Chapter 4

Heat transfer and diffusion of particles

Random walk

Take 10 minutes to prepare this exercise.

Then, if you lack ideas to begin, look at the given clue and start searching for the solution.

A detailed solution is then proposed to you.

If you have more questions, feel free to ask them on the forum.

Because of the thermal motion, the particles in a gas or a liquid have broken lines trajectories.

Starting from the initial point O at , a particle makes moves ( to ).

After each collision, it goes towards another direction, independent from the previous one.

Question

What is the average distance between O and the particle after ( is huge), where is the average time between to following collisions ?

Express with , (free mean path) and (mean square velocity).

Hint

Solution

After moves , the particle is in :

So, in modulus :

And, as the definition of the free mean path states :

Furthermore, for really huge, the random characteristic of scattering makes cosinus equally distributed between and , so :

It remains, with  :

So :

Question

Numerical application : estimate the time required by a fragrance to diffuse after the opening to a distance ; comment on it.

Hint

Solution

Numericale application : .

Scattering phenomenons are really slow.

In reality, the molecules are heavier than air, so is smaller and is longer, and here is the interest of convection.

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Scattering of peak concentrations
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To test the understanding of the lesson