# 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:
- The volume of an individual gas molecule is negligible
- The average kinetic energy of a gas molecule ($1/2mv^2$) is proportional to temperature.
- 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:
- 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.