Question

In kinetic molecular theory, the average velocity of gas particle is zero since the molecule move in different directions, and the overall effect is zero. Howeever, you can calculate different speeds.

Specifically, RMS (Root Mean Square) speed is the speed of a particle in a gas with the average kinetic energy. This is
μᵣₘₛ = √ 3RT / M
Conceptually, if v1,v2,...,vn are the speed of n particles in a gas, the RMS is
μᵣₘₛ = √ ∑v²ᵢ / n
As the Wikipedia page on RMS points out, RMS is a special case of standard deviation when the mean is zero. Hence:

1. Is it correct to say that μᵣₘₛ is the standard deviation of the magnitudes of the velocity (not speed).
2. If it is correct, is there any significance/use of this interpretation?

Answer 1

The root mean square (rms) speed, represented by μᵣₘₛ, is the square root of the average of the squares of the speeds of gas particles. It is a measure of speed that corresponds to the average kinetic energy of the particles in a gas but is not a standard deviation of the velocities. The concept is valuable in physics for understanding the behavior of gases and the effects of temperature changes.

Step-by-step explanation:

The root mean square (rms) speed, denoted as μᵣₘₛ, is a statistical measure and one conceptualization of the speed of gas particles in a collective sense, and it is related to the average kinetic energy of the gas particles. Specifically, μᵣₘₛ is equal to the square root of the average of the squares of the individual gas particles' speeds, and it is used in the context of kinetic molecular theory (KMT) to describe the motion of particles within an ideal gas.

For clarity, the rms speed is distinct from the average speed or the most probable speed given by the Maxwell-Boltzmann distribution; rather, it's a measure of the magnitude of velocity, and it represents the speed of a particle that embodies the average kinetic energy of the gas particles. It's important to note that rms speed is not a standard deviation; as standard deviation measures the spread or dispersion of a set of values around the mean, and in the case of gas molecules in KMT, the mean velocity is zero due to the random motion in different directions.

However, it can be conceptually likened to a standard deviation in a certain sense. If all velocities were distributed symmetrically around zero, then the rms speed would be equivalent to the standard deviation of the velocities. The key here is that we're discussing velocities, which have a direction, as opposed to speeds, which do not. The rms speed has significant implications in physics, particularly when examining the effects of temperature on gases, predicting the distribution of molecular speeds, and understanding thermodynamic properties of gaseous systems. The knowledge of rms speed is crucial for understanding gas behavior under different physical conditions.