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Are $K_P$ and $K_C$ dependent on initial concentration of reactants? So far I had known that $K_P$ and $K_C$ are only dependent on temperature which is explained by thermydnamic approaches perhaps. I didn't learn the derivations of those equilibrium constants. I just learnt that they are the ratio of product of active masses of respective products and reactants. But I am asking this question because I was stuck at a point while solving a problem regarding $K_P$ and $K_C$

The problem:

In the dissociation reaction of Nitrogen tetraoxide$(N_2O_4)$ $100\alpha$% of the reactant is dissociated at $P atm$ pressure and at $TK$ temperature[suppose volume=$V$]

so as per the problem if the initial concentration of the reactant is $C M$ then concentration of product is $\alpha C M$ at equilibrium $$N_2O_4<=>2NO_2$$ $$initial:C <=>0 $$ $$equilibrium:C-C\alpha<=>2C\alpha$$

so $$K_C=\frac{[NO_2]^2} {[N_2O_4]}=\frac{4C^2\alpha^2} {C(1-\alpha)}=\frac{4C\alpha^2} {(1-\alpha)}$$ $$K_P=\frac{P_{(NO_2)}^2} {P{(N_2O_4)}^2}=\frac{\frac{4C^2P^2\alpha^2V^2} {CV(1-\alpha)+2CV\alpha}} {\frac{CPV(1-\alpha)} {CV(1-\alpha)+2CV\alpha}}=\frac{4P\alpha^2} {(1-\alpha)}$$

so It can be seen that in the equation of $K_C$ there is $C$ involved in the numerator. since equilibrium constants are not dependent on intial concentration, I suppose that the degree of dissociation($\alpha$) varies depending on initial conditions. But if $\alpha$ changes depending on intial concentration does not that make $K_P$ dependent on intial concentration? Where am I wrong here?

Thank you.

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The constants 𝐾𝑝 and 𝐾𝑐 have dimensions, which are in bar and mol/L. So if the pressure or the concentration changes, the parameter 𝛼 adapts its value so as to obtain the same value of the equilibrium constants.

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  • $\begingroup$ ohh I messed up by thinking one dimensionally that is considered the change of $\alpha$ only to maintain consistency of $K_C$ but ignored the necessity of maintaining numerical consistency of $K_P$ alongside. $\endgroup$
    – MSKB
    Commented Dec 7, 2021 at 9:25
  • $\begingroup$ a small point but I think you mean bar and mol/L in your first line. $\endgroup$
    – porphyrin
    Commented Dec 7, 2021 at 18:48
  • $\begingroup$ @porphyrin. Thank you for the comment. I have edited it in my text. $\endgroup$
    – Maurice
    Commented Dec 7, 2021 at 19:44
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    $\begingroup$ It's not generally the case that Kp and Kc have dimensions. That depends on the convention used by the textbook or paper. See chemistry.stackexchange.com/questions/107374/… Your conclusion—that Kp and Kc don't depend on initial concentrations—is correct, but the reason you give for this is not: The reason Kp and Kc are independent of initial concentrations is not that they have dimensions, since they are also independent of initial concentrations when the dimensionless convention is used. $\endgroup$
    – theorist
    Commented Dec 7, 2021 at 20:06
  • $\begingroup$ @theorist. I agree about your critics. I must also state that the initial question is not clear. First, the data do not specify whether this reaction is carried out at constant volume, or at constant pressure. Furthermore, the value $K_c$ should not be used with gas reactions. Only the $K_p$ constant makes sense, thermodynamically speaking. Equilibrium constants $K_c$ are introduced in thermodynamics with solutes, because it is obvious that their partial pressures cannot be measured. $\endgroup$
    – Maurice
    Commented Dec 7, 2021 at 21:36

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