures.
12. Experimental Verification of Thomson’s Theory.—In order to justify the assumption involved in the application of the second law of thermodynamics to the theory of the thermocouple in the manner above specified, it would be necessary and sufficient, as Thomson pointed out (Phil. Mag., December 1852), to make experiments to verify quantitatively the relation P/T=dE/dT between the Peltier effect and the thermoelectric power. A qualitative relation was known at that time to exist, but no absolute measurements of sufficient accuracy had been made. The most accurate measurements of the heat absorption due to the Peltier effect at present available are probably those of H. M. Jahn (Wied. Ann., 34, p. 755, 1888). He enclosed various metallic junctions in a Bunsen ice calorimeter, and observed the evolution of heat per hour with a current of about 1.6 amperes in either direction. The Peltier effect was only a small fraction of the total effect, but could be separated from the Joule effect owing to the reversal of the current. The values of dE/dT for the same specimens of metal at 0° C. were determined by experiments between +20° C. and −20° C. The results of his observations are contained in the following table, heat absorbed being reckoned positive as in Table I.:—
Table II.
| Thermo- couple. | dE/dT Microvolts per deg. | P=Td E/dT Microvolts at C. | Heat calc. Calories per hour. | Heat observed Calories per hour. |
| Cu-Ag | +2.12 | +579 | +0.495 | +0.413 |
| Cu-Fe | +11.28 | +3079 | +2.640 | +3.163 |
| Cu-Pt | −1. 40 | −382 | −0.327 | −0.320 |
| Cu-Zn | +1.51 | +412 | +0.353 | +0.585 |
| Cu-Cd | +2.64 | +721 | +0.617 | +0.616 |
| Cu-Ni | −20.03 | −5468 | −4.680 | −4.362 |
The agreement between the observed and calculated values in the last two columns is as good as can be expected considering the great difficulty of measuring such small quantities of heat. The analogous reversible heat effects which occur at the junction of a metal and an electrolyte were also investigated by Jahn, but he did not succeed in obtaining so complete an agreement with theory in this case.
13. Tait’s Hypothesis.—From general considerations concerning minimum dissipation of energy (Proc. R. S. Edin., 1867–68), Tait was led to the conclusion that “the thermal and electric conductivities of metals varied inversely as the absolute temperature, and that the specific heat of electricity was directly proportional to the same.” Subsequent experiments led him to doubt this conclusion as regards conductivity, but his thermoelectric experiments (Proc. R. S. Edin., December 1870) appeared to be in good agreement with it. If we adopt this hypothesis, and substitute s= 2cT, where c is a constant, in the fundamental equation (9), we obtain at once d$^2$E/dT$^2$= −2 (c′−c″), which is immediately integrable, and gives
| $dE/dt=p=2(t_{0}-t)(c'-c'')$ | (10) |
| $E_{t-t'}=(t-t')(c'-c'')\lbrace 2t_{0}-(t+t')\rbrace$ | (11) |
where to is the temperature of the neutral point at which dE/dt=0. This is the equation to a parabola, and is equivalent to the empirical formula of Avenarius, with this difference, that in Tait’s formula the constants have all a simple and direct interpretation in relation to the theory. Tait’s theory and formula were subsequently assimilated by Avenarius (Pogg. Ann., 140, p. 372, 1873), and are now generally attributed to Avenarius in foreign periodicals.
* Fig. 2.—Temperature by Thermocouple. Difference from Tait’s Formula.
In accordance with this hypothesis, the curves representing the variations of thermoelectric power, dE/dt, with temperature are straight lines, the slope of which for any couple is equal to the difference of the constants 2(c′−c″). The diagram constructed by Tait on this principle is fully explained and illustrated in many text-books, and has been generally adopted as embodying in a simple form the fundamental phenomena of thermoelectricity.
