|MadSci Network: Physics|
Dear Martijn: You have asked a question which is difficult to answer in any simple way. It falls into the subject matter of Statistical Mechanics and the Kinetic Theory of Gases. I can only anwer you in a very general way. The mathematics behind the answer is advanced and probably beyond the scope of your present training. Rather than struggling here to present this information to you I refer you to any thorough reference which teaches the Kinetic Theory of Gases. The following reference is one that I have which explains the mathematical derivation of these terms for average speed, and root mean square speed, which illustrates the complex mathematical derivation. Basically everything starts from the ideal gas law (pV=NkT); the classical equipartition law (the average kinetic energy of a system of noninteracting particles is given by 1/2kT per degree of freedom); and the simple kowledge that the kinetic energy of a particle of mass(m) moving at velocity (v)is given by (1/2)mv(square). Since the kinetic energy is related to the "square" of the velocity the root-mean-square concept gets involved when one wishes to obtain velocity information when starting with total energy information. Normally given a system of particles to analyze the initial information are temperature, pressure, and volume from which total energy can be calculated. Then if you want to determine velocity properties you get involved with averages of squares going from energy to velocity. The book which I have is: Handbook of Physics; Authors E. U. Condon and H. Odishaw. McGraw-Hill Book Company, New York,Toronto, London. 1958; Chapter 2, pp 5-11 ;"Principles of Statistical Mechanics and Kinetic Theory of Gases". I am sure there are more modern books available but the theories discussed here are very old and well founded. The differences between the two computations relate to the accuracy of the velocity distribution function used. The mean speed which you cite is based upon the famous Maxwell velocity distribution function. The RMS speed which you cite is based merely on the computation of the mean speed from the following facts: a.U(total kinetic energy)=(1/2)Nm
sq (average) b. average=3kT/2;and c. U=(3/2)NkT Substitute c. into a. and you get sq (average)=3kT/m (simple root mean square velocity).The difference is one of ultimate accuracy! I hope this has been helpful! R. Bersin......
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