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IV.  Changing the Perspective About Forces

Having in mind a broadening of our perspective of judgment, let us do it in a more radical way. The noncentral forces knocked many times at the door of our mind, in order to open it for a conceptual approach of them, but we were not prepared for this approach. One notorious example is the theory of light. The initial approach of Biot and Fresnel was a mechanical one, by forces. However, because one couldn’t conceive but exclusively central forces, the theory of light suffered a dichotomy. On one hand the light was imagined as a phenomenon of local motion; on the other hand this phenomenon of motion propagates in straight lines in space. In this image the place of force remains uncertain. The only theory referring to the forces with the certainty required by experimental possibilities is the well known electromagnetic theory of light. We do not analyze this theory for the moment, but remain strictly in the realm of Mechanics. Obviously we have to take the reasoning outside the realm of Classical Mechanics of a single particle, and recall some results from the kinetic theory of gases. As known, for an ideal gas the definitions of the pressure and temperature are related to the mean kinetic energy of its molecules. We describe now these definitions in a little more detail for reasons that will become clear shortly.

     Cauchy and Poisson have generalized the definition of the pressure to that of the stress tensor, no matter of the forces acting between molecules. Thus, for the first time in history, the state of a mechanical system in static equilibrium could be seen as very rich indeed, for the static equilibrium is only the appearance at our scale of perception of an internal Universe in motion. The very same should obviously go for an ideal gas. However here we have an explicit way to apply the dynamical principles in order to characterize the static equilibrium in question. Start with the observation that, in the case of ideal gases some sound experimental facts are gathered together in the form of the equation of state,

(29)

where p is the pressure, V is this time the volume of gas container, T is the temperature, ν is the number of mols of gas and R is the gas constant. In order to understand the members of this equation, we must explain them in terms of Mechanics. Of course V does not need any explanation, but p and T do. Thus we start from Newton’s second law of motion, which we write as

(30)

where is the force acting on a molecule of mass m, and is its position vector. Now, from this equation we can obtain the following relation, valid for the averages over suitable periods of time (Jeans, 1954)

(31)

The right-hand side of this last equation is called by Clausius the virial of forces. This is why the equation (31) is sometimes called the virial theorem. The sum in the right hand side of (31) is done over all molecules surrounding one of them, a generic one. This theorem shows that the mean kinetic energy of the molecules equals in magnitude the virial of forces acting on them.

     The forces acting on a generic molecule from the bulk of gas are of three types: the forces due to impact between molecule and the walls of the enclosure, the forces of collision between molecules and the forces of cohesion or repulsion between molecules. Obviously the virial from the right hand side of equation (31) has then three parts corresponding to those forces. The part of the virial due to collisions between molecules is usually eliminated on the grounds of molecular chaos, which assures that it is null. The part due to the impact with the walls of the enclosure can be estimated from the very definition of the pressure. Namely, the elementary force exerted by the gas acting on area (dA) of the wall is given by

(32)

where is the outward unit normal to the area (dA). Replacing the sum in (31) by the integral over the area of enclosure (assuming that we have this right), we have

(33)

This integral can be estimated by Gauss’ theorem and gives (- 3pV), so that the wall contribution to the virial is

(34)

Let’s estimate now the part of virial due to the action at distance between molecules. Suppose, in the classical fashion, that the forces between molecules are central forces and their magnitude does not depend but on the distance between them. According to our previous discussion such forces are also conservative. As a consequence, a pair of molecules will give a contribution amounting to

(35)

where f(r) is the magnitude of the interaction forces. Collecting (34) and (35) in (31), gives

(36)

where the double summation is over all the pairs of molecules. This last relation gave Maxwell the opportunity of an interesting speculation (Maxwell, 1965 vol. 2, p.422). Namely it shows that the pressure must be built on both the kinetic energy of the molecules and their forces of interaction at distance. At a given temperature we can check the truth of (36) by comparing it with experimentally sound equation (29). If the pressure is exclusively given by forces, then these must be repulsion forces. Newton in his Principia had already given such an explanation for pressure. It is unsustainable as the following arguments show: if, at a given temperature, the product between pressure and volume is constant as (29) shows, and the pressure is exclusively due to forces, then the double sum from (36) must be constant. But this condition can only be achieved if the repelling forces have a magnitude inversely proportional with the distance between the molecules. As Jeans poses it:

“This is, however, an impossible law for a gas; it would make the action of the distant parts of the mass preponderate over that of contiguous parts, and would not give a pressure which, for a given volume and temperature, would be constant as we passed […] from one part to another of the surface of the same vessel ” (Jeans, 1954, p. 127ff. Our Italics).

So, the conclusion imposes itself that the intermolecular repelling forces must be inexistent, and thus the contribution to the virial is solely given by the kinetic energy term:

(37)

From this moment on the story goes as well known: a comparison between (29) and (37) gives then the mean kinetic energy of a molecule as

(38)

where k is a constant (Boltzmann’s constant) and thus we have an explanatory definition of the temperature, making it a measure of the energy of translation of the molecules in a gas. This is the basis for the kinetic theory of ideal gases and the consequent Thermodynamics – and implicitly Physics – as we know them today. We owe to this banishing of those forces the rational definition of absolute temperature, which allowed the tremendous growing of our positive knowledge in the century past. Without the definition of the absolute temperature the idea of quantum for instance would not be possible. However…

     …there is nothing to add to Jeans’ words cited above, in order to understand that the situation described as inadmissible for a gas is perfectly suitable for a solid. Indeed, in a solid we cannot expect but non-homogeneous pressures (modernly termed stresses) and, as a result of these, also non-isotropic pressures. In a solid, long distance actions among the members of its structure are to be expected; otherwise the cohesion of the solids cannot be explained. Finally, let us remember that mathematics found here a way to deal with internal pressures in a solid as if they were bulk forces. Consequently it may not be quite so surprising to find forces in a solid that control mainly the distant parts and are also non-central. From time to time one can hear voices calling for a rational analysis of this situation (Luban, Novogrodsky, 1972). However, here we push the claim even further: by what just has been said before we need to stress that the elimination of these forces is not justified even from the realm of ideal gases.

     Notice that the key of the classical theory of gases is the calculation of the virial according to the expression

(39)

The last term here takes the form involved in equation (36) only in the case of central forces, which were the only forces classically accepted as intermolecular forces in a gas. This is the main argument in saying that these intermolecular forces don’t exist. However that term is naturally zero for some forces acting always perpendicularly to the line joining the molecules. As we have shown that this kind of forces is a priori realistic, the ideal gas equation of state can very well be seen as a consequence of this type of interaction between the molecules of an ideal gas. Therefore, based on the ideal gas experience we are entitled to say that the forces of interaction between the molecules of a gas are noncentral forces having null components along the direction joining the molecules.

     Not only this, but still maintaining the classical framework, with central forces and all that, we are still close to the historical experience related to the equation of state of the ideal gas, if we assume that these central forces have a magnitude inversely proportional with the distance between the molecules of the gas. This fact will induce a constant term in the equation of state of the ideal gas which, again, seems to be a necessary one, in view of the historical perspective regarding this problem. This will be discussed in the next section. For the moment notice that, in the good spirit of Poincaré and Jeans, the classical kinetic theory of the ideal gas was applied to the dynamics of stars (Jeans, 1913, 1915, 1916). In this connection it is worth mentioning that there are ideas according to which the forces with a magnitude inversely proportional to the distance are the natural forces explaining inertia as an interaction (Sciama, 1969).

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