0.1 SECTION III

0.2 Principle of the communication of heat.

From The Analytical Theory of Heat by Joseph Fourier

57. We now proceed to examine what experiments teach us concerning the communication of heat.

If two equal molecules are formed of the same substance and have the same temperature, each of them receives from the other as much heat as it gives up to it; their mutual action may then be regarded as null, since the result of this action can bring about no change in the state of the molecules. If, on the contrary, the first is hotter than the second, it sends to it more heat than it receives from it; the result of the mutual action is the difference of these two quantities of heat. In all cases we make abstraction of the two equal quantities of heat which any two material points reciprocally give up; we conceive that the point most heated acts only on the other, and that, in virtue of this action, the first loses a certain quantity of heat which is acquired by the second. Thus the action of two molecules, or the quantity of heat which the hottest communicates to the other, is the difference of the two quantities which they give up to each other.

58. Suppose that we place in air a solid homogeneous body, whose different points have unequal actual temperatures; each of the molecules of which the body is composed will begin to receive heat from those which are at extremely small distances, or will communicate it to them. This action exerted during the same instant between all points of the mass, will produce an infinitesimal resultant change in all the temperatures: the solid will experience at each instant similar effects, so that the variations of temperature will become more and more sensible.

Consider only the system of two molecules, m and n, equal and extremely near, and let us ascertain what quantity of heat the first can receive from the second during one instant: we may then apply the same reasoning to all the other points which are near enough to the point m, to act directly on it during the first instant.

The quantity of heat communicated by the point n to the point m depends on the duration of the instant, on the very small distance between these points, on the actual temperature of each point, and on the nature of the solid substance; that is to say, if one of these elements happened to vary, all the other remaining the same, the quantity of heat transmitted would vary also. Now experiments have disclosed, in this respect, a general result: it consists in this, that all the other circumstances being the same, the quantity of heat which one of the molecules receives from the other is proportional to the difference of temperature of the two molecules. Thus the quantity would be double, triple, quadruple, if everything else remaining the same, the difference of the temperature of the point n from that of the point m became double, triple, or quadruple. To account for this result, we must consider that the action of n on m is always just as much greater as there is a greater difference between the temperatures of the two points: it is null, if the temperatures are equal, but if the molecule n contains more heat than the equal molecule m, that is to say, if the temperature of m being v, that of n is v + Δ, a portion of the exceeding heat will pass from n to m. Now, if the excess of heat were double, or, which is the same thing, if the temperature of n were v + 2Δ, the exceeding heat would be composed of two equal parts corresponding to the two halves of the whole difference of temperature 2Δ; each of these parts would have its proper effect as if it alone existed: thus the quantity of heat communicated by n to m would be twice as great as when the difference of temperature is only Δ. This simultaneous action of the different parts of the exceeding heat is that which constitutes the principle of the communication of heat. It follows from it that the sum of the partial actions, or the total quantity of heat which m receives from n is proportional to the difference of the two temperatures.

59. Denoting by v and v′ the temperatures of two equal molecules m and n, by p, their extremely small distance, and by dt, the infinitely small duration of the instant, the quantity of heat which m receives from n during this instant will be expressed by (v′−v)ϕ(p)dt. We denote by ϕ(p) a certain function distance p which, in solid bodies and in liquids, becomes nothing when p has a sensible magnitude. The function is the same for every point of the same given substance; it varies with the nature of the substance.

60. The quantity of heat which bodies lose through their surface is subject to the same principle. If we denote by σ the area, finite or infinitely small, of the surface, all of whose points have the temperature v, and if a represents the temperature of the atmospheric air, the coefficient h being the measure of the external conducibility, we shall have σh(v − a)dt as the expression for the quantity of heat which this surface σ transmits to the air during the instant dt.

When the two molecules, one of which transmits to the other a certain quantity of heat, belong to the same solid, the exact expression for the heat communicated is that which we have given in the preceding article; and since the molecules are extremely near, the difference of the temperatures is extremely small. It is not the same when heat passes from a solid body into a gaseous medium. But the experiments teach us that if the difference is a quantity sufficiently small, the heat transmitted is sensibly proportional to that difference, and that the number h may, in these first researches [1], be considered as having a constant value, proper to each state of the surface, but independent of the temperature.