14. Experimental Verification.—Tait’s verification of this hypothesis consisted in showing that the experimental curves of E.M.F. were parabolas in most cases within the limits of error of his observations. He records, however, certain notable divergences, particularly in the case of iron and nickel, and many others have since come to light from other observations. It should also be remarked that even if the curves were not parabolas, it would always be possible to draw parabolas to agree closely with the observations over a restricted range of temperature. When the question is tested more carefully, either by taking more accurate measurements of temperature, or by extending the observations over a wider range, it is found that there are systematic deviations from the parabola in the majority of cases, which cannot be explained by errors of experiment. A more accurate verification of these relations, both at high and low extremes of temperature, has become possible of late years owing to the development of the theory and application of the platinum resistance thermometer. (See Thermometry The curves in fig. 2 illustrate the differences from the parabolic formula, measured in degrees of temperature, as observed by H. M. Tory (B.A. Report, 1897). The deviations for the copper iron couple, and for the copper cast-iron couple over the range 0° to 200° C., appear to be of the order of 1° C., and were careful y verified by repeated and independent series of observations. The deviations of the platinum and platinum-rhodium 10 per cent. couple over the range 0° to 1000° C. are shown on a smaller scale, and are seen to be of a similar nature, but rather greater in proportion. It should be observed that these deviations are continuous, and differ in character from the abrupt changes observed by Tait in special cases. A number of similar deviations at temperatures below 0° C. were found by the writer in reducing the curves re resenting the observations of Dewar and Fleming (Phil. Mag., July 1895) to the normal scale of temperature from the platinum scale in which they are recorded. In many cases the deviations do not appear to favour any simple hypothesis as to the mode of variation of s with temperature, but as a rule the indication is that s is nearly constant, or even diminishes with rise of temperature. It may be interesting therefore to consider the effect of one or two other simple hypotheses with regard to the mode of variation of s with T .
15. Other Assumptions.—If we take the analogy of a perfect gas and assume s=constant, we have
| $dE^{2}/dT^{2}=-s/T,dE/dT=s$ log$_{e}T_{0}/T$ | (12) |
| $E_{(T-T')}=sTlog,T_{0}/T--sT'$ log$_{e}T_{0}/T$ | (13) |
where $T$ and $T'$ are the temperatures of the junctions, and $T_{0}$ is the neutral temperature. These formulae are not so simple and convenient as Tait’s, though apparently founded on a more simple assumption, but they frequently represent the observations more closely. If we suppose that s is not quite constant, but increases or diminishes slightly with change of temperature according to a linear formula $s=s_{0}+2cT$ (in which 50 represents the constant part of s, and c may have either sign), we obtain a more general formula which is evidently the sum of the two previous solutions and may be made to cover a greater variety of cases. Another simple and possible assumption is that made by A. Stansfield (Phil. Mag., July 1898), that the value of s varies inversely as the absolute temperature. Putting $s=c/T$, we obtain
| $E_{(T-T')}=c$ log$_{e}T/T'-c(T-T')/T_{0}$ | (14) |
which is equivalent to the form given by Stansfield, but with the neutral temperature $T_{0}$ explicitly included. According to this formula, the Peltier effect is a linear function of the temperature. It may appear at first sight astonishing that it should be possible to apply so many different assumptions to the solution of one and the same problem. In many cases a formula of the last type would be quite inapplicable, as Stansfield remarks, but the difference between the three is often much less than might be supposed. For instance, in the case of 10 per cent. Rh. Pt.—Pt. couple, if we calculate three formulae of the above types to satisfy the same pair of observations at 0°—445° and 0°—1000° C., we shall find that the formula s=constant lies midway between that of Tait and that of Stansfield, but the difference between the formulae is of the same order as that between different observers. In this particular case the parabolic'formula appears to be undoubtedly inadequate. The writer’s observations agree more nearly with the assumption s=constant, those of Stansfield with s=c/T Many other formulae have' been suggested L. F. C. Holborn and A. Day (Berl. Akad., 1899) have one back to Tait's method at high temperatures, employing arcs of parabolas for limited ranges. But since the parabolic formula is certainly erroneous at low temperatures, it can hardly be trusted for extrapolation above 1000° C.
* Fig. 3.—Thomson Effect. Batelli (Le Roux’s Method).
