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According to my textbook (and intuitively) certain changes when the aforementioned 3 variables are altered occur in accordance with Le Chatelier's Principle. However, what I don't understand is what makes temperature change alter the equilibrium constant while the other two do not. All three cause changes in the equilibrium position (the state of concentrations at which equilibrium is achieved with lowest possible stress in the system), so why don't they all also change $K_{eq}$?

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Please have a look at this answer of mine. It contains the derivation of the defining formula of the equilibrium constant.

\begin{equation} \log \underbrace{\prod_i [a_{i}]^{\nu_{i}}}_{= \, K} = -\frac{\Delta G^{0}}{RT} \qquad \Rightarrow \qquad \log K = -\frac{\Delta G^{0}}{RT} \ . \end{equation}

This formula shows the temperature dependence explicitely, so it is clear that $K$ must depend on $T$.

Apart from the temperature this formula also relates $K$ to the standard Gibbs free energy of reaction $\Delta G^{0}$. This quantity is defined for specific standard conditions involving standard pressure and equilibrium concentrations. Since there aren't any other quantities contained in the formula (apart from the constant $R$) it follows that $K$ is defined for this specific pressure and these specific concentrations too and so it has no pressure or concentration dependence.

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