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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

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Summary of lecture 3 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:

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Summary of lecture 3 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:

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**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

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**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):

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**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.

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

For example, let us calculate average kinetic energy: Let us use the integral:

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

For us, a=2 and Hence:

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**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

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**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:

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**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:

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

Let us calculate the average speed; this integral again has standard form With a=1 and

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

Note once again that: This difference is very important

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**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!)

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**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.

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**Wall Collision Rates (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:

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Wall Collision Rates 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:

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**Molecular Collision Rates**

(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

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**Molecular Collision Rates**

(the rate at which molecules collide with each other) An exact calculation (assuming the other molecules move as well, as is correct) yields

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Mean Free Path (how far a molecule travel before colliding with another) This is simply the average speed divided by the collision rate

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**Diffusion coefficient**

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

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**Diffusion coefficient**

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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We consider a dilute gas of N atoms in a box in the classical limit Kinetic theory de Broglie de Broglie wave length with density classical limit high.

We consider a dilute gas of N atoms in a box in the classical limit Kinetic theory de Broglie de Broglie wave length with density classical limit high.

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