16. Absolute Measurement of Thomson Effect.-Another method of verifying Tait’s hypothesis, of greater difficulty but of considerable interest, is to measure the absolute value of the heat absorbed by the Thomson effect, and to observe whether or not it varies with the temperature. Le Roux (Ann. Chim. Phys., x. p. 201, 1867) made a number of relative measurements of the effect in different metals, which agreed qualitatively with observations of the thermoelectric power, and showed that the effect was proportional to the current for a given temperature gradient. Batelli has applied the same method (Accad. Sci. Turin, 1886) to the absolute measurement. He observed with a thermocouple the difference of temperature (about .01° C.) produced by the Thomson effect in twenty minutes between two mercury calorimeters, B$_1$ and B$_2$, surrounding the central portions of a pair of rods arranged as in Le Roux’s method (see fig. 3). The value of the Thomson effect was calculated by multiplying this difference of temperature by the thermal capacity of either calorimeter, and dividing by the current, by they number of seconds in twenty minutes, and by twice the difference of temperature (about 20°) between the ends a and b of either calorimeter. The method appears to be open to the objection that, the difference of temperature reached in so long an interval would be more or less independent of the thermal ​ capacities of the calorimeters, and would also be difficult to measure accurately with a thermocouple under the conditions described. The general results of the work appeared to support Tait's hypothesis that the effect was proportional to the absolute temperature, but direct thermoelectric tests do not appear to have been made on the specimens employed, which would have afforded a valuable confirmation by the comparison of the values of d$^2$E/dT$^2$, as in Jahn’s experiments.
* Fig. 4.—Potential Diagrams of Thermocouple on the Contact Theory.
17. King’s Experiments.—The method employed by the writer, to which allusion has already been made, consisted in observing the change of distribution of temperature in terms of the resistance along a wire heated by an electric current, when the heating current was reversed. It has been fully described by King (Proc. Amer. Acad., June 1898), who applied it most successfully to the case of copper. Although the effect in copper is so small, he succeeded in obtaining changes of temperature due to the Thomson effect of the order of 1° C., which could be measured with satisfactorily accuracy. He also determined the effect of change of temperature distribution on the rate of generation of heat by the current; and on the external loss of heat by radiation, convection and conduction. It is necessary to take all these conditions carefully into account in calculating the balance due to the Thomson effect. According to King’s experiments, the value of the effect appears to diminish with rise of temperature to a slight extent in copper, but the diminution is so small that he does not regard it as established with certainty. The value found at a temperature of 150° C. was +2.5 microjoules per ampere-second per degree, or +2.5 microvolts per degree in the case of copper, which agrees very fairly with the value deduced from thermoelectric tests. The value found by Batelli for iron was −5.0 microvolts per degree at 108° C., which appears too small in comparison. These measurements, though subject to some uncertainty on account of the great experimental difficulties, are a very valuable confirmation of the accuracy of Thomson’s theory, because they show that the magnitude of the effect is of the required order, but they cannot be said to be strongly in support of Tait’s hypothesis. A comparison of the results of different observers would, also suggest that the law of variation may be different in different metals, although the differences in the values of d $^2$E/dT$^2$ may be due in part to differences of purity or errors of observation. It would appear, for instance, according to the observations of Dewar and Fleming, that the value of s for iron is positive below −150° C., at which point it vanishes. At ordinary temperatures the value is negative, increasing rapidly in the negative direction as the temperature rises. This might be appropriately represented, as already suggested, by a linear formula s=s$_0$−CT.
18. Potential Diagrams on the Contact Theory.—It is instructive to consider the distribution of potential in a thermoelectric circuit, and its relation to the resultant E.M.F. and to the seat of the In fig. 4, which is given as an illustration, the cold junctions are supposed to be at 0° C. and the hot junctions at 100° C. Noll’s values (Table I.) are taken for the E.M.F., and it is supposed that the coefficient of the Thomson effect is zero in lead, Le. that there is no E.M.F. and that the potential is uniform throughout the length of the lead wire. Taking the lead-iron couple as an example, the value of dE/dt at the hot junction 100° C. is 10.305 microvolts per degree, and the value of the Peltier coefficient P = TdE/dT is +3844 microvolts. In other words, we may suppose that there is an E.M.F. of that magnitude situated at the junction which causes positive electricity to flow from the lead to the iron. If the circuit is open, as represented in the diagram, the flow will cease as soon as it has raised the potential of the iron 3844 microvolts above that of the lead. In the substance of the iron itself there is an E.M.F. due to the Thomson effect of about 10 microvolts per degree tending to drive positive electricity from hot to cold, and raising the cold end of the iron 989 microvolts in potential above the hot end on open circuit. At the cold junction the iron is supposed to be connected to a piece of lead at 0° C., and there is a sudden drop of potential due to the Peltier effect of 3648 microvolts. If the circuit is cut at this point, there remains a difference of potential E = 1184 microvolts, the resultant E.M.F. of the circuit, tending to drive positive electricity from the iron to the lead across the cold junction. If the circuit is closed, there will be a current C =E/R, where R=R′ +R″, the sum of the resistances of the lead and iron. The flow of the current will produce a fall of potential ER′/R in the lead from cold to hot, and ER″/R in the iron from hot to cold, but the potential difference due to the Peltier effect at either 'unction will not be affected. For simplicity in the diagram the temperature gradient has been taken as uniform, and the specific heat s=constant, but the total P.D. would be the same whatever the gradient.
