Is it valid to calculate concentrations given equilibrium constant and initial concentration? Suppose we have the following reaction:
$$\ce{2A &lt;=&gt; B + C}\tag{1}$$
which can be thought the sum of the following reactions:
$$\ce{A &lt;=&gt; B}\tag{2}$$
$$\ce{A &lt;=&gt; C}\tag{3}$$
with equilibrium constants $K_1$ and $K_2$ respectively. The $K$ for the first reaction will be the product of $K_1$ and $K_2$ i.e. $K=K_1K_2 $. Now if we are given that initial we have $x$ mol of $\ce{A}$ and no amount of $\ce{B}$ and $\ce{C}$, one can calculate the amount of B and C present at equilibrium using the equilibrium constant $K$:
$$K=\frac{y\cdot y}{(x-2y)^2}\tag{4}$$
If we solve (4) for a given $x$ (initial amount) then concentrations of $\ce{B}$ and $\ce{C}$ must be equal. But how can this be possible if $K_1$ and $K_2$ have different values? I mean the following equality must holds:
$$[\ce{C}]= [\ce{B}] \Longleftrightarrow K_1{[\ce{A}]} = K_2\ce[{A}] \tag{5}$$
Edit I am asking if it is valid to use the equilibrium constant $K$ to find the concentrations of $\ce{B}$ and $\ce{C}$ because every reaction can be thought of as sum of other reactions. It is a common exercise in many general Chemistry textbooks where you are given a reaction with an equilibrium constant and you must find the concentrations at equilibrium. Well the substitutions into equilibrium constant that they make is the one that I have also did (assuming 0 initial concentration for both $\ce{B}$ and $\ce{C}$. So what is going wrong?
The equation
$$\ce{2A &lt;=&gt; B + C}$$
implies a fixed stoichiometry between B and C. If the two separate reactions (2) and (3) were happening, the correct net equation would be
$$\ce{(x + y) A &lt;=&gt; x B + y C}$$
Here is an example that would be well-described by reaction (1):
$$\ce{2 H2O &lt;=&gt; H3O+ + OH-}$$
One water molecule cannot make a hydronium ion (goes against the conservation of mass and charge). The same goes for a water reacting to form a hydroxide. Instead, for every hydronium that forms, exactly one hydroxide is formed. So the separate reactions do not exist, and the separate equilibrium constants do not either.
Of course, there are other reactions where two independent products exist, such as linear glucose forming the alpha or beta anomer of the pyranose ring. In this case, there are two equilibrium constants, but there is no combined equilibrium constant because the two reactions are not the steps of an overall reaction, but rather two distinct reaction paths.