Chemostatistics-The RMS Velocity.

[2810712]

Why does root mean square velocity has importance in Kinetic theory?

Whats its specific importance? I mean why not just mean of magnitudes or cube-root mean cube velocity etc.???

 

I don't know more abut statistics, so please help...

 

hrushikesh

[swansont]

Does the term kinetic have an implications at all, for the energy?

[Klaynos]

Takeing a guess I think he may be refereing to statistical mechanics, and possible the kinetic theory of gasses.... the root mean square speed is important because it is the most probable speed of an individual atom.

[5614]

You can't just take the average because it would include all the particles with negative velocity. ie. say you said 'particles going right have a positive velocity' therefore things going left have a negative velocity. As gas particles move randomly if you took the average they would all cancel each other. The average speed would 0, this is obviously not correct.

 

By squaring all of the numbers you remove the negative signs, and by square rooting you get back where you came from (but this time positive numbers only), you can then work out the average of this, this gives you the true average speed.

 

It is important in Kinetic Theory because kinetic means movement or motion, so obviously the velocity of particles is quite important when studing the motion of them aka Kinetic Theory.

[2810712]

that really helped me think better...

@5614 & all

but about 'true average.' we can have magnitudes of velocities added and then averaged. This will be the true magnitudal av. But rootmean is greater that it mostly. And with the same reasoning u used, it'll be alright with using fourthroot mean fourth power velocity and further even powers of velocities.

They all don't come equal to each other and to the magnitudal average.So just removing negative sign can't be the sole reason...

What do you think?

 

 

hrushikesh

[5614]

From: http://www.analytictech.com/mb313/rootmean.htm

The root mean square is a measure of the magnitude of a set of numbers. It gives a sense for the typical size of the numbers

 

Also:

 

[math]x_{rms}^2 = \bar{x}^2 + \sigma_{x}^2[/math]

 

Where [math]\sigma_{x}^2[/math] is the standard deviation.

[2810712]

Yeah, the site helped. But it didn't tell why is this way of removing negative signs preffered over the others... because all methods lead to different results...

 

 

 

 

 

 

 

hrushikesh

[Klaynos]

If you don't remove all the signs then a gas with no bulk movement then all the velocities would cancel out and the gas would have a 0 KE.

[swansont]

For ideal gases the values come from the Maxwell-Boltzmann speed distribution, so the negative sign is moot; there is none. RMS is used because that's a relevant term for kinetic energy to relate in a simple way to temperature, which is what I was hinting at earlier.

 

rms is not the most probable value; Klaynos was incorrect in that statement. However, for the M-B distribution, rms average and most-probable have definite relationships to each other. more

[2810712]

so as used in the gas law eqn

1/2mv1^2+1/2mv2^2+.....+1/2mvn^2 will give total K.E. go gas so taking out 1/2 m common we get sum of squares of all velocities which is constant due to perfectly elastic collosions of molecules at a constant temp. If we divide it by n and then take a root we get out RMS velocity. SO, substituting squared RMS velocity in the eqn seems correct. And as RMS is constant at given temp. it can be used to relate the temp. and energy in a better way right?

 

Thanks for those links and replies.

Further opinions welcome.

 

 

hrushikesh

[Klaynos]

[quote name=swansont

rms is not the most probable value; Klaynos was incorrect in that statement. However' date=' for the M-B distribution, rms average and most-probable have definite relationships to each other. more

 

 

My appologies, hope this didn't mislead the topic too much :|