2.2 GAS LAW AND THE MOLECULE
The first attempt at demonstrating the explanatory power of the corpuscular conception of matter was made by Daniel Bernoulli in his 1738 book on Hydrodynamics. Chapter Ten of that book addresses the properties and motions of elastic fluids, but especially of air. In it he considers an air-filled cylindrical vessel covered with a movable lid upon which rests a given weight. He then conceives of the air within as a collection of “small bodies agitated in a very rapid motion”, bodies which support the weight by their “continually repeated impacts on the lid.” He points out that the collection of bodies behaves as an elastic fluid which expands if the weight is diminished and is compressed if the weight is increased and then proceeds to evaluate the ratio between the weights which would be supported by the fluid at two different volumes. To that end he observes that the lid will suffer a greater pressure when the fluid is compressed because the number of bodies within striking distance to it would increase and each of those bodies would strike the lid more times per second and then proceeds to evaluate the resulting pressure as the product of these two factors. Simple estimates based on a spherical shape for the constituent bodies and the assumption that the bodies are equally likely to be anywhere in the container at any given time, show the ratio of the compressing weights to be inverse proportional to the ratio of the corresponding volumes, a relationship which “manifold experience has confirmed.” In fact, Robert Boyle had experimentally proved it in his New Experiments Physico-Mechanical, Touching the Spring of the Air and their Effects published in 1660.
An evaluation of the absolute pressure exerted by a gas upon the walls of its container, rather than merely of its relative value, became possible later when Ludwig Boltzmann married the corpuscular conception with Newton’s mechanics under the kinetic theory of gases. By averaging over the force exerted per unit time by all gas molecules as they impinge the unit area of the walls, that theory concludes:
where represents the pressure, the volume, the total number of molecules in the volume, and the average kinetic energy of the molecules. Notwithstanding the increased explanatory reach of the formalism, the equation continues to rest upon the same statistical assumption Bernoulli had made: molecules are equally likely to be anywhere within the volume. In fact, it is not only a molecule’s location, but its speed as well, which must be treated statistically if one is to gain an explanation of the temperature dependence in Boyle’s law. Statistical treatment of the details appertaining to the individual molecule is a structural feature of the entire theory. This however was a small price to pay for the additional explanatory power thus gained. For, with the kinetic theory of matter developed by Boltzmann as with the more general statistical physics developed by Gibbs, one can explain not only Boyle’s law but all of thermodynamics as well.
This need to have recourse to statistics appeared acceptable from the very beginning because an exact evaluation of instantaneous pressure would have been neither practicable nor necessary. Not practicable because one would have had to solve as many Newton’s equations of motion as there were molecules, and would have had to provide just as many initial conditions from which the resulting configuration informing the instantaneous pressure could evolve. Not necessary because, in any event, actual measurements could not register the instantaneous pressure exerted by the gas but would rather perform a time average of sorts over these instantaneous values. Palatable as this justification for introducing statistics into the corpuscular theory may be, the fact of having to do so foreshadows the kind of compromises one has to make when dealing with applications of the corpuscular concept. It is a way of avoiding the necessity of accounting for the individual molecule and its behavior and, as we shall indicate further on in the essay, it is therefore directly related to our ontological question.