Kinetic Theory 2

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Kinetic Theory 2

• Linking Kinetic Theory and Temperature

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Key Ideas

• We are going to relate the rms speed of gas
  molecules to the temperature of the gas.
• Temperature is a measure of the internal energy
  of a material.
• The higher the temperature in a gas, the faster
  the molecules will travel.

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Some Assumptions

• The internal energy is represented by the
  potential energy and kinetic energy in the bonds
  of any material.
• In a gas, the molecules are so far apart that
  there are no intermolecular interactions. So
  there is no potential energy.
• Therefore the energy is entirely kinetic.

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Kinetic Energy in a Gas

• Kinetic energy is shared randomly throughout the
  gas.
• This means that any given molecule can be moving
  at any velocity.
• Because there are lots of molecules, there is a
  random distribution of speeds. A few molecules
  will travel slowly, a few very fast, while most
  will be somewhere in between.

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We can show this on a graph
100 K
1000 K
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Thermal Equilibrium

• If no energy is being transferred as heat between
  an object and its surroundings, we say that it is
  in thermal equilibrium.
• If a gas is in thermal equilibrium and it is not
  being compressed or expanded, the average kinetic
  energy will remain constant, so will the
  temperature.

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Two Relationships from Before
pV nRT and pV 1/3 Nmc2
It does not take a genius to see that nRT 1/3
Nmc2
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Lets look at this relationship further
nRT 1/3 Nmc2
Mean square speed
Mass of each molecule
No of molecules
Total mass of the molecules
It reminds us of the relationship that describes
kinetic energy Ek 1/2 mv2
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So we can write
nRT 2/3 N1/2mc2 since 2/3 1/2 1/3.
Rearranging gives us 1/2mc2 3/2nRT 3/2RT
N N/n
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Since N is the total number of molecules and n is
the number of moles, N/n will always be NA, which
is Avagadros number, 6.02 x 1023. So we can
write 1/2mc2 3/2RT NA
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Boltzmanns Constant
R/NA is called Boltzmanns constant and is given
the code k. We can easily work out the value for
k. k 8.31 1.38 x 10-23 J K-1
6.02 x 1023
And this gives us our final relationship 1/2mc2
3/2kT