Solving for the average velocity of gas particles gives us the average velocity of zero, assuming that all particles are moving equally in different directions

from https://www.boundless.com/chemistry/textbooks/boundless-chemistry-textbook/gases-5/kinetic-molecular-theory-55/root-mean-square-speed-265-1270/

For any list of numbers holds: The root mean square (rms) is always equal or higher than the average (avg). Only if all numbers in the list are positive and equal then rms = avg.

The reason is that higher values in the list have a higher weight (because you average the squares) in the calculation of a rms compared to the calculation of the avg.

So only if all molecules have the same velocity we would have: V(rms) = V(avg).

However, at least in ideal gases, the velocities of the molecules are distributed according to the Maxwell-Boltzmann distribution. For the Maxwell-Boltzmann distribution holds: V(rms) = V(ave)*sqrt(3/8*pi)

More information: https://en.wikipedia.org/wiki/Maxwell-Boltzmann_distribution

Dear Osamah Nawfal Oudah,

that rms is larger or equal avg can be easiy seen applying the different averaging methods to two number series. Teun Sweere already mentioned that the difference is caused by the higher weight of the faster molecules. Where does this come from? Assume a gas moving along the x-axes in one direction where the half of the molecules has velocity v1 and the second half velocity v2 and v1 < v2. The molecules hit a wall at some point x0 where there are ideally reflected. The number densities of the molecules are sufficiently low that we can neglect any collisions. The average momentum change per molecule at the wall ist given by p_avg=0.5 m v1+0.5 m v2 := 0.5 p1+0.5 p2 with the molecule mass m. If we look for the force onto the wall we have also to take into account the frequency of collisions. Since the molecules of v2 are faster these molecules hit the wall more often since we assumed identical number densities. Therefore, the total impact onto the wall is proportional to 0.5 v1 p1 + 0.5 v2 p2, this is proportional to the rms velocity of the molecules.

The average value has to be taken when we ask for the mean of the event size/impact, while the rms includes that the probability for the event occurence is also proportional to the event size.

I hope this shows a little bit the idea what is behind the maths concept.

Best regards,

Hans Werner Müller