root mean square speed of helium atom

dlthompson81

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

Vrms = sqrt(3RT/M)

3. 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

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

kuruman

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.

dlthompson81

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

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

dlthompson81

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

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.
N_A 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

dlthompson81

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

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

ehild

kuruman

This is where you went wrong. Actually, N_A=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.

cupid.callin

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)

dlthompson81

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

N_A=M/m

6.02e23 = 4.0026e-3/m

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

Back to the other formulas:

V_rms = sqrt( 3RT / M )

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

And

V_rms = sqrt( 3K_bT / 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?

cupid.callin

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 !!!!

dlthompson81

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

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.

cupid.callin

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

Choise

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