Lord Kelvin: An account of his scientific life and work

by Andrew Gray

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.