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Because an inert gas does not react with the reactants or products, it is essentially not changing the concentration of the products and reactants. In addition, the volume is constant so the concentrations remain untouched. Therefore, this will not effect equilibrium.
A pressure change will only have an effect on a reaction if it is caused by a change in volume. If you increase pressure by reducing the volume, then the molecules within the reaction are compressed and because they want to be stable with the lowest energy, the reaction will favor whichever side has less moles because that means there will be less molecules interacting. If you add an inert gas to the reaction, you aren't actually changing the volume and the reaction is still contained within the same space. The inert gas will increase the pressure, but the reaction has no reason to change because there isn't a different space to compress or spread out into.
When a reaction vessel has an inert gas pumped into it at a constant volume to increase the total pressure, then the total pressure will increase but the partial pressures of the reactants and products are unchanged and therefore the K value will not be affected.
Adding an inert gas does change the pressure, but since the change in pressure was not caused by a change in volume and since the inert gas does not have any effect on the reaction, there will be no change in equilibrium.
Adding an inert gas doesn't affect the reaction equilibrium because the gas addition doesnt alter the volume, volume is the factor that has an effect on pressure and if volume is unaffected the pressure is unaffected as well.
The only time when a change of pressure will have an effect is when the volume changes. A decrease in volume creates an increase in pressure. However the addition of inert gas does not change the volume and it does not react with the present reactants or products.
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