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

• Kaushik
In summary: The rms is a way of converting the average of the squares of the speeds into a number that is roughly equivalent to the speed.In summary, the reason for using ##v_{rms}## as a new kind of average in the Kinetic theory of gases is because the regular arithmetic mean of velocities would be zero, making it less useful. The root mean square velocity accounts for the sign of the velocities and is also associated with the equipartition of energy, allowing for a better measure of the average kinetic energy of the molecules. This method also allows for the measurement of temperature, which is not possible with the average velocity.
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?

Kaushik said:
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).

Kaushik
Kaushik said:
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.

Kaushik

## What is ##v_{rms}## in the Kinetic Theory Of Gases?

##v_{rms}## stands for root-mean-square velocity and it is a measure of the average speed of gas particles in a system. It takes into account the velocities of all gas particles, not just the most probable velocity.

## How is ##v_{rms}## calculated?

To calculate ##v_{rms}##, you can use the equation ##v_{rms} = \sqrt{\frac{3RT}{M}}##, where R is the gas constant, T is the temperature in Kelvin, and M is the molar mass of the gas.

## What is the significance of ##v_{rms}## in the Kinetic Theory Of Gases?

##v_{rms}## is important because it helps us understand the behavior of gas particles in a system. It can be used to calculate other important properties, such as pressure and temperature.

## How does ##v_{rms}## change with temperature and molar mass?

As temperature increases, the average speed of gas particles also increases, leading to a higher ##v_{rms}##. On the other hand, as molar mass increases, the average speed of gas particles decreases, resulting in a lower ##v_{rms}##.

## What is the relationship between ##v_{rms}## and the kinetic energy of gas particles?

The kinetic energy of gas particles is directly proportional to ##v_{rms}##. This means that as ##v_{rms}## increases, so does the kinetic energy of gas particles.

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