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When you are looking at changes in the value of K you want to look at whether the reaction is endothermic or exothermic because that will give you a hint as to how K will change if temperature is increased or decreased. For example, if temperature is increased for an endothermic reaction, K increases because products are favored as their product will "use up" some of the heat from the increased temperature. Hope that adds a bit to the conversation :).
The reason why only temperature changes the K value is because the other variables e.g. pressure only really change the initial concentration of reaction or product. However, because of this, the Q value changes which will shift the chemical equation.
Only temperature will change the K value. Temperature changes the value of K depending on if the reaction is endothermic or exothermic. For a change in the concentration and pressure in the products or reactants, the system will eventually come back to equilibrium over time so the ratio of the products to reactants after the added products or reactants will be the same as the initial equilibrium concentrations/pressures.
You could also potentially change K by multiplying the reaction equation by a whole number. If you double the stoichiometric coefficients of everything in the equation, the K will become K^2, which will either increase K is K > 1 or decrease K if K < 1.
Temperature is the only modification that can affect the K value. Pressure, volume, and concentration does not affect the K value but it does affect which side of the reaction is favored more.
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