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Phy 232
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RMSvelocityproblem
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The masses of 1 mole of various gases are as follows: hydrogen about 2 grams, helium about 4 grams, nitrogen about 28 grams, oxygen about 32 grams and carbon dioxide about 44 grams.
1) If the average (rms) molecular speed of helium atoms in a mixed gas containing all four of these gases is 308 m/s, then what is the average (rms) molecular speed of molecules of each of the other gases?
2) How much total translational KE is there in a mole of this gas?
3) What other forms of KE might there be in this gas?
I ran into this problem and have a question about solving for the translation KE. I understand how to obtain the values in the first part of the question about the other velocities of other elements, however I believe I am missing some steps for the KE. I know that the I can calculate the the kinetic energy by doing (6.02*10^23 particles)*(.5*mass*v^2). And I have all of the info to calculate that KE. Is that what I am solving for or do I have to incorporate the equation KE = (3/2)nRT, and if so, how can I do that?
My work so far has been calculating the KE = (6.02*10^23 particles/mole)*(.5 (.004/6.02*10^23) (308^2)). I believe I have the correct mass for helium and I used its rms velocity in the equation. This KE came out to equal = 189.7 J. Also, is my work here correct?
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You need to include units throughout your calculations. The units reveal the meanings.
The KE of a single particle at temperature T is 3/2 k T.
The KE of a mole of particles is therefore 3/2 R T, since R = N_A * k (N_A being Avagodro's Number).
However that's not the point here.
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The point here is that the average KE of every particle must be the same.
So if the speed of a helium atom is 308 m/s, what would be the speed of an oxygen molecule, which is 8 times as massive?
What is the KE of a single helium atom at 308 m/s (you appear to understand everything you need to calculate the mass of a single atom)?
What therefore is the KE of a mole of these atoms?
You should find that the speed of an oxygen atom is 1 / sqrt(8) times that of the helium, with hydrogen sqrt(2) times the speed of helium?
KE = (6.02*10^23 particles/mole)*(.5 (.004/6.02*10^23) (308^2)) is good, but should have units.
Easier:
KE = N_A * 1/2 * molar mass / N_A * v^2
simplifies to
KE = 1/2 * molar mass * v^2.
4 grams of atoms all moving at 308 m/s^2 have total KE of 1/2 * .004 kg * (308 m/s)^2.
189.7 J looks about right.
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