if I consider the reaction : $$A \longrightarrow B+C$$ by what we call conservation of the amount of substance, we have $$nA_{0}=nA+nB=nA+nC$$ But for me the conservation says that $$nA_{0}=nA+nB+nC$$, because we have to consider everything in our closed system and we form A and B. I konw it is false, but I can't figure why we only consider one product (or one reactiv if we have 2 reactivs)
Where is my missunderstanding...?
The general Law is the conservation of MASS-ENERGY. In chemistry this breaks down into two separate Laws because the mass change in most chemical reactions is much too small to measure with the present instrumentation: Conservation of MASS, and Conservation of Energy. A balanced chemical equation means that the MASS on each side is equal and the ENERGY on each side is equal. Since the mass is contained in the nuclei and electrons of the atoms each side of the equation must contain the same number of ATOMS and the same CHARGE regardless of their arrangement; the number of MOLECULES is NOT conserved. Energy, the oft-ignored reactant or product, is the chemical potential energy on one side of the equation or the release or absorbed energy on the other.
A correct chemical equation is: A = B + Energy. A and B are EQUAL NUMBERS of ATOMS that are in different molecular arrangements. Energy is the energy released to or obtained from the environment. These values are determined by careful observations and measurements sometimes on paper but eventually in the laboratory. It has taken many chemists, physicists, and other scientists many centuries to get it to what we think is almost right.
You are suffering from a severe case of equationitis; The feat of writing equations before understanding concepts and then writing equations without defining the terms and relating them to what concepts you have defined. Your last premise is correct! Everything must be considered. I suggest that you start with a very simple equation: 2H = H2 + energy. Start with 2 atoms then escalate to one mole of atoms then 2 moles of atoms. After that is understood you can worry about the mechanism and equilibrium.
Amount conservation must respect the stoichiometry of the reaction.
If the sum of stoichiometric coefficients on both reaction sides is not equal then the total amount of substances is not constant and changes with the reaction progress.
If you cut a heap of apples to halves then the total count of fruit pieces is not constant either.
For the reaction $\ce{A -> B + C}$
and initial respective amounts $n_\mathrm{A0}$, $n_\mathrm{B0}$, $n_\mathrm{C0}$
and for the reaction completion degree $0 \lt \alpha \lt 1$,
the respective amounts of components follow this schema:
$$n_\mathrm{A}=n_\mathrm{A0}(1-\alpha)$$ $$n_\mathrm{B}=n_\mathrm{B0} + n_\mathrm{A0}\alpha$$ $$n_\mathrm{C}=n_\mathrm{C0} + n_\mathrm{A0}\alpha$$
In summary:
$$n_\text{tot} = n_\text{A} + n_\text{B} + n_\text{C} = \\ n_\mathrm{A0}(1-\alpha) + n_\mathrm{B0} + n_\mathrm{A0}\alpha + n_\mathrm{C0} + n_\mathrm{A0}\alpha= \\ = n_\mathrm{A0}(1 + \alpha) + n_\mathrm{B0} + n_\mathrm{C0} $$
If there were different reaction stoichiometric coefficients, these would be reflected as multiplication coefficients.
For the reaction $\ce{aA + bB -> cC + dD}$
initial respective amounts $n_\mathrm{A0}$, $n_\mathrm{B0}$, $n_\mathrm{C0}$, , $n_\mathrm{D0}$,
assuming the reactant $\ce{A}$ is in excess,
and for the reaction completion degree $0 \lt \alpha \lt 1$,
the respective amounts of components follow this schema:
$$n_\mathrm{A}=n_\mathrm{A0}-\alpha (\frac ab) n_\mathrm{B0}$$ $$n_\mathrm{B}=n_\mathrm{B0}(1-\alpha)$$ $$n_\mathrm{C}=n_\mathrm{C0} + \alpha(\frac cb)n_\mathrm{B0}$$ $$n_\mathrm{D}=n_\mathrm{D0} + \alpha(\frac db)n_\mathrm{B0}$$
