Estimating Air Molecule Motion at STP

[mlostrac]

Homework Statement

From the average separation between air molecules at STP, and their mean speed, estimate how long it would take one molecule to move into the region occupied by another. Assume that air consists mainly of nitrogen molecules.

Homework Equations

1/2mv^2 = 3/2KT

The Attempt at a Solution

Distance between molecules at STP:

(6.02 x 10^23) x D^3 = 22.4 x 10^-3 m^3
D^3 = 3sqrt 22.4x10^-3 / 6.02 x 10^23
D = 3.34 x 10^-9 m

Since mass of N2 = 28 g/ mol, I used this to find the average velocity of the molecules:

1/2mv^2 = 3/2KT
v(rms) = sqrt(3KT/m)
= sqrt (3K (273) / .028 kg
= 6.35 x 10^-10 m/s

I then just plugged my two answers from above into the velocity equation,
velocity = distance/time
time = D/V
time = 3.34 x 10^-9 / 6.25 x 10^-10 = 5.3 seconds

Does what I did make sense? The velocity looks awfully small, so I'm not sure. Any help appreciated!

 

[dulrich]

The "m" is the mass of each molecule. You've noted that the mass for each mole is 0.028 kg. You need to divide this by 6.02E23 to get the mass per molecule.

 

[mlostrac]

Ah ha! Good eye, thank you for that.

My new velocity is 493 m/s, and my new time is therefore 6.77 x 10^-12 seconds.

Do these numbers make sense?

 

[dulrich]

I think so. At first I was thinking your distance was way too small (it's on the order of the atomic width). But I can't see anything wrong with logic. So I googled "average separation of atoms in a gas" and got the following link:

http://www.practicalphysics.org/go/Guidance_34.html;jsessionid=alZLdQlAHb1?topic_id=4&guidance_id=1

and one from here:

https://www.physicsforums.com/showthread.php?t=43728

Both confirm your number.

 

[mlostrac]

Thanks very much for all your help