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lecture 32 - Diffusion A molecule in a gas or a...

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Unformatted text preview: Diffusion A molecule in a gas or a liquid frequently collides with other molecules and, as a result, its trajectory looks like a random walk. Here we will derive some properties of such motion. First, consider a random walker in one dimension. At each step, our walker can jump a distance λ either left (l) or right (r), with equal probabilities. We would like to find out what is the mean distance x traveled by our random walker in say n steps. We will also calculate the mean distance square <x2>. Below is a table showing our results for n = 1, 2, and 3: n 1 2 3 possibilities r, l rr, ll, lr, rl rrr, rll, rlr, rrl, lrr, lll, lrl, llr <x> 0 0 0 <x2> λ 2 2 λ 2 3 λ 2 The mean distance traveled is always zero. This is not hard to guess because our walker is equally likely to go right or left, so , on the average, it doesn’t go anywhere. However the mean square of the deviation is not zero: the more steps our particle makes, the more likely to find it far from the origin. From this table, we can guess: <x2> = n λ 2 or (<x2>)1/2 = n1/2 λ This is interesting: the mean distance traveled is proportional to the square root of the number of steps. If I assume that the walker makes ν steps per second, then n = νt and <x2> = λ 2 νt (1) The mean distance traveled is proportional to the square root of the time t, unlike the case of a particle moving at a constant velocity, where the displacement would be proportional to time. One writes Eq. 2 as <x2> = 2Dt (2) and calls D the diffusion coefficient (don’t worry about the factor of 2, it is really a matter of definition). In three dimensions, the total displacement r from the initial position is given by Mean free path λ. To relate the above model to dynamics of molecules, we need to know the length of the “step” λ. More precisely, we would like to know the average distance (aka mean free path) that a molecule travels without colliding with another molecule. To estimate it, we will treat molecules as spheres of a diameter d. Suppose our molecule travels a distance l. That is, the total length along its trajectory is l. It will collide with another molecule whose center comes within a distance d to the center of the molecule we are considering. The number of such molecules along the trajectory is equal to l (N/V) πd2 where (N/V) is the number of molecules divided by the total volume (physicists usually call N/V the concentration but in chemistry we measure the concentration in mol/liter). This the mean free path is the length of the trajectory divided by the number of encounters: λ = l/( l (N/V) πd2 ) = 1/((N/V) πd2 ) (3) r 2 = x 2 + y 2 + z 2 = 6 Dt ...
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