Understanding Kinetic Molecular Theory and Graham's Law

In summary, the equation for average kinetic energy of an ideal gas is as follows: E = 1/2 mv2. First, the equation is based in the kinetic energy of a point particle, molecular level. Second, the equation jumps to average kinetic energy of an ideal gas when n=1. Third, the equation for root-mean-square speed is used to calculate average kinetic energy. Fourth, the constant 8.314 is used in all equations. Fifth, the units of the radicand are J/g, the square of speed, and kg/m^2, the square of mass. Sixth, the units of the rms equation is J/g, the square of speed, and PSI, pressure in
  • #1
revacious
11
0

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 Eaverage, 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 ratex = constant/sqrt(molar massx)?

thanks!
 
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  • #2
There's a property of the electron called the Root-mean-square speed. The formula for root-mean-square speed is something like this:

[tex]v_{rms} = \sqrt{\frac{3RT}{M_m}}[/tex]

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.
 
  • #3
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?
 
  • #4
Well, the constant is always 8.314, and here's the reason...

The units of the radicand are ...

[tex](\frac{J}{mol-K})(K)(\frac{mol}{g}) =
\frac{J-mol-K}{mol-K-g} = \frac{J}{g}[/tex]

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.
 
  • #5
Char. Limit said:
There's a property of the electron called the Root-mean-square speed

This is not property of the electron :bugeye:

revacious said:
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.

--
 
  • #6
I meant to say gas...
 
  • #7
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  • #8
okay, thanks for the input, i think i get this concept better now :)
 

1. What is Kinetic Molecular Theory?

Kinetic Molecular Theory is a scientific theory that explains the behavior of gases at the molecular level. It states that gases consist of small particles in constant motion and that the properties of gases can be explained by the motion of these particles.

2. What are the postulates of Kinetic Molecular Theory?

The postulates of Kinetic Molecular Theory include: 1) Gas particles are in constant, random motion; 2) Gas particles have negligible volume and are far apart from each other; 3) Gas particles do not exert attractive or repulsive forces on each other; 4) The average kinetic energy of gas particles is directly proportional to the temperature of the gas; and 5) Collisions between gas particles are elastic.

3. How does Kinetic Molecular Theory explain gas pressure?

According to Kinetic Molecular Theory, gas pressure is caused by the collisions of gas particles with the walls of the container. The more frequent and energetic the collisions, the higher the gas pressure. This is also why increasing the temperature or decreasing the volume of a gas will increase its pressure, as it results in more collisions between particles and the container walls.

4. What is Graham's Law and how does it relate to Kinetic Molecular Theory?

Graham's Law states that the rate of effusion (the escape of gas particles through a tiny opening) of a gas is inversely proportional to the square root of its molar mass. This means that lighter gas particles will effuse faster than heavier ones. This law is based on the principles of Kinetic Molecular Theory, as lighter gas particles have higher average kinetic energy and therefore move faster than heavier particles.

5. How does Kinetic Molecular Theory explain the behavior of gases at different temperatures and pressures?

Kinetic Molecular Theory explains that at higher temperatures, gas particles have more kinetic energy and therefore move faster, resulting in higher pressure and a larger volume. At lower temperatures, gas particles have less kinetic energy and move slower, resulting in lower pressure and a smaller volume. Similarly, at higher pressures, gas particles are forced closer together and have more frequent collisions, resulting in a decrease in volume and an increase in pressure.

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