# 1911 Encyclopædia Britannica/Thermoelectricity

**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

(1) |

2. *Thermoelectric Power*, *Series*, *Inversion*.—The limiting
value, *d*E/*d*t, 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

&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*°+2*cT*, and*c*is small. See sec. 15.)

- (Where

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*/2*c*, which gives
the neutral temperature. We have also the relations *dE*/*dt*=*b*+2*ct*
and *d* ^{2}*E*/*dt*^{2}=2*c*. 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.

*p*=

*dE*/

*dt*, in microvolts at 50° C. of pure metals with respect to lead. (The mean change,

2

*c*=

*d*

^{2}

*E*/

*dt*

^{2}, of the thermoelectric power per degree C. over the range covered by the experiments, is added in each case.)

Metal. | Tait (0° to 300°) | Steele (0° to 100°). | Noll (0° 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:—

, at any temperature , Cent. | (3) |

(4) |

for the E.M.F. between any temperature and .

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,

whence | (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

(6) |

If we apply the second law, regarding the couple as a reversible engine, and considering only the reversible effects, we obtain

(7) |

Eliminating (*s*′−*s*″) we find for the Peltier effect

(8) |

Whence we obtain for the difference of the specific heats

(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/dTMicrovolts per deg. | P=Td E/dTMicrovolts 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*= 2*cT*, 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

(10) |

(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

log | (12) |

log | (13) |

where and are the temperatures of the junctions, and 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 (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 , we obtain

log | (14) |

which is equivalent to the form given by Stansfield, but with the
neutral temperature 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,

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*=
−θ*dT*, in each element, where *k* is the thermal conductivity and the specific resistance. The “thermoelectric constant,” θ, of
Kohlrausch, is evidently the same as the thermoelectric power, ,
in Thomson's theory. In order to explain the Peltier effect, Kohlrausch
further assumes that an electric current, , carries a heat-flow,
, with it, where “ is a constant which can be made
equal to unity by a proper choice of units.” If and 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 per second by the current .
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
, we have . If we also assume the ratio
of the current to the heat-flow to be the same in both postulates,
we have , whence . 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,
, instead of the equation given
by Thomson, namely, . 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 ,
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.*

- = Numerical constants in formulae.
- = Electric Current.
- = E.M.F. = Electromotive Force.
- = Thermal Conductivity.
- = Coefficient of Peltier Effect.
- = Thermoelectric Power.
- = Heat-flow due to Conduction.
- = Electrical Resistance; r, Specific Resistance.
- = Specific Heat, or Coefficient of Thomson Effect.
- = Temperature on the Centigrade Scale.
- = Temperature on the Absolute Scale.

(H. L. C.)