Maxwell Boltzmann Speed Distribution

[ramsharmjarm]

Homework Statement

The three quantities Vmost probable, Vaverage, and Vrms, are not the same for the Maxwell speed distribution in 3D. If you restrict the gas to only be one dimensional, are these three quantities equal to each other? Justify your answer with a short explanation.

Homework Equations

The Attempt at a Solution

The curve looks like gaussian curve. I know Vmostprobable is going to be the peak of the curve. And Vaverage would also be at the peak of the curve which is at V=0. I am not sure how to get Vrms

 

[vela]

What's the definition of $v_\text{rms}$?

 

[ramsharmjarm]

Root mean square velocity

 

[vela]

I was looking for perhaps a mathematical formula or some other indication that you know what those words mean.

 

[Nickie]

though.The Maxwell-Boltzmann speed distribution describes the distribution of speeds of particles in a gas at a given temperature. In a 3-dimensional gas, the three quantities Vmost probable, Vaverage, and Vrms are not equal to each other. Vmost probable represents the speed at which the most particles in the gas are moving, Vaverage represents the average speed of all particles in the gas, and Vrms represents the root mean square speed of the particles in the gas. These three quantities are not equal because they are calculated using different methods.

In a one-dimensional gas, however, the three quantities would be equal to each other. This is because the particles in a one-dimensional gas can only move along a single axis, and therefore their speeds are all in the same direction. In this case, the most probable speed, average speed, and root mean square speed would all be calculated using the same method and would have the same value.

To calculate Vrms in a one-dimensional gas, we can use the equation Vrms = √(3RT/M), where R is the gas constant, T is the temperature, and M is the molar mass of the gas. This equation is derived from the kinetic theory of gases and applies to both one-dimensional and three-dimensional gases. However, in a one-dimensional gas, the velocity components in the x, y, and z directions are all equal, so the equation simplifies to Vrms = √(3RT/M).

In conclusion, in a one-dimensional gas, the three quantities Vmost probable, Vaverage, and Vrms would be equal to each other because the particles can only move along a single axis and their speeds are all in the same direction. This is not the case in a three-dimensional gas, where the particles can move in any direction and their speeds are distributed in a more complex manner.