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A small equilibrium equilibrium constant (K<10^-3) means that there is a large equilibrium concentration of reactants (reactants are in the denominator of the K expression, so when the concentration of reactants is large, K will be small). Because there is a greater concentration of reactants at equilibrium, we say that equilibrium favors the reactants in the reaction.
If a K is that small (less than 10^-3), then at equilibrium, there will be way more reactants than products, indicating only a small amount of product was formed before the reaction reached equilibrium. This is because the reactant state has greater stability, so the reactants will want to stay reactants.
In the equilibrium expression, reactants are int he denominator so if the reactants are favored there are way more of them being produced because reactants want to stay as reactants, similarly, if the K was greater than 10^-3 the products would be favored, producing more products as products want to stay as products
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