Kinetic Theory

Kinetic Theory of Gases

Overview • Assume atomic picture of gases – Simpler than solids/liquids, as interactions can be neglected

• Predict behavior – E.g., relations between P and V, P and T…

• Test in lab experiments

Basic Picture • Gas consists of noninteracting particles • They move around randomly • Temperature corresponds to (average) speed of particles – Hotter ↔ faster

• Pressure a manifestation of collisions with container walls

Basic Processes • Thermal expansion • Evaporation – A cooling process

• Dissolving solids in liquids • Reaction rates • …

More on Temperature • Prediction of kinetic theory: 1 2 3 mv = k BT 2 2

v is the average speed T is the temperature (in Kelvins) m is the mass of a gas particle kB is Boltzmann’s constant

• Note that

T ∝v

2

More on Pressure Weight W

• Canonical example: container wih movable piston • P is the average force per unit area due to collisions with walls – Average because it fluctuates

• Weight on piston balances this force, in equilibrium – W tells us P of gas

Now change something… • E.g. add weight to the piston (T = const) • Forces out of equilibrium; piston drops • Collision rate increases until forces again balance • P has increased, V decreased • In fact,

P ∝1 V

(Boyle)

Computer Simulation • Allows changing N, W, v • Replaces tedious mathematical analysis • Explore all relations encoded in the Ideal Gas Law: PV = NkBT • Most of these relations are qualitatively obvious, some even quantitatively so!

Another Example • Increase T keeping P fixed – Note: doubling T means increasing v by 2

• Faster particles means harder collisions and more rapid • Piston rises, reducing collision rate • Equilibrium is restored • Model gives

V ∝T

(constant P)

Another Example • Increase N with P and T held fixed • More particles means more collisions, piston rises • Reduced collision rate restores equilibrium • In detail:

V∝N

(constant T, P)

A slightly more complicated one… • Increase T with V and N held constant • Do it in two steps: – Increase T with P unchanged – Increase W to return V to its original value

• Result:

P ∝T

(constant V, N)

Verifying the Predictions • These relations are simple predictions of atomic/kinetic theory • If they are found to hold in experiments, we gain confidence that the atomic picture is correct! • Several of them are easily checked in lab exercises

Sample Exercises • Calculate v for gas at room temperature • It may take a few seconds for a smell to reach you from across a room, e.g. from a perfume bottle. What does this suggest about the path taken by the perfume particles?

Reference • R. P. Feynman, et al., The Feynman Lectures on Physics, v. I (Addison Wesley, 1970)