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Ch 24 Notice: we are following a different route from the book Getting to the same end-point though

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Lecture 4 – Kinetic Theory of Ideal Gases

Ch 24

Notice: we are following a different route from the book

Getting to the same end-point though

- The ideal Gas Law: PV=nRT is an equation of state deduced from experiments (i.e. Boyle’s and Charles’ Laws). It introduces pressure as a strictly macroscopic concept
- Classical mechanical principles express pressure as a mechanical (microscopic) property in the Kinetic Theory of Gases:

- Statistical thermodynamics provided yet another description of an ideal gas from the statistical mechanical point of view

- All together, we have provided a microscopic interpretation of temperature in terms of the average kinetic energy per molecule:

Maxwell-Boltzmann speed distribution

- Maxwell-Boltzmann speed distribution function provides the probability for each molecule in the gas having a certain speed

- We will see how to use this distribution in the next exercise

Maxwell-Boltzmann speed distribution

- What is the fraction of O2 molecules in a container held at T=1000K that have a speed of 1500m/s?

- Use the expression:

Calculate the mass m for a single O2 molecule (in kilograms):

Average Kinetic Energy, Speed, Root-Mean-Square Speed

- According to the basic principles of statistical mechanics, we can calculate the average of any quantity that depends on the speed c using the Maxwell-Boltzmann distribution function, e.g.

Average Kinetic Energy, Speed, Root-Mean-Square Speed

- For example, let us calculate average kinetic energy:

- Let us use the integral:

Average Kinetic Energy, Speed, Root-Mean-Square Speed

We have found again that the average kinetic energy:

Since:

This is yet another (statistical mechanical) derivation of the ideal gas law

Average Kinetic Energy, Speed, Root-Mean-Square Speed

- Let us derive the energy of an ideal gas using the Boltzmann distribution function. The energy of an ideal gas is the average molecular kinetic energy:

Hence:

Average Kinetic Energy, Speed, Root-Mean-Square Speed

- We can provide other quantities that will become useful in the next few lectures
- For example, the root-mean squared speed (square root of the average squared speed)

The rms speed is not equal to the average speed:

Average Kinetic Energy, Speed, Root-Mean-Square Speed

- Let us calculate the average speed; this integral again has standard form

With a=1 and

Average Kinetic Energy, Speed, Root-Mean-Square Speed

- Note once again that:

This difference is very important

Collision Rates, Mean Free Path, Diffusion

- How molecules in a gas move

The zig-zag motion of the molecules in a gas is called diffusion. It explains the characteristic slowness of gases (straight line speed for O2 at room T is about 400 m/s!)

Collision Rates, Mean Free Path, Diffusion

- Statistical techniques and the velocity distribution function can be used to calculate properties of gases:

wall collision rate Zw

molecular collision rate Z1

mean-free path l

diffusion constant D

We will give only approximate derivations for these quantities; more thorough derivations yield only small correction factors.

(rate at which molecules collide with wall)

We expect this to be proportional to:

density N/V

average speed

surface area

A better derivation accounting for the directionality of molecular motion would provide:

We can also derive this expression from M-B distribution by realizing that if a particle has speed c and is within a distance c(dt) from a surface A will hit the surface within time dt

All particle within a volume Acdt will hit the wall in time dt; how many particles are there?

The number of collisions per unit time will be:

(the rate at which molecules collide with each other)

If a molecule has diameter d and speed c, it moved cdt within a time dt=1s tracing a cylindrical volume:

The number of collisions per unit time will be simply the number of molecules found in that volume

(the rate at which molecules collide with each other)

An exact calculation (assuming the other molecules move as well, as is correct) yields

(how far a molecule travel before colliding with another)

This is simply the average speed divided by the collision rate

The zig-zag motion of the molecules in a gas is called diffusion. It explains the characteristic slowness of gases

The mean square displacement associated with this motion is given by:

D is called diffusion coefficient and has units of m2/s

D is proportional to the mean free path l and the average speed:

The proportionality constant is difficult to calculate, and depends on the properties of the gas; for a single component gas, for example:

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