Constituents of Matter / 4 / 4.6 / 4.6.1

4.6.1 Corpuscular Theory Of Boyle’s Law

The simplest way to make our argument is to consider the relationship that obtains between the pressure of an ideal gas in equilibrium at constant temperature and the volume of the container in which it is being contained. Following a suggestion made by Daniel Bernoulli in 1738, the pressure exerted by an ideal gas upon the walls of the vessel which contains it was to be explained as the force per unit area communicated by the molecules constituting the gas to the wall when they happened to hit those walls. Since molecules were taken to be like small mechanical objects of experience, it was natural to employ Newton’s laws of motion to describe their behavior, laws which would normally have been used to explain the motion of any other mechanical object of experience impinging upon a surface.

However, by thus taking molecules to be Newtonian particles, we denied them their power to explain our experience with ideal gases. Indeed, one can show that the expression for the instantaneous gas pressure one obtains by applying Newtonian mechanics to a collection of elastic particles that are constantly agitating within a rectangular container, depends upon the kinematic configuration at time of all the molecules which constitute the gas, upon the shape and size of the container, upon the location of the point at which the pressure is calculated, and upon the size of the area over which it is exerted. Most importantly, however, the pressure does not explicitly depend on the container volume as our experience with ideal gases would have led us to expect. In fact, according to Boyle’s law, the pressure should have been inversely proportional to that volume if the ideal gas was at equilibrium within the container and its temperature was held constant. Since nothing of the kind emerges from our description of an ideal gas as a collection of mechanical molecules obeying Newton’s laws, we must conclude that the molecular hypothesis, in and of itself, fails to explain our thermodynamic experience.

The conventional way out of this impasse was to observe that any equipment we may use to measure the pressure will not be able to record the instantaneous value; rather, it will measure the time average of that pressure over intervals which are large compared with the time scale of molecular motion. Therefore, during the measurement, molecules will have had time to “explore” the entire container and each and every one of the molecules will thus have a finite probability of finding itself within striking distance of the point on the wall where the pressure is being measured. Under the circumstances, the time average could be replaced with an average over the probability distribution of the locations of the molecules of the ideal gas, the validity of which replacement is asserted by the ergodic hypothesis. Performing such an average while keeping the velocity configuration fixed, leads directly to the volume dependence required by Boyle’s law if molecules are assumed to be statistically independent of each other and the location of each molecule is uniformly distributed over the container. We find therefore that one must talk of probabilities if one is to confer upon Newtonian mechanics the power to explain thermodynamic experience. But what kind of probability is this that we were forced to use here and what does the need to use it say about the ontology of a molecule?

To understand that, consider by way of illustration a collection of people of various heights. This collection has a structure characterized by the height of its individual members. One can describe this structure by partitioning the entire scale of heights into a given number of equal intervals and then specifying how many of its members are found to have heights that fit into each interval. Note, that in this description of things, each member of our collection, as indeed the entire collection itself, has the Being of an object of experience which possesses the measurable property of height. As long as we do not change this ontological situation, we are free to describe this structure in terms of an a posteriori probability distribution by saying: the probability that, upon measurement, the height of a member randomly picked out of the collection falls between h and h + Δh is equal to the ratio of how many members were actually found to fall in that interval and the total number of members in the collection. This can also be taken to mean that, if we picked out of our collection one member at random, measured his height, and then, having returned him to the collection without prejudice, repeated this operation a very large number of times, the number of times we get a member with height between h and h + Δh relative to the total number of times we performed the procedure would eventually be equal to the probability defined above.

One can however envision collections of entities whose properties one cannot measure as readily as one could the height of people. Clearly, in such cases, it would be difficult to empirically secure an a posteriori probability for any property characterizing the entities under consideration. But one could still go ahead and assign a probability for the occurrence of any value of a given property a priori. The question is, what would this assignment do to the ontological status of the collection and indeed to that of each of its members? As long as the property could be ascertained through measurement, the ontological status would remain unchanged because the a priori probability distribution could then be taken to be nothing but a guess at the results one would have obtained if one went through the sorting process by actual measurement, inconvenient as such a sorting process may turn out to be. But if the measurement were impossible to perform even in principle, the assignment of an a priori probability could no longer be ontologically neutral. One can no longer say: the a priori probability P(x) that the given property will have a value between x and x + Δx represents the fraction of the collection found to possess that value if one were to have measured that property for all entities in the collection, because no such measurement was possible in principle. And neither can one say that it represents the fraction of times one would have measured the property of a randomly chosen entity to be between x and x + Δx had one performed a large number of such measurements because to say so continues to suggest that one can, in principle at least, obtain access to each individual entity in the collection.

Since in that case there cannot be any talk of a statistical counting on which to rest the definition of a probability, the statement that the given property has a probability P(x) of possessing a value between x and x + Δx must be a statement about each entity in the collection separately. But then, because the property characterizing each entity only has a probability of its value being in that interval, the value cannot fully be in that interval; it must also, with some probability, be outside that interval. It sounds therefore as if the property does not have a well defined value but rather that its value is spread over the space of all its possible values in proportion with the a priori probability distribution P(x), in which case the ontological status of the entities involved would be fundamentally altered.

Applied to molecular theory, this observation means that the ontological character of a molecule might depend on whether or not one can, in principle, access it. If we took the position that one can, the fact that explaining thermodynamic experience by means of Newtonian mechanics requires the introduction of location probabilities for each molecule would carry no ontological content. If, on the other hand, we took the position that in principle molecules are not accessible to direct measurement, the need to introduce probabilities for their location would have profound ontological consequences. As we know, the scientific community decided to accept the first position and therefore took the probability involved to be a guess at the form of the posterior probability which would have been obtained if only we were willing to take the time and effort to ascertain the spatial configuration of the collection.

Our ontological question of whether a gas molecule is an object of experience or one of explanation hangs therefore upon the question of whether we take the probabilities required by the corpuscular theory’s explanation of Boyle’s law to be a guess at an a posteriori probability which we chose not to ascertain by actual observation or whether we take that probability to be truly an a priori one. However, given the reasonableness of the statistical argument involved in the kinetic theory of gases and the ever expanding realm of thermodynamic experience which statistical mechanics explained, it would clearly have been quite preposterous to insist on settling the question by actual observation. We see then, that the ontological question was there, in the theory, from the very beginning, and so was the opportunity to discover a few centuries earlier that objects of explanation are not objects of experience.