Similar diagrams are given in fig. 4 for cadmium in which both the specific heat and the Peltier effect are positive, and also for platinum and nickel in which both coefficients are negative. The metals are supposed to be all joined together at the hot junction, and the circuit cut in the lead near the cold junction. The diagram will serve for any selected couple, such as iron-nickel, and is not restricted to combinations with lead. The following table shows the component parts of the E.M.F. in each case:—
TABLE III. :
| Thermocouple. | P$_{100}$ − | −P$_0$ | −100×s$_{50}$ | E$_{0⋅100}$ |
| Iron-lead | +3844− | +3648− | −988 = | +1184 |
| Cadmium-lead | +2389− | +823− | +1095 = | +471 |
| Platinum-lead | −1919− | −828− | −682 = | −409 |
| Nickel-lead | −8239 − | −5206− | −975 = | −2058 |
The components for any other combination of two are found by taking the algebraic difference of the values with respect to lead.
19. Relation to the Volta Effect.—It is now generally conceded that the relatively large differences of potential observable with an electrometer between metals on open circuit, as discovered by Volta, are due to the chemical affinities of the metals, and have no direct relation to thermoelectric phenomena or to the Peltier effect. The order of the metals in respect of the two effects is quite different. The potential difference, due to the Volta effect in air, has been shown by Thomson (Lord Kelvin) and his pupils to be of the same order of magnitude, if not absolutely the same, as that produced in a dilute electrolyte in which two metallic ally connected plates (e.g. zinc and copper) are immersed. (On this hypothesis, it may be explained by regarding the air as an electrolyte of infinite specific resistance.) It is also profoundly modified by the state of the exposed surfaces, a coating of oxide on the copper greatly increasing the effect, as it would in a voltaic cell. The Peltier effect and the thermo-E.M.F., on the other hand, do not depend on the state of the surfaces, but only on the state of the substance. An attempt has been made to explain the Volta effect as due to the affinity of the metals for each other, but that would not account for the variation of the effect with the state of the surface, except as affecting the actual surface of contact. It is equally evident that chemical affinity between the metals cannot be the explanation of the Peltier E.M.F. This would necessitate chemical action at the junction when a current passed through it, as in an electrolytic cell, whereas the action appears to be purely thermal, and leads to a consistent theory on that hypothesis. The chemical action between metals in the solid state must be infinitesimal, and could only suffice to produce small charges analogous to those of frictional electricity; it could not maintain a permanent difference of potential at a metallic junction through which a current was passing. Although it is possible that differences of potential larger than the Peltier effect might exist between two metals in contact on open circuit, it is certain that the only effective E.M.F. in practice is the ​ Peltier effect, and that the difference of potential in the substance of the metals when the circuit is complete cannot be greater than the coefficient P. The Peltier effect, it may be objected, measures that part only of the potential difference which depends upon temperature, and can therefore give no information about the absolute potential difference. But the reason for concluding that there is no other effective source of potential difference at the junction besides the Peltier effect, is simply that no other appreciable action takes place at the junction when a current passes except the Peltier generation or absorption of heat.
20. Convection Theory.—The idea of convection of heat by an electric current, and the phrase “specific heat of electricity” were introduced by Thomson as a convenient mode of expressing the phenomena of the Thomson effect. He did not intend to imply that electricity really possessed a positive or negative specific heat, but merely that a quantity of heat was absorbed in a metal when unit quantity of electricity flowed from cold to hot through a difference of temperature of 1°. The absorption of heat was considered as representing an equivalent conversion of heat energy into electrical energy in the element. The element might thus be regarded as the seat of an E.M.F., dE=sdT, where dT is the difference of temperature between its ends. The potential diagrams already given have been drawn on this assumption, that the Thomson effect is not really due to convection of heat by the current, but is the measure of an E.M.F. located in the substance of the conductor. This view with regard to the seat of the E.M.F. has been generally taken by the majority of writers on the subject. It is not, however, necessarily implied in the reasoning or in the equations given by Thomson, which are not founded on any assumptions with regard to the seat of the E.M.F., but only on the balance of heat absorbed and evolved in all the different parts of the circuit. In fact, the equations themselves are open to an entirely different interpretation in this respect from that which is generally given.
