Thermoelasticity and Thermoelectricity
In the second definition of the scale of absolute temperature just discussed, stress of any type may be substituted for pressure, and the corresponding displacement s for the change of volume. Thus for a piece of elastic material put through a cycle of changes we may substitute dS for dp and Ads for dv; where A is such a factor that AdSds is the work done in the displacement ds by the stress dS. As an example consider a wire subjected to simple longitudinal stress S. Longitudinal extension is produced, but this is not the only change; there is at the same time lateral contraction. However, s within certain limits is proportional to S.
Let heat dH in dynamical measure be given to the wire while the stress S is maintained constant, and let the extension increase from s to s + ds. The stress S will do work ASds on the wire, and the work ratio will be − ASds ⁄ dH. Now let the stress be increased to S + dS while the extension is kept constant, and the absolute temperature raised from T to T + dT. The stress ratio (as we may call it) is dS ⁄ S and the temperature ratio dT ⁄ T. Thus we obtain (p. [134] above)
In his Heat article Thomson used the alteration e of strain under constant stress (that is ds ⁄ l, where l is the length of the wire) corresponding to an amount of heat sufficient to raise the temperature under constant stress by 1°. Hence if K be the specific heat under constant stress, and le be put for ds in the sense just stated, we have
where ρ is the density, since dH = KρlA.
The ratio of dH to the increase ds of the extension is positive or negative, that is, the substance absorbs or evolves heat, when strained under the condition of constant stress, according as dS ⁄ dT is negative or positive. Or we may put the same thing in another way which is frequently useful. If a wire subjected to constant stress has heat given to it, ds is negative or positive, in other words the wire shortens or lengthens, according as dS ⁄ dT is positive or negative, that is, according as the stress for a given strain is increased or diminished by increase of temperature.
It is known from experiment that a metal wire expands under constant stress when heat is given to it, and thus we learn from the equation (F) that the stress required for a given strain is diminished when the temperature of the wire is raised. Again, a strip of india-rubber stretched by a weight is shortened if its temperature is raised, consequently the stress required for a given strain is increased by rise of temperature.
These results, from a qualitative point of view, are self-evident. But from what has been set forth it will be obvious that an equation exactly similar to (F) holds whether the change ds of s is taken as before under constant stress, or at uniform temperature, or whether the change dS of S is effected adiabatically or at constant strain.
In all these cases the same equation
applies, with the change of meaning of dT involved.
This equation differs from that of Thomson as given in various places (e.g. in the Encyclopædia Britannica article on Elasticity which he also wrote) in the negative sign on the right-hand side, but the difference is only apparent. According to his specification a pressure would be a positive stress, and an expansion a positive displacement, and in applying the equation to numerical examples this must be borne in mind so that the proper signs may be given to each numerical magnitude. As an example of adiabatic change, a sudden extension of the wire already referred to by an increase of stress dS may be considered. If there is not time for the passage of heat from or to the surroundings of the wire, the change of temperature will be given by equation (G).
This equation was applied by Thomson (article Elasticity) to find the relation between what he called the kinetic modulus of elasticity and the static modulus, that is, between the modulus for adiabatic strain and the modulus for isothermal strain.
The augmentation of the strain produced by raising the temperature 1° is e, and therefore edT, that is, − Te2dS ⁄ Kρ, is the increase of strain due to the sudden rise of temperature dT. This added to the isothermal strain produced by dS will give the whole adiabatic strain. Thus if M be the static or isothermal modulus, the adiabatic strain is dS ⁄ M − Te2dS ⁄ Kρ. If M' denote the kinetic or adiabatic modulus its value is dS divided by the whole adiabatic strain, that is, M' = M ⁄ (1 − MTe2 ⁄ Kρ) and the ratio M' ⁄ M = 1 ⁄ (1 − MTe2 ⁄ Kρ).
