# Root mean square speed of helium atom

## Homework Statement

Helium is used to fill a balloon of diameter 44 cm at 22 degrees C and 0.9 atm.

The mass of a helium atom is 4.0026 u, the conversion factor from u to kg is 1.66 x 10^-27 kg/u, the conversion factor from atm to Pa is 1.013 X 10^5 Pa/atm, the universal gas constant is 8.31451 J/K mol, the Boltzmann's constant is 1.38066 x 10^-23 J/K.

What is the average speed (ie, the root mean square speed) of each Helium atom? Answer in units of m/s.

## Homework Equations

Vrms = sqrt(3RT/M)

## The Attempt at a Solution

I think my problem is with finding the mass. I did this:

4.0026 x 1.66 x 10^-27 = 6.644316 x 10^-27

Then:

T = 22C = 295K
R = 8.31451

Sqrt( (3 x 8.31451 x 295)/6.644316 x 10^-27)

I got:

1.0523612 x 10^15

That is wrong. Any help on where I messed up? Thanks.

ehild
Homework Helper
M means the mass of 1 mole of a substance. The mass of 1 mole of He is 4.0026 g. Edit: So M=4.0026 x 10-3 kg/mol.

ehild

Last edited:
kuruman
Homework Helper
Gold Member
2021 Award
If you use the mass of one atom in the denominator, you have to use the Boltzmann constant k in the numerator, not the universal gas constant R.

So it should be:

Vrms = sqrt( 3KbT/m )

And I got:

3 x 1.38066e-23 x 295 / 4.0026 = 3.052725978e-21

sqrt(3.052725978e-21) = 5.52514794e-11

I haven't tried to see if that's right yet. Does it look right to you guys? Thanks again.

ehild
Homework Helper
No, it is wrong again. Accept only one piece of advice :). Either use Kb with the real mass of the He atom (6.644316 x 10^-27 kg)
or the molar mass with R, but not both.

ehild

So I just need to use 4.0026 like I was doing it in the first way correct?

So:

Sqrt( (3 x 8.31451 x 295)/4.0026 )

This one is with the mass of a single molecule.

Or:

Sqrt( (3 x 1.38066e-23 x 295)/6.644316e-27 )

And this one is the mass of the whole atom.

Am I following correctly?

Thanks for the help. I'm pretty much having to teach this to myself. Don't know what I would do without this place. The internet is great.

ehild
Homework Helper
4.0026 is not the mass but the ratio of the mass of one He atom to that of 1/12th of the mass of C12 atom.
NA atoms ( the Avogadro number) have the mass of one mole of substance. In case of He, it is 4.0026 g, equal to 4.0026x10-3 kg.

ehild

Ok. I've been looking up on how to do molar mass stuff, and I think I may have it now.

The formula for molar masses is:

Na = m/M or Avogadro's Number = Number of Moles / Molar Mass

So, for the example of helium, the M in grams is: 4.0026 g/mol

Converting this to kg gives: 4.0026e-3 kg/mol

Using the above formula:

Na = m/M

6.02e23 = m / 4.0026e-3 m = 2.4095652e21

So in the original problem:

Vrms = sqrt(3RT/M) M = 4.0026e-3

Vrms = sqrt( 3KbT/m ) m = 2.4095652e21

Where am I going wrong?

When I was working this out earlier I thought this was right, but I am getting two different answers when I plug them into the equations, so I guess I have made another mistake somewhere.

Thanks for the help.

ehild
Homework Helper
The formula for molar masses is:

Na = m/M or Avogadro's Number = Number of Moles / Molar Mass

No, the molar mass is the mass of NA (about 6x 1023 atoms). Just think, how small an atom is. Can it be 1021kg?

ehild

kuruman
Homework Helper
Gold Member
2021 Award
Na = m/M
This is where you went wrong. Actually, NA=M/m.

For hydrogen, M = 1 gram (to a good approximation); it is no coincidence that the inverse of the proton's mass (in grams) is a number that matches Avogadro's number.

I guess rms & most probable speed are different for group of molicules

rms speed = sqrt (3RT/M)
most probable speed = sqrt (2RT/M)
average speed = sqrt (8RT/πM)

Last edited:
This is where you went wrong. Actually, NA=M/m.

Ok. So I reworked everything out since I had my formula backwards:

NA=M/m

6.02e23 = 4.0026e-3/m

M = 4.0026e-3
m = 6.64883721e-27

Back to the other formulas:

Vrms = sqrt( 3RT / M )

sqrt( (3 x 8.31451 x 295) / 4.0026e-3 ) = 1355.872554

And

Vrms = sqrt( 3KbT / m )

sqrt( (3 x 1.38066e-23 x 295) / 6.64883721e-27 ) = 1355.633077

Maybe I'm closer to right this time? Would those two answers be in m/s if they are right?

Why you tried the two formulas, they are same!!!

3KT/m = 3Rt/mNa = 3RT/M

and for m/s ... it depends on what unites you used in the constants ...

You used all values in SI so speed will be in SI ... i.e. m/s !!!!

Why you tried the two formulas, they are same!!!

3KT/m = 3Rt/mNa = 3RT/M

I used both formulas because I wanted to be sure I had the correct understanding of what m and M were.

and for m/s ... it depends on what unites you used in the constants ...

You used all values in SI so speed will be in SI ... i.e. m/s !!!!

Thanks. Was trying to figure out where the m/s came in. I guess I should have been writing my units down in all my calculations in order to see.

well if you really want to see then this will be a very good homework!!! LOL!!!

kg/mol meanig, 4/6.02 x 10^-27 and it might be changed with 6.64*10^-27kg/mol