2024 Pom Lecture 8 Annotated PDF

LECTURE 8 Kinetic theory of gases Maxwell speed distribution Microscopic definition of temperature Reading: Cutnell & Johnson section 14.3 1 Kinetic theory of gases Remember ideal gas: – Low density – but lots of particles – Pa...

LECTURE 8 Kinetic theory of gases Maxwell speed distribution Microscopic definition of temperature Reading: Cutnell & Johnson section 14.3 1 Kinetic theory of gases Remember ideal gas: – Low density – but lots of particles – Particles modelled as hard spheres – Elastic collisions between particles and with walls Lots of particles, all in random motion à each particle undergoes many collisions each second – Collisions change speed and direction of particles Particles will all have different speeds Can still talk about average particle speed 2 Maxwell speed distribution 19th century Scottish physicist James Clerk Maxwell calculated the distribution of particle speeds within a large collection of atoms or molecules at a constant temperature 3 Maxwell speed distribution What does it tell us? Example below: O2 gas at two different temperatures Peak of distribution tells us the most probable speed, vmp, of a particle Peak shifts to the right with increasing temperature 4 Maxwell speed distribution What does it tell us? Another useful benchmark: root-mean-square speed, vrms – Always slightly higher than most probable speed In addition: the average speed, v, which lies in between v = 445 m/s v = 891 m/s 5 Pressure of an ideal gas Gas: Particles bounce off the walls of the container – Similarly to balls thrown at it (or billiard balls off the edge of a pool table) Particles exert force on the wall Large number of particles; frequent collisions with the wall à a continuous force on the wall Pressure by gas on the wall = force of all the collisions divided by area of the wall 6 Pressure of an ideal gas Model: Ideal gas – Elastic collisions – No other interactions N identical particles, each of mass m Cubical container; side has length L 7 Pressure of an ideal gas Approach: Consider one particle – 1D change of momentum Consider N particles – 3D, different speeds – Average force Pressure is force per unit area Average kinetic energy Ideal gas law – Relate pressure to temperature 8 Pressure of an ideal gas One particle, 1D: 9 Pressure of an ideal gas N particles, 3D: 10 Pressure of an ideal gas Pressure = force per unit area: 11 Pressure of an ideal gas Ideal-gas law: 12 Pressure of an ideal gas Note on average speed-squared: 13 Microscopic definition of temperature 1 2 3 KE = mv rms = kT 2 2 € Figure from http://en.wikipedia.org/wiki/Kinetic_energy Temperature is a measure of the average energy of motion, or kinetic energy, of particles in matter 20 SUMMARY Kinetic theory models takes a microscopic look at ideal-gas model – Mechanics of hard-sphere particles making elastic collisions Maxwell speed distribution: v rms = v 2 € Temperature is a measure of the average kinetic energy of particles in matter: 1 2 3 KE = mv rms = kT 2 2 21