# Root mean square speed

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Hey guys, another question:

In one of my text books it says to note that the root mean square speed (Crms) of the molecules of a gas is not the same as the mean speed. However, in the other it says that the square root of the mean square speed (Crms)^2 gives the typical speed. I am really confused can someone clarify the definitions of Root mean square speed and mean square speed?

I really don't understand why we use Root mean square speed and not just mean speed or something.

In one of my text books it says to note that the root mean square speed (Crms) of the molecules of a gas is not the same as the mean speed. However, in the other it says that the square root of the mean square speed (Crms)^2 gives the typical speed. I am really confused can someone clarify the definitions of Root mean square speed and mean square speed?

I really don't understand why we use Root mean square speed and not just mean speed or something.

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The so-called "typical speed" in your second textbook is not intended to mean "mean speed". The root-mean-squared (RMS) speed is found by squaring each speed, adding them together, dividing by the number of values and square-rooting the answer. The result is a quantity with the same units of speed as your inputs, but it is not, in general, numerically equal to the mean speed (because the squaring operation is non-associative).

You wouldn't be the first to have some confusion over RMS speeds. Why do we use that?

Think of some gas in a box. The gas is composed of many molecules or atoms moving around in the box. The speed of each individual molecule will vary: some will be moving fast, some slow, so you can say that the collection of molecules (comprising the gas) has a

So far all we talked about were properties of individual molecules and distributions. But to charachterise the gas in a box, we don't want a 10^23-rowed list of the individual molecular speeds and energies, we want to compress it into a small number of statistics. You could take the average of the speeds, but that isn't really very useful, and statistics are only good if they are useful. It is much more useful if we square the speeds before averaging them: that gives us the mean-squared speed of the distribution. That's useful because, as long as all the molecules have the same mass, the average kinetic energy of the particles is proportional to the average of the squares of the speeds. The average kinetic energy is a useful statistic because it is closely related to the temperature of the gas (an even more useful statistic: we can actually measure that).

So you've got a really useful single number that gives you all the required information about the speeds of the particles (for temperature calculations), and that's the mean-squared speed, but it's not very aesthetically-pleasing because it is the sum of a load of squares of speeds and thus has units of m^2/s^2, and therefore it really isn't an actual "speed". So, to obtain a quantity with dimensions of speed, you take the square root. That is the root-mean-squared speed.

You wouldn't be the first to have some confusion over RMS speeds. Why do we use that?

Think of some gas in a box. The gas is composed of many molecules or atoms moving around in the box. The speed of each individual molecule will vary: some will be moving fast, some slow, so you can say that the collection of molecules (comprising the gas) has a

*distribution*of speeds. The kinetic energy of each molecule is proportional to the square of its speed (=0.5*m*v^2), so the gas will also have a (different) distribution of kinetic energies.So far all we talked about were properties of individual molecules and distributions. But to charachterise the gas in a box, we don't want a 10^23-rowed list of the individual molecular speeds and energies, we want to compress it into a small number of statistics. You could take the average of the speeds, but that isn't really very useful, and statistics are only good if they are useful. It is much more useful if we square the speeds before averaging them: that gives us the mean-squared speed of the distribution. That's useful because, as long as all the molecules have the same mass, the average kinetic energy of the particles is proportional to the average of the squares of the speeds. The average kinetic energy is a useful statistic because it is closely related to the temperature of the gas (an even more useful statistic: we can actually measure that).

So you've got a really useful single number that gives you all the required information about the speeds of the particles (for temperature calculations), and that's the mean-squared speed, but it's not very aesthetically-pleasing because it is the sum of a load of squares of speeds and thus has units of m^2/s^2, and therefore it really isn't an actual "speed". So, to obtain a quantity with dimensions of speed, you take the square root. That is the root-mean-squared speed.

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