physical chemistry - $K_{eq} $ vs $K_c$

I was wondering if there was a difference between $K_{eq}$ and $K_c$. I think they both refer to the Equilibrium Constant. If I'm wrong, could you please tell me the difference between the two? Thanks!

Answer

$K_{eq}$ is the generic equilibrium constant.

$K_c$ is the equilibrium constant for a reaction explicitly in solution, for which the mass action expression can be written using concentrations only. $$\ce{aA + bB <=> cC + dD}$$ $$K_c = Q_c (\text{at equilibrium}) = \frac{[\ce{C}]^c [\ce{D}]^d}{[\ce{A}]^a [\ce{B}]^b }$$

$K_p$ is the equilibrium constant for a reaction explicitly in the gas state, for which the mass action expression can be written using partial pressures only:

$$K_p = Q_p (\text{at equilibrium}) = \frac{P_{\ce{C}}^c P^d_{\ce{D}}}{P^a_{\ce{A}} P^b_{\ce{B}}}$$

$K$ is the generalizd equilibrium constant, and is otherwise used when reactions are in mixed states. $K$ should perhaps be written with mole fractions $\chi_i$.

$$K = Q (\text{at equilibrium}) = \frac{\chi_{\ce{C}}^c \chi^d_{\ce{D}}}{\chi^a_{\ce{A}} \chi^b_{\ce{B}}}$$

Since concentration ($[i]=\frac{n_i}{V}$) is related to pressure from the ideal gas law:

$$\frac{n}{V}=\frac{P}{RT}$$

$K_p$ amd $K_c$ are related by powers of $RT$, depending on the stoichiometric coefficients.

$$K_c = \frac{[\ce{C}]^c [\ce{D}]^d}{[\ce{A}]^a [\ce{B}]^b}=\frac{(\frac{P_{\ce{C}}}{RT})^c (\frac{P_{\ce{D}}}{RT})^d}{(\frac{P_{\ce{A}}}{RT})^a (\frac{P_{\ce{B}}}{RT})^b}= \frac{P_{\ce{C}}^c P^d_{\ce{D}}}{P^a_{\ce{A}} P^b_{\ce{B}}} (RT)^{a+b-c-d}=K_p (RT)^{a+b-c-d}$$

Similar relations exist between mole fraction ($\chi_i=\frac{n_i}{n_T}$) and concentration ($\chi_i=\frac{c_i}{c_T}$) or pressure ($\chi_i=\frac{P_i}{P_T}$), and $K_c$ or $K_p$ can similarly be converted to $K$ or vice versa. $$K= \frac{\chi_{\ce{C}}^c \chi^d_{\ce{D}}}{\chi^a_{\ce{A}} \chi^b_{\ce{B}}} = \frac{(\frac{P_{\ce{C}}}{P_T})^c (\frac{P_{\ce{D}}}{P_T})^d}{(\frac{P_{\ce{A}}}{P_T})^a (\frac{P_{\ce{B}}}{P_T})^b} =\frac{P_{\ce{C}}^c P^d_{\ce{D}}}{P^a_{\ce{A}} P^b_{\ce{B}}} (P_T)^{a+b-c-d}=K_p (P_T)^{a+b-c-d}$$