Lecture 5 – Transport pr operties of gases Ch 24 pages 625-627 Summar y of lecture 4 We can use Maxwell-Boltzmann distribution to calculate the average values for any property that depends on speed, e.g. Summar y of lecture 4
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away from where it started. The mean displacement is always 0, because the probability of moving back or forward is the same. For any random walk, the root mean square displacement is the microscopic unit displacement times the square root of the total number of hops. This is a fundamental property of random walk.
Let us consider now the random diffusion of a molecule in a gas; each molecule will move in a straight line until it encounters another molecule with which it collides
The length of each step the random walk is the mean free path l,
The rate at which the step will be interrupted is determined by the collisional rate, the number of collision per second
For the random diffusion of molecules in a gas, the mean square displacement of each molecule is:
Thus, although the average speed of a particle in a gas is very high (>100 m/s) it takes a very long time for molecules to diffuse in a gas because the root mean square displacement is inversely proportional to the rate at which molecules collide
The probability W of a random walk of N steps having taken m steps forward is:
Once we know the probability, we know everything (in a statistical sense). For example, the average number of steps forward is:
The mean of the square of the number of steps forward is:
We have expressed the probability in terms of a number displacement, but it is useful to do so in terms of the net distance displacement x=lm where l is the unit displacement (the length of each step taken). Through a simple substitution:
Suppose the number of hops or steps per unit time is N’. Then the number of hops N=N’t. Therefore we can also express this probability in terms of frequency of steps and time: