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I can't remember the way he taught it in class, but the way I've always thought about it is the gas mixture is attempting to alleviate the increase in pressure as a result of the compression. Because fewer molecules of gas would mean less pressure, the reaction favors the side with fewer moles of gas.
When a reaction is compressed, you would be lowering the volume therefore increasing the concentration of each reactant and product. Let's say that the reactants have less mols of gas. Since K is products over reactants, the concentration of products would increase a lot more than the increase in reactants therefore the equation would shift towards the reactants.
Compressing a gas means increasing the concentration of each gas. The reaction would favor the side with less moles of gas because it reduces the number of particles in the gas. Since volume and pressure are inversely related, when compressing a gas you are decreasing the volume and this is why the reaction favors the side with the least amount of moles.
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