Thermodynamics

1
Thermodynamics

• Kinetic Theory of Gases
• (Section 10.6)

2
Physicists try to understand heat and energy in
terms of basic physics principles

• Momentum conservation
• Energy Conservation
• Newtons Laws
• 1023 particles moving randomly

3
Molecular Model for Pressure of an Ideal
GasASSUMPTIONS

1. The number of molecules is large and the
   particles are small.
2. The particles are all classical, obeying Newtons
   laws, but their motion is random (no synchronized
   swimming)
3. Particles undergo completely elastic collisions
   with the walls of the container. Thus kinetic
   energy is constant.
4. The forces between the molecules are negligible
   except during a collision.
5. The gas is a pure substance. All molecules are
   identical

4
One at a time

• Instead of trying to track all the particles,
  lets look at just one particle.
• Just the x component
• Making an ideal collision with just one wall
• Use this average collision and extrapolate to all
  three directions and all N particles.

5

• One collision ?px pf pi -2(pi) -2mvx
• F1?t ?p 2mvx For each collision
• . The next collision it makes with that same
  wall, it travels a distance 2d at speed vx so
•  
• vx 2d/?t
• so ?t 2d/vx is the time interval between
  collisions.
•  
• The average force on a wall is therefore
• F1 2mvx/?t 2mvx/(2d/vx) mvx2/d
• F1 (m/d) vx2
•  
•  

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• If F1 (m/d) vx2 is the force for 1 object,
• then N object have a total Force of
• F (m/d)?(vx12 vx22 vx32 vx42 vxN2)
• But all the particles are really randomly moving
  about, so lets look at the average value of the
  squares of the velocities.
• (vx2)av (vx12 vx22 vx32 vx42 vxN2)/N
• F (m/d) N(vx2)av

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• This was Just the x direction
• We assume that every direction is the same, and
  using Pythagorean theorem in 3-dimensions, we
  get
• So all molecules together have an average
  velocity
• But, all three directions are equivalent,
• and so we get

• v2 vx2 vy2 vz2
• v2av(vx2)av(vy2)av (vz2)av
• (vx2)av (vy2)av (vz2)av
• v2av 3(vx2)av

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Putting it together

• Total Average Velocity squared of a particle
• Force on a wall by N particles
• Total Force on the wall
• Pressure PF/A F/d2
• Pressure of an Ideal Gas

• v2av 3(vx2)av
• F (m/d) N(vx2)av
• F (m/d)N(v2av/3)
• (N/3)(mv2av)/d
• P (N/3)(mv2av)/d3
• P 2(N/3)(½ mv2av)/V
• P (2/3)(N/V)(½ mv2av)

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PRESSURE OF AN IDEAL GAS

• P (2/3)(N/V)(½ mv2av)
• Proportional to the number of molecules per unit
  volume
• Proportional to the average translational kinetic
  energy of the molecules
• We have linked the
• LARGE WORLD to the SMALL WORLD

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BUT WAIT!! (theres more)

• Now we can understand temperature
• P (2/3)(N/V)(½ mv2av)
• PV(2/3)(N)(½ mv2av)
• PV NkBT
• T (2/3kB)(½ mv2av)
• Temperature is a direct measure of average
  molecular kinetic energy!
• The energy of a molecule tells us its
  temperature!
• (½ mv2av) (3/2)kBT

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Temperature is energy

• (½ mv2av) (3/2)kBT
• Note 3 stands for 3 dimensions.
• Each component contributed equally to the total
  translational kinetic energy
• vrms ?(v2av) ?(3kBT/m )
• This is known as the rms speed of a molecule.
• Notice that as Temp goes up. vrms goes up.
• Notice that as mass goes up. vrms goes down.

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rms Speeds and escape velocity
Gas Molar Mass Vrms at 20oC
H2 2.02 x 10-3 1902
He 4.0 x 10-3 1352
H2O 18 x 10-3 637
N2 or CO 28 x 10-3 511
CO2 44 x 10-3 408