In analogy with the thermal conductivity, derive an approximate formula for the diffusion coefficient of an ideal gas in terms of the mean free path and the average thermal speed. Evaluate your formula numerically for air at room temperature and atmospheric pressure, and compare to the experimental value quoted in the text. How does D depend on T, at fixed pressure?
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Diffusion is the phenomenon by which the concentration of some substance varies over time due to concentration gradient by the means of drifting of molecules through a volume.
In the given case, if we treat diffusion in ideal gases in the similar way as we used in thermal conductivity. In ideal gas , diffusion is due to the movement of molecular concentration rather than energy transport in thermal conductivity. The derivation of the crude formula for diffusion however follows the same steps.
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Let us consider a thin slab of gas with cross sectional area A. Which we divide into half, so that the number of molecules in the one side of partition is and on the other side . As shown in the figure below, we assume that the whole box is at same temperature T.
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For the given case, by approximation the mean free path and average thermal speed is given by,
Where r is molecular radius and m is mass of one molecule.
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As only half of the molecules will , on an average moves towards the partition. Therefore, the net number of molecules that cross the partition in time is given by,
By using gradient in terms of molecular number as ,