It is well known and easy to prove, without the use of any theorem which can be properly called thermodynamic, that this ratio of moduli is equal to the ratio of the specific heat K of the substance, under the condition of constant stress, to the specific heat N under the condition of constant strain of the corresponding type. This, indeed, is self-evident if two changes of stress, one isothermal the other adiabatic, which produce the same steps of displacement ds, be considered, and it be remembered that the step ∂T of temperature which accompanies the adiabatic change may be regarded as made up of a step − dT of temperature, accompanying a displacement ds effected at constant stress, and then two successive steps dT and ∂T effected, at constant strain, along with the steps of stress dS. The ratio M' ⁄ M is easily seen to have the value (∂T + dT) ⁄ dt, and since − KdT + N (∂T + dT) = 0, by the adiabatic condition, the theorem is proved.
Laplace's celebrated result for air, according to which the adiabatic bulk-modulus is equal to the static bulk-modulus multiplied by the ratio of the specific heat of air pressure constant to the specific heat of air volume constant, is a particular example of this theory.
Thomson showed in the Elasticity article how, by the value of M' ⁄ M, derived as above from thermodynamic theory, the value of K ⁄ N could be obtained for different substances and for different types of stress, and gave very interesting tables of results for solids, liquids, and gases subjected to pressure-stress (bulk-modulus) and for solids subjected to longitudinal stress (Young's modulus).
The discussion as to the relation of the adiabatic and isothermal moduli of elasticity is part of a very important paper on "Thermoelastic, Thermomagnetic, and Thermoelectric Properties of Matter," which he published in the Philosophical Magazine for January 1878. This was in the main a reprint of an article entitled, "On the Thermoelastic and Thermomagnetic Properties of Matter, Part I," which appeared in April 1855 in the first number of the Quarterly Journal of Mathematics. Only thermoelasticity was considered in this article; the thermomagnetic results had, however, been indicated in an article on "Thermomagnetism" in the second edition of the Cyclopædia of Physical Science, edited and in great part written by Professor J. P. Nichol, and published in 1860. For the same Cyclopædia Thomson also wrote an article entitled, "Thermo-electric, Division I.—Pyro-Electricity, or Thermo-Electricity of Non-conducting Crystals," and the enlarged Phil. Mag. article also contained the application of thermodynamics to this kind of thermoelectric action.
This great paper cannot be described without a good deal of mathematical analysis; but the student who has read the earlier thermodynamical papers of Thomson will have little difficulty in mastering it. It must suffice to say here that it may be regarded as giving the keynote of much of the general thermodynamic treatment of physical phenomena, which forms so large a part of the physical mathematics of the present day, and which we owe to Willard Gibbs Duhem, and other contemporary writers.
Thomson had, however, previous to the publication of this paper, applied thermodynamic theory to thermoelectric phenomena. A long series of papers containing experimental investigations, and entitled, "Electrodynamic Qualities of Metals," are placed in the second volume of his Mathematical and Physical Papers. This series begins with the Bakerian Lecture (published in the Transactions of the Royal Society for 1856) which includes an account of the remarkable experimental work accomplished during the preceding four or five years by the volunteer laboratory corps in the newly-established physical laboratory in the old College. The subjects dealt with are the Electric Convection of Heat, Thermoelectric Inversions, the Effects of Mechanical Strain and of Magnetisation on the Thermoelectric Qualities of Metals, and the Effects of Tension and Magnetisation on the Electric Conductivity of Metals. It is only possible to give here a very short indication of the thermodynamic treatment, and of the nature of Thomson's remarkable discovery of the electric convection of heat.
It was found by Seebeck in 1822 that when a circuit is formed of two different metals (without any cell or battery) a current flows round the circuit if the two junctions are not at the same temperature. For example, if the two metals be rods of antimony and bismuth, joined at their extremities so as to form a complete circuit, and one junction be warmed while the other is kept at the ordinary temperature, a current flows across the hot junction in the direction from bismuth to antimony. Similarly, if a circuit be made of a copper wire and an iron wire, a current passes across the warmer junction from copper to iron. The current strength—other things being the same—depends on the metals used; for example, bismuth and antimony are more effective than other metals.
