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For the following gas phase reaction: $$ A\leftrightarrow B $$ The concentration equilibrium constant ($K_c$) can be expressed as: $$ K_c = \frac{c_B}{c_A} = \frac{\frac{F_B}{\vartheta}}{\frac{F_A}{\vartheta}} = \frac{F_{A0}X_{eq}}{F_{A0}(1-X_{eq})} = \frac{X_{eq}}{1-X_{eq}} $$ Where $c_i$ is the concentration and $F_i$ is the molar flow rate of component i ($F_{A0}$ is the initial molar flow rate of A, assuming no B is present initially), $\vartheta$ is the volumetric flow rate, and $X_{eq}$ is the equilibrium conversion of the reaction.

Following the same methodology for the following gas-phase reaction: $$ A \leftrightarrow 2B $$ $$ K_c = \frac{c_B^2}{c_A} = \frac{(\frac{F_B}{\vartheta})^2}{\frac{F_A}{\vartheta}} = \frac{(2F_{A0}X_{eq})^2}{\vartheta F_{A0}(1-X_{eq})} = \frac{4F_{A0}X_{eq}^2}{\vartheta (1-X_{eq})} $$ Would this be correct?

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In your first step, I don't understand what you mean by a chemical reaction of one reactant (A) converting to another reactant (B) with no other reactant or other outside influence.

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