Returning again to the equations already given in § 11 for an elementary thermocouple, we have the following equivalent expressions for the E.M.F. dE, namely,
$dE=dP+(s'-s'')dT=(P/T)dT=pdT=<p''-p')dT,$
in which the coefficient, P, of the Peltier effect, and the thermoelectric power, p, of the couple, may be expressed in terms of the difference of the thermoelectric powers, p′ and p″, of the separate metals with respect to a neutral standard. So far as these equations are concerned, we might evidently regard the seat of the as located entirely in the conductors themselves, and not at all at the junctions, if p or (p′ −p′) is the difference of the E.M.F.s per degree in corresponding elements of the two metals. In this case, however, in order to account for the phenomenon of the Peltier effect at the junctions, it is necessary to suppose that there is a real convection of heat by an electric current, and that the coefficient P or pr is the difference of the quantities of heat carried by unit quantity of electricity in the two metals. On this hypothesis, if we confine our attention to one of the two metals, say p″, in which the current is supposed to flow from hot to cold, we observe that p″dT expresses the quantity of heat converted into electrical energy per unit of electricity by an E.M.F. p″ per 1° located in the element dT. It happens that the absolute magnitude of p″ cannot be experimentally determined, but this is immaterial, as we are only concerned with differences. The quantity of heat liberated by convection as the current flows from hot to cold is represented in the equation by dP=d(pT). Since d(p″T)=p″dT+Tdp″, it is clear that the balance of heat liberated in the element is only Tdp″=s″dT, namely, the Thomson effect, and is not the equivalent of the E.M.F. p″dT, because on this theory the absorption of heat is masked by the convection. If p is constant there is no Thomson effect, but it does not follow that there is no E.M.F. located in the element. The Peltier effect, on the other hand, may be ascribed entirely to convection. The quantity of heat p″T is brought up to one side of the junction per unit of electricity, and the quantity of heat p′T taken away on the other. The balance (p″−p′)T is evolved at the junction. If, therefore, we are prepared to admit that an electric current can carry heat, the existence of the Peltier effect is no proof that a corresponding is located at the junction, or, in other words, that the conversion of heat into electrical energy occurs at this point of the circuit, or is due to the contact of dissimilar metals. On the contact theory, as generally adopted, the is due entirely to change of substance (dP−Tdp); on the convection theory, it is due entirely to change of temperature (pdT). But the two expressions are equivalent, and give the same results.
21. Potential Diagrams on Convection Theory.—The difference between the two theories is most readily appreciated by drawing the potential diagrams corresponding to the supposed locations of the E.M.F. in each case. The contact theory has been already illustrated in fig. 4. Corresponding diagrams for the same metals on the convection theory are given in fig. 5. In this diagram the metals are supposed to be all joined together and to be at the same time potential at the cold junction at 0° C. The ordinate of the curve at any temperature is the difference of potential between any point in the metal and a point in lead at the same temperature. Since there is no contact E.M.F. on this theory, the ordinates also represent the E.M.F. of a thermocouple metal-lead
* Fig. 5.—Curves of Thermo-E.M.F., or Potential Diagrams, on the Convection Theory.
in which one junction is at 0° C. and the other at t° C. For this reason the potential diagrams on the convection theory are more simple and useful than those on the contact theory. The curves of E.M.F. are in fact the most natural and most convenient method of recording the numerical data, more particularly in cases where they do not admit of being adequately represented by a formula. The line of lead is taken to be horizontal in the diagram, because the thermoelectric power, p, may be reckoned from any convenient zero. It is not intended to imply that there is no E.M.F. in the metal-lead with change of temperature, but that the value of p in this metal is nearly constant, as the Thomson effect is very small. It is very probable that the absolute values of p in different metals are of the same sign and of the same order of magnitude, being large compared with the differences observed. It would be theoretically possible to measure the absolute value in some metal by observing with an electrometer the P.D. between parts of the same metal at different temperatures, but the difference would probably be of the order of only one-hundredth of a volt for a difference of 100° C. It would be sufficiently difficult to detect so small a difference under the best conditions. The difficulty would be greatly increased, if not rendered practically insuperable, by the large difference of temperature.