It was found by Peltier that when a current, say from a battery, is sent round such a circuit, that junction is cooled and that junction is heated by the passage of the current, which, being respectively heated and cooled, would without the cell have caused a current to flow in the same direction. Thus the current produced by the difference of temperature of the junctions causes an absorption of heat from the warmer junction, and an evolution of heat at the colder junction.
This naturally suggested to Thomson the consideration of a circuit of two metals, with the junctions at different temperatures, as a heat engine, of which the hot junction was the source and the cold junction the refrigerator, while the heat generated in the circuit by the current and other work performed, if there was any, was the equivalent of the difference between the heat absorbed and the heat evolved. Of course in such an arrangement there is always irreversible loss of heat by conduction; but when such losses are properly allowed for the circuit is capable of being correctly regarded as a reversible engine.
Shortly after Seebeck's discovery it was found by Cumming that when the hot junction was increased in temperature the electromotive force increased more and more slowly, at a certain temperature of the hot junction took its maximum value, and then as the temperature of the hot junction was further increased began to diminish, and ultimately, at a sufficiently high temperature, in most instances changed sign. The temperature of maximum electromotive force was found to be independent of the temperature of the colder junction. It is called the temperature of the neutral point, from the fact that if the two junctions of a thermoelectric circuit be kept at a constant small difference of temperature, and be both raised in temperature until one is at a higher temperature than the neutral point, and the other is at a lower, the electromotive force will fall off, until finally, when this point is reached, it has become zero.
Thus it was found that for every pair of metals there was at least one such temperature of the hot junction, and it was assumed, with consequences in agreement with experimental results, that when the temperature was the neutral temperature there was neither absorption nor evolution of heat at the junction. But then the source provided by the thermodynamic view just stated had ceased to exist. The current still flowed, there was evolution of heat at the cold junction, and likewise Joulean evolution of heat in the wires of the circuit in consequence of their resistance. Hence it was clear that energy must be obtained elsewhere than at the junctions. Thomson solved the problem by showing that (besides the Joulean evolution of heat) there is absorption (or evolution) of heat when a current flows in a conductor along which there is a gradient of temperature. For example, when an electric current flows along an unequally heated copper wire, heat is evolved where the current flows from the hot parts to the cold, and heat is absorbed where the flow is from cold to hot. When the hot junction is at the temperature of zero absorption or evolution of heat—the so-called neutral temperature—the heat absorbed in the flow of the circuit along the unequally heated conductors is greater than that evolved on the whole, by an amount which is the equivalent of the energy electrically expended in the circuit in the same time.
It was found, moreover, that the amount of heat absorbed by a given current in ascending or descending through a given difference of temperature is different in different metals. When the current was unit current and the temperature difference also unity, Thomson called the heat absorbed or evolved in a metal the specific heat of electricity in the metal, a name which is convenient in some ways, but misleading in others. The term rather conveys the notion that electricity has a material existence. A substance such as copper, lead, water, or mercury has a specific heat in a perfectly understood sense; electricity is not a substance, hence there cannot be in the same proper sense a specific heat of electricity.
However, this absorption and evolution of heat was investigated experimentally and mathematically by Thomson, and is generally now referred to in thermoelectric discussions as the "Thomson effect."
Part VI (Trans. R.S., 1875) of the investigations of the electrodynamic qualities of metals dealt with the effects of stretching and compressing force, and of torsion, on the magnetisation of iron and steel and of nickel and cobalt.
One of the principal results was the discovery that the effect of longitudinal pull is to increase the inductive magnetisation of soft iron, and of transverse thrust to diminish it, so long as the magnetising field does not exceed a certain value. When this value, which depends on the specimen, is exceeded, the effect of stress is reversed. The field-intensity at which the effect is reversed is called the Villari critical intensity, from the fact, afterwards ascertained, that the result had previously been established by Villari in Italy. No such critical value of the field was found to exist for steel, or nickel, or cobalt.
In some of the experiments the specimen was put through a cycle of magnetic changes, and the results recorded by curves. These proved that in going from one state to another and returning the material lagged in its return path behind the corresponding states in the outward path. This is the phenomenon called later "hysteresis," and studied in minute detail by Ewing and others. Thomson's magnetic work was thus the starting point of many more recent researches.