Understanding Kinetic Molecular Theory and Graham's Law

[revacious]

Homework Statement

Ok, firstly, I apologise for posting something which is probably trivial to any physics student, but my understanding of physics is pretty poor, so baby steps would be appreciated!

Homework Equations

PV=nRT
E = 1/2 mv²

The Attempt at a Solution

My understanding of it so far:

At the molecular level, a point particle of an ideal gas (ignoring rotational/vibrational components and forces between particles) has kinetic energy equal to:
E = 1/2 mv²

But particles are constantly colliding with each other as well as the walls of the container. Hence an expression for average kinetic energy tells us more information. My course notes give this as:
Ebar = 1/2 mv(bar)^-2

an explanation as to what that means and how they got there would be nice :S

And then there's another jump to average molar kinetic energy of an ideal gas, which is given in a different form by different sources. If I have reasoned correctly, the form in my notes for E_{average, molar} = 3/2 RT = 3/2 PV when n=1 in the ideal gas equation.

but how did they jump from 1/2 mv² to this? No doubt avogadro's constant comes into play, but its clearly not as simple as taking the second expression and multiplying.

And finally, the jump to graham's law. I understand and accept that kinetic energy is only dependant on the temperature; hence two gases at equal temperatures have equal kinetic energy.

If the rate of diffusion/effusion is a velocity, then how did we get to rate_x = constant/sqrt(molar mass_x)?

thanks!

 

[Char. Limit]

There's a property of the electron called the Root-mean-square speed. The formula for root-mean-square speed is something like this:

$$v_{rms} = \sqrt{\frac{3RT}{M_m}}$$

Where v is the rms speed, M_m is the molar mass of the gas, T is temperature, R is 8.314, and 3 is 3.

Plug the rms speed into the kinetic energy of a particle equation and you should get the average kinetic energy of an ideal gas. But don't trust me. (seriously, don't.) Try it.

 

[revacious]

Ok i figured half of it out, it was meant to be a v with a bar instead of v^-2 for the second equation. gah, i feel stupid now.

And with the rms equation, is the constant always 8.314? Would we ever need to use the other versions of the gas constant?

 

[Char. Limit]

Well, the constant is always 8.314, and here's the reason...

The units of the radicand are ...

$$(\frac{J}{mol-K})(K)(\frac{mol}{g}) = \frac{J-mol-K}{mol-K-g} = \frac{J}{g}$$

And J/g is near m^2/s^2, the square of speed, and then the square root takes care of everything.

But, you ask, Joules aren't g-m^2/s^2, they're kg-m^2/s^2. Well, I'm pretty sure that you can ignore this for the same reason that you can throw a 3 in the equation... it's all about proportionality.

 

[Borek]

There's a property of the electron called the Root-mean-square speed

This is not property of the electron

And with the rms equation, is the constant always 8.314? Would we ever need to use the other versions of the gas constant?

Depends on what units are other values expressed in. If you are given pressure in PSI and volume in cubic feet you may prefer other R value. It is all about convenience. But it is different just because it is expressed in different units, physical sense it still the same.

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[Char. Limit]

I meant to say gas...

 

[Borek]

A little bit better, but I still think you are partially wrong. It is a property of any set of moving objects. Molecules of gas in particular. Compare http://en.wikipedia.org/wiki/Root_mean_square

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[revacious]

okay, thanks for the input, i think i get this concept better now :)