22. Conduction Theory.—In Thomson's theory it is expressly assumed that the reversible thermal effects may be considered Separately without reference to conduction. In the conduction theory of F. W. G. Kohlrausch (Pogg. Ann., 1875, Vol. 156, p. 601), the fundamental postulate is that the thermo-E.M.F. is due to the conduction of heat in the metal, which is contrary to Thomson s theory. It is assumed that a flow of heat Q, due to conduction, tends to carry with it a proportional electric current C =aQ. This is interpreted to mean that there is an E.M.F. dE= −akr dT= −θdT, in each element, where k is the thermal conductivity and ​$r$ the specific resistance. The “thermoelectric constant,” θ, of Kohlrausch, is evidently the same as the thermoelectric power, $p$, in Thomson's theory. In order to explain the Peltier effect, Kohlrausch further assumes that an electric current, $C$, carries a heat-flow, $Q=A\theta C$, with it, where “$A$ is a constant which can be made equal to unity by a proper choice of units.” If $A$ and $Q=A\theta C$ are constant, the Peltier effects at the hot and cold junctions are equal and opposite, and may therefore be neglected. The combination of the two postulates leads to a complication. By the second postulate the flow of the current increases the heat-flow, and this bi the first postulate increases the E.M.F., or the resistance, which therefore depends on the current. It is difficult to see how this complication can be avoided, unless the first postulate is abandoned, and the heat-How due to conduction is assumed to be independent of the thermoelectric phenomena. By applying the first law of thermodynamics, Kohlrausch deduces that a quantity of heat, C6dT, is absorbed in the element $dT$ per second by the current $C$. He wrongly identifies this with the Thomson effect, by omitting to allow for the heat carried. He does not make any application of the second law to the theory. If we apply Thomson's condition $P=TdE/dT=Tp$, we have $A=T$. If we also assume the ratio of the current to the heat-flow to be the same in both postulates, we have $a=1/T\theta$, whence $\theta ^{2}=kr/T$. This condition was applied in 1899 by C. H. J. B. Liebenow (Wied. Ann., 68, p. 316). It simplifies the theory, and gives a possible relation between the constants, but it does not appear to remove the complication above referred to, which seems to be inseparable from any conduction theory.
L. Boltzmann (Sitz. Wien. Akad., 1887, vol. 96, p. 1258) gives a theoretical discussion of all possible forms of expression for thermoelectric phenomena. Neglecting conduction, all the expressions which he gives are equivalent to the equations of Thomson. Taking conduction into account in the application of the second law of thermodynamics; he proposes to substitute the inequality, $Td/dET-p\leq {\sqrt {T}}({\sqrt {k'r'}}+{\sqrt {k''r''}})$, instead of the equation given by Thomson, namely, $P=TdE/dT$. Since, however, Thomson's equation has been so closely verified by Jahn, it is probable that Boltzmann would now consider that the reversible effects might be treated independently of conduction.
23. Thermoelectric Relations.—A number of suggestions have been made as to the possible relations between heat and electricity, and the mechanism by which an electric current might also be a carrier of heat. The simplest is probably that of W. E. Weber (Wied. Ann., 1875), who regarded electricity as consisting of atoms much smaller than those of matter, and supposed that heat was the kinetic energy of these electric atoms. If we suppose that an electric current in a metal is a flow of negative electric atoms in one direction, the positive electricity associated with the far heavier material atoms remaining practically stationary, and if the atomic heat of electricity is of the same order as that of an equivalent quantity of hydrogen or any other element, the heat carried per ampere-second at 0° C., namely $P$, would be of the order of .030 of a joule, which would be ample to account for all the observed effects on the convection theory. Others have considered conduction in a metal to be analogous to electrolytic conduction, and the observed effects to be due to “migration of the ions.” The majority of these theories are too vague to be profitably discussed in an article like the present, but there can be little doubt that the study of thermoelectricity affords one of the most promising roads to the discovery of the true relations between heat and electricity.
Alphabetical Index of Symbols.
    $a,b,c$ = Numerical constants in formulae.
    $C$ = Electric Current.
    $E$ = E.M.F. = Electromotive Force.
    $k$ = Thermal Conductivity.
    $P$ = Coefficient of Peltier Effect.
    $p=dE/dt$ = Thermoelectric Power.
    $O$ = Heat-flow due to Conduction.
    $R$ = Electrical Resistance; r, Specific Resistance.
    $s$ = Specific Heat, or Coefficient of Thomson Effect.
    $t$ = Temperature on the Centigrade Scale.
    $T$ = Temperature on the Absolute Scale.
(H. L. C.)