MadSci Network: Physics

Re: The difference between the root mean square and the mean speed of molecules

Date: Mon Sep 14 23:32:16 1998
Posted By: Richard Bersin, Other (pls. specify below), Senior Technical Staff Member, Emergent Technologies
Area of science: Physics
ID: 905348331.Ph

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 

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)Nmsq (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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