# Rms speed

## Homework Statement

Calculate the rms speed of one nitrogen molecule at 27 Celcius

## Homework Equations

sqrt(v^2) = sqrt((3kT)/(m))

## The Attempt at a Solution

sqrt(v^2) = sqrt((3kT)/(m))
k = 8.62*10^-5 eV/K
T = 300 K
3kT = 0.078 eV
m = 14 g/mol
m = 0.014 kg/mol

3kT/m = 0.078 eV/0.014 kg/mol = 5.571
if 5.571 was 557.1 m/s I would assume it was correct but I doing something completely wrong here, any help will be appreciated. edit nevermind that statement since I have to take the sqaure root also

sqrt(5.571)

Last edited:

## Answers and Replies

Bystander
Science Advisor
Homework Helper
Gold Member
27 Kelvin

Yea sorry it is 27 Celcius which is 300 K :)

Bystander
Science Advisor
Homework Helper
Gold Member
nitrogen molecule
Diatomic molecule.

Diatomic molecule.
increasing the weight of the molecule does not solve the problem when you would expect it going about 500 m/s on average.

Bystander
Science Advisor
Homework Helper
Gold Member
300 m/s: eV conversion, or something equally fatal.

DrClaude
Mentor
3kT/m = 0.078 eV/0.014 kg/mol = 5.571
Units?

Units?

3kT/m = 0.078 eV/0.014 kg/mol = 5.571 eV*kg/mol

DrClaude
Mentor
3kT/m = 0.078 eV/0.014 kg/mol = 5.571 eV*kg/mol
And how do you go from there to (m/s)2?

And how do you go from there to (m/s)2?
1 eV = 1.6 * 10^-19 J, so adding this to the 3kT/m = (0.078 eV *(1.6 *10*-19 J))/0.014 kg/mol = 8.9264^-19 kg*J/mol
J = N*m = (kg*m^2)/s^2
8.9264^-19 kg^2*m^2/s^2*mol
hmm something fishy is going on here, no idea how kg^2/mol relate if they do at all

Oh wait.
it is 8.9264^-19 m^2/s^2*mol
Still have the 1/mol problem though

DrClaude
Mentor
Still have the 1/mol problem though
Yes, because you are taking m as the mass of one mole of atomic N, not as the mass of one molecule of N2.

According to my book Boltzmann's constant is 8.62*10^-5 eV/K
but every place online I look at problems similar they use 8.3145 J/mol*K
and if I do this problem method I have seen online, I get 3*(8.3145 J/mol*K)(300 K) / 0.014 kg/mol = 5.345^5 J/kg which makes the units a lot easier to process for me, kg *m^2/ kg*s^2
which gives sqrt(5.345^5 m^2/s^2) = 66.049 m/s , which is still quite slow but better.

DrClaude
Mentor
According to my book Boltzmann's constant is 8.62*10^-5 eV/K
but every place online I look at problems similar they use 8.3145 J/mol*K
Your value for the Boltzmann constant is correct. The other you cite is the gas constant, defined as ##R = N_A k_B##, where ##N_A## is the Avogadro constant. It is used mostly by chemists.

and if I do this problem method I have seen online, I get 3*(8.3145 J/mol*K)(300 K) / 0.014 kg/mol = 5.345^5 J/kg which makes the units a lot easier to process for me, kg *m^2/ kg*s^2
You are still not taking into account that you have diatomic molecules of N2.

which gives sqrt(5.345^5 m^2/s^2) = 66.049 m/s , which is still quite slow but better.
That should be 5.345×105, not 5.3455.

Yes, because you are taking m as the mass of one mole of atomic N, not as the mass of one molecule of N2.
Oh yea im missing Avogadro's constant and I also forgot 1 atom is not 1 molecule.

(28.0134 g/mol) / (6.02*10^23 mol^-1) = 4.65*10^-23 g = 4.65*10^-26 kg

3*(8.62*10^-5 eV/K)*(300 K) / (4.65*10^-26 kg) = 1.6092^29 m^2/s^2

v = sqrt(1.6092^29 m^2/s^2) = 990.479 m/s
that is more like it, now it is super fast. thank you very much for all your help :)

Your value for the Boltzmann constant is correct. The other you cite is the gas constant, defined as ##R = N_A k_B##, where ##N_A## is the Avogadro constant. It is used mostly by chemists.

You are still not taking into account that you have diatomic molecules of N2.

That should be 5.345×105, not 5.3455.
took me some time to write the above, but thank you for clarifying :)

DrClaude
Mentor
3*(8.62*10^-5 eV/K)*(300 K) / (4.65*10^-26 kg) = 1.6092^29 m^2/s^2
The number is not correct for units of m2/s2

v = sqrt(1.6092^29 m^2/s^2) = 990.479 m/s
that is more like it, now it is super fast. thank you very much for all your help :)
That speed is not the correct one, it is too big. Are you converting from eV to J correctly?

• KUphysstudent
The number is not correct for units of m2/s2

That speed is not the correct one, it is too big. Are you converting from eV to J correctly?

No it looks like I made a mistake with the power.
28.0134/6.02*10^23 = 4.6533^-23
(3(8.62*10^-5)(300))/(4.65*10^-26) = 1.66838*10^24
sqrt(1.66838^24) = 465.09 m/s