### 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
Message:
```
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

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)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......

```

Current Queue | Current Queue for Physics | Physics archives