Analytical Methods for Problems of Molecular Transport
The transport of a given species (atoms, molecules, neutrons, photons, etc. ), either through its own kind or through some other host medium, is a problem of considerable interest. Practical applications may be found in many technologically and environmen
- PDF / 234,391 Bytes
- 14 Pages / 439.37 x 666.142 pts Page_size
- 78 Downloads / 197 Views
1.
SOME INTRODUCTORY REMARKS.
Systems consisting of what are usually called rarefied gases will be considered in this book. Such systems contain large numbers of molecules; specifically, about 2.7 u 1019 cm–3 for a gas at standard temperature and pressure (STP) defined to be 0 °C (273.15 K) and 1.0 atm (760 torr). How may the behavior of such huge collections of particles be described? If one were to try to apply the methods of classical mechanics, one would have to construct and solve a system of equations of motion containing as many equations and initial conditions as there are numbers of interacting particles. Obviously, such a problem could not be solved today even with the help of the most advanced computers. On this basis, one might naively suspect that the greater the number of particles, the more difficult the problem. This is not strictly true, however, as it turns out that useful results can be obtained by applying statistical descriptions to such systems. The state of a gas can be analyzed by employing statistical laws which allow one to determine average values for the different macroscopic quantities that characterize the behavior of the gas. It has been proven in statistical mechanics [1-3] that the relative fluctuations of additive quantities (i.e. quantities whose values for the body as a whole are equal to the sum of the values for its separate parts) are proportional to N 1 2 , where N is the number of molecules of the gas. In accordance with this theorem, the additive quantities are really equal to their average values to an extremely high degree of accuracy and, therefore, deviations of the actual quantities
2
Analytical Methods for Problems of Molecular Transport
from the average values do not have a practical influence on the trustworthiness of the statistical description. The mean values mentioned above may be found by using a probability framework in which the main interest is in the statistical distribution function for the molecules. It is very important to note that the extreme accuracy of this type of probabilistic analysis, due to the relatively small deviations of the macroscopic quantities from their mean values, is much greater than the accuracy of actual experimental measurements and, hence, small deviations of the macroscopic quantities from their mean values are typically neglected.
2.
DENSITY AND MEAN MOTION.
First, some macroscopic quantities for a ‘simple’ gas composed entirely of identical molecules [4-6] will be determined. Let the mass of any molecule be m and let dr denote a small volume element surrounding the point, r . This volume element is assumed to be large enough to contain a great number of molecules while still possessing dimensions small compared to the scale of variation of the macroscopic quantities of interest. Let the mass contained in dr be averaged over a time interval, dt , which is long compared to the average time needed for a molecule to traverse dr yet short compared to the scale of the time variations in the macroscopic properties of th
Data Loading...
