Have you ever wondered about how fast molecules are moving
in the air when it’s 75 degrees (Fahrenheit that is) outside? It might seem on a calm day that the air is
“still” and therefore the molecules aren’t moving very much. But I think you will be surprised just how
fast molecules are moving (even on a calm day) by walking you through a basic
formula that will help us calculate, at any temperature, what the average speed
of molecules is.

Don’t be intimidated by the math or the symbols, it’s all
very understandable, and when you understand how to use it can be used to get
very interesting results.

However, before we learn how to use this equation and what
the symbols mean, we need some background on the physics of molecules in
general.

Molecules are almost unimaginably small and can exist in
different phases: mainly solid, liquid, gas and plasma phases. To give us a feel for how small molecules
are, if you had a one foot cube of solid oxygen, it would contain approximately
7.25 x 10

^{26}molecules! That is 72,500,000,000,000,000,000,000,000 molecules in a one foot by one foot by one foot box!
Now if that same box was filled with oxygen in gas phase, it
would still contain 6.97 x 10

^{23}molecules! This is about a thousand times less dense than solid phase, but there are still a lot of molecules in that box. And what is even crazier, is that even though there are a lot of molecules in this one foot cube, it’s still mostly empty space because atoms are mostly empty space. The main point I’m trying to drive home here is that molecules are very, very tiny and it is very difficult for us to imagine what must be going on down at that micro level.
Although, it is hard for us to know exactly what is going on
at the molecular level, we do know some things. One is that molecules are constantly
“vibrating” and “jostling” about and when they are in gas phase they are
constantly moving around and colliding with one another. In fact, pressure is the result of both the
mass and motion of molecules; what keeps a balloon inflated is the movement and
collision of molecules, in the gas phase, with the walls of the balloon. And what we experience as temperature, is
also the result of molecules on the move.
In short, the faster molecules move the more temperature and the greater
the pressure they create.

The root mean square velocity equation captures the
relationship between the mass of molecules, the temperature and their average
velocity. In other words, it tells us
that temperature, and the mass and average velocity of molecules are
proportional; as temperature increases, so does the velocity of molecules; it
also tells us that the smaller the mass of the molecules, the greater their
velocity will be relative to other more massive molecules. For example, helium gas (less massive) will
on average be moving at a greater velocity than oxygen gas (more massive) at
any temperature.

We say “average” here because at a given temperature, at any
given moment there will be molecules that are moving very fast and molecules
that are moving very slow. Molecules
that are moving in a straight line, molecules that have just bumped into others
and veered off, and still others that have just hit head on with another
molecule and reversed direction.

V

_{rms}is the symbol for the average velocity of some molecules of interest (in meters per second or meters/second), T is the temperature (in Kelvin), M_{m}is the molar mass of the molecule of interest (in kilograms per mole or kg/mole), and R is the Ideal Gas Constant (8.314 Joules/moles x Kelvin). Kelvin is just another unit of temperature like Fahrenheit or Celsius. Each temperature scale is based upon a different “zero.” Celsius is zero at the freezing point of water while the Kelvin scale is zero at absolute zero. The molar mass of a substance is how much mass a mole (or 6.022 x 10^{23}) of them possesses. If you wanted to know how much the molar mass of basketballs was, you’d have to weigh out 6.022 x 10^{23}basketballs. A mole is just a number of something much like a dozen. It’s a number that gets used a lot by chemists and physicists because most things that we deal with at our level are composed of so many atoms or molecules that it’s much easier to say, “this baseball is composed of three moles of molecules” than it is to say “this baseball is composed of eighteen point eight zero six six times ten to the twenty third power of molecules.”
So how fast are molecules, lets say oxygen molecules, moving
in the air on a 75 degree Fahrenheit day?

First we have to convert 75 degrees Fahrenheit to Celsius
and then to Kelvin, because the equation will only work with temperature in
these units. We can do this all on the Internet
by just typing Fahrenheit to Celsius or Fahrenheit to Kelvin.

75 degrees Fahrenheit = 23.9 degrees Celsius = 297.15 Kelvin

Then we have to look up the molar mass of a molecule of
oxygen on the periodic table and convert it to kilograms.

The mass is 15.999 atomic mass units (amu) which is also equal to 15.999 grams/mole, but in order to do our calculations correctly we need to know that the oxygen in the air we breathe exists as a diatomic molecule (as two oxygen atoms bonded together by a double bond).

Therefore the mass needs to be doubled, 15.999 x 2 = 31.998 atomic mass units (31.998 grams/mole). To get it into kilograms we have to divide by one thousand because there are a thousand grams in a kilogram. If we divide 31.998 by 1000 we get 0.031998 kilograms/mole.

The mass is 15.999 atomic mass units (amu) which is also equal to 15.999 grams/mole, but in order to do our calculations correctly we need to know that the oxygen in the air we breathe exists as a diatomic molecule (as two oxygen atoms bonded together by a double bond).

Therefore the mass needs to be doubled, 15.999 x 2 = 31.998 atomic mass units (31.998 grams/mole). To get it into kilograms we have to divide by one thousand because there are a thousand grams in a kilogram. If we divide 31.998 by 1000 we get 0.031998 kilograms/mole.

Let’s remember our equation we’ll be using.

When we plug in all of our numbers we get an expression that looks something like this,

When we plug in all of our numbers we get an expression that looks something like this,

V

_{rms}= √ 3 x (8.314 Joules/moles x Kelvin) x (297.15 Kelvin) / (0.031998 kg/mole)
More simply, we just need to multiply 3, 8.314 and 297.15. Then divide by 0.031998. And finally, square root our answer.

If we do all of these calculations we get an answer of 481.3
meters/second or 1077 miles per hour!
This means that on a “still” day when the temperature is 75 degrees
Fahrenheit, the oxygen molecules in the air, on average, are moving the same
speed as .22 caliber bullets………which is, by the way, faster than the speed of
sound!

So, why, if molecules are moving so fast, don’t we feel a
constant hurricane? How can there be
anything like a calm day?

Well as it turns out, even though on average molecules are
moving pretty fast, they collide so often with one another (it is estimated, in
gas phase at normal atmospheric pressure, that any given molecule will
experience about a billion collisions per second!) and as result change course
so often, that in essence they move very fast, but in a very confined space and
don’t get very far. They are running in
place if you like.

So there you have it.
Hopefully, you endured the math and saw how a very simple tool like the
root mean square velocity equation can tell us something very profound and
amazing about the world around us.

--Seth Commichaux

Sources: Principles of Chemistry. Nivaldo Tro.