##v_{rms}## in the Kinetic Theory Of Gases

[Kaushik]

In Kinetic theory of gases, what is the reason behind introducing a new kind of average known as root mean square velocity ($v_{rms}$)?

I read the following: The molecules in a container are in constant random motion. So when we add all the velocity vectors to find the average it cancels out and gives $v_{av} = 0$. So to avoid this we square the velocities (to get rid of the sign) and then add them.

Is there any other reason? I read that there is another reason which is associated with equipartition (but it did not mention the reason). What could that reason be?

 

[sophiecentaur]

Summary:: Why do we use $v_{rms}$

a new kind of average

The arithmetic mean of velocities of all molecules would be zero (assuming the mass of gas is not moving). That would be of little use. RMS is only zero when all molecules are stationary and, as a measure of Energy, for instance, it is useful. The formula for RMS is the same as for statistical Standard Deviation (which again avoids the problem of the mean of a distribution about zero can be zero).

 

[Herman Trivilino]

I read that there is another reason which is associated with equipartition (but it did not mention the reason). What could that reason be?

At a given temperature, on average, each molecule has the same amount of kinetic energy. When you average the kinetic energies you are averaging the squares of the speeds of each molecule because the kinetic energy is proportional to the square of the speed. If you then take the square root of that average you get a number that is roughly equivalent to the speed.

The reason we do it this way is because we can measure the temperature of the gas and from that we can infer the average of the squares of the speeds. We have no way of measuring or inferring the speed.