carter tomlenovich

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Kinetic Theory of Gases

  • The “macroscoptic obervations of Boyle, Charles, Avogadro and others may be explained from molecular properties.
  • This approach is called “The Kinetic Theory of Gases” and based on some reasonable assumptions:
    1. The volume of an individual gas molecule is negligible
    2. The average kinetic energy of a gas molecule ($1/2mv^2$) is proportional to temperature.
    3. All gas molecule collisions are completely “elastic”

Qualitative Explanations

  • Boyle’s Law: Pressure-Volume ($PV=c$)
    • Decreased V by 2 –> Double gas molecules per V –> Double rate of collisions on wall –> Double force on wall per unit of area
  • Charles’s Law: Volume-Temperature ($V=cT$)
    • Increase T –> Increase kinetic energy per molecule –> increase rate of collision, and force of collision on wall –> Must inscrease volume to bring pressure back to ambient value.
  • Avogadro’s Law: Amount-Volume ($V=cn$)
    • Double n –> double rate of collisions at wall –> double pressure –> double volume to bring pressure back to ambient value.

Molecular Velocities in a Gas

  • Gas molecules move at different volecities: Molecular Velocities
    • Lighter molecules and atoms move more quickly on average ???

Root Mean Square Velocities

  • One estimate of the average velocity of gas molecules is called the RMS velocity
    • $u_{RMS} = \sqrt{\bar{u}^2}$
  • Here, $\bar{u}^2$ is the “average (or mean) of the squares of the particle velocities”.
Velocity Velocity Squarted
5 25
9 81
3 9
10 100
2 4
  • SUM: 219
  • AVG: 43.8
  • SQRT(AVG): 6.6

### Average Kinetic Energy

  • The average kinetic energy of a mole of gas molecule is:
    • $KE_{avg} = N_a\frac{1}{2}m\bar{u}^2$
  • From the qui-partitioning theory of mechanics, we have that the average energy of a gas molecule at T is also:
    • $KE_{avg} = \frac{3}{2}RT$
  • Bringing these together, we have:
    • $\sqrt{\bar{u}^2} = u_{RMS} = \sqrt{\frac{3RT}{N_am}}$

RMS Velocity Consequences

  • If we recognize that $N_a*m = M$,
    • $u_{RMS} = \sqrt{3RT}{M}$
  • From this, we see that the average velocity is:
    • Directly proportial to $\sqrt{T}$: For a given type of molecule, hot molecules move faster than cooler ones.
    • Inversely proportial to $\sqrt{M}$: At a given timperature, heavier molecules move slower than light molecules.