# carter tomlenovich

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• 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:
• 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.