61. These propositions relative to the quantity of heat communicated have been derived from different observations. We see first, as an evident consequence of the expressions in question, that if we increased by a common quantity all the initial temperatures of the slid mass, and that of the medium in which it is placed, the successive changes of temperature would be exactly the same as if this increase had not been made. Now this result is sensibly in accordance with experiment; it has been admitted by the physicists who first have observed the effects of heat.

62. If the medium is maintained at a constant temperature, and if the heated body which is placed in that medium has dimensions sufficiently small for the temperature, whilst falling more and more, to remain sensibly the same at all points of the body, it follows from the same propositions, that a quantity of heat will escape at each instant through the surface of the body proportional to the excess of its actual temperature over that of the medium. Whence it is easy to conclude, as will be seen in the course of this work, that the line whose abscissae represent the temperatures corresponding to those times, is a logarithmic curve: now. observations also furnish the same result, when the excess of the temperature of the solid over that of the medium is a sufficiently small quantity.

63. Suppose the medium to be maintained at the constant temperature 0, and that the initial temperatures of different points a,b,c,d&c. of the same mass are α,β,γ,δ&c., that at the end of the first instant they have become α′,β′,γ′,δ′&c., that at the end of the second instant they have become α′′,β′′,γ′′,δ′′&c., and so on. We may easily conclude from the propositions enunciated, that if the initial temperatures of the same points had been gα,gβ,gγ,gδ&c. (g being any number whatever), they would have become, at the end of the first instant, by virtual of the action of the different points, gα′,gβ′,gγ′,gδ′&c., and at the end of the second instant, gα′′,gβ′′,gγ′′,gδ′′&c., and so on. For instance, let us compare the case when the initial temperatures of the points, a,b,c,d&c. were α,β,γ,δ&c. with that in which they are 2α, 2β, 2γ, 2δ&c., the medium preserving in both cases the temperature 0. In the second hypothesis, the difference of the temperatures of any two points whatever is double what it was in the first, and the excess of the temperature of each point, over that of each molecule of the medium, is also double; consequently the quantity of heat which any molecule whatever sends to any other, or that which it receives, is, in the second hypothesis, double of that which it was in the firs. The change of temperature which each point suffers being proportional to the quantity of heat acquired, it follows that, in the second case, this change is double what it was in the first case. Now we have supposed that the initial temperature of the first point, which was α, became α′ at the end of the first instant; hence if this initial temperature had been 2α, and if all the other temperatures had been doubled, it would have become 2α′. The same would be the case with all the other molecules b,c,d, and a similar result would be derived, if the ratio instead of being 2, were any number whatever g. It follows then, from the principle of the communication of heat, that if we increase or diminish in the same ratio all the successive temperatures.

This, like the two preceding results, is confirmed by observation. It could not have existed if the quantity of heat which passes from one molecule to another had not been, actually, proportional to the difference of the temperatures.

64. Observations have been made with accurate instruments, on the permanent temperatures at different points of a bar or of a metallic ring, and on the propagation of heat in the same bodies and ins several other solids of the form of spheres or cubes. The results of these experiments agree with those which are derived from the preceding propositions. They would be entirely different if the quantity of heat transmitted from one solid molecule to another, or to a molecule of air, were not proportional to the excess of temperature. It is necessary first to know all the rigorous consequences of this proposition; by it we determine the chief part of the quantities which are the object of the problem. By comparing then the calculated values with those given by numerous and very exact experiments, we can easily measure the variations of the coefficients, and perfect our first researches.

0.3 References

This chapter is a derivative from the public domain work in [2].

[1] More exact laws of cooling investigated experimentally by Dulong and Petit will be found in the Journal de l’Ecole Polytechnique, Tome XI. pp. 234-294, Paris, 1820, or in Jamin, Cours de Physique, Lecon 47.

[2] Fourier, Joseph. ”The Analytical Theory of Heat” Translation by Alexander Freeman, Cambridge: At the University Press, London, 1878.