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In introductory chemical kinetics, an elementary reaction is taken for instance $$\ce{A -> B}$$ Then, we go on to derive integrated rate kinetics equations for them by assuming $$\text{rate} = k [\ce{A}]^n$$
Could it possible in a more generic case that : $\text{rate}=k[\ce{A}]^n [\ce{B}]^m$ and the total order of the reaction be $m+n$? In other words, can the rate of reaction depend on both the molar concentrations of reactants and products independently?
There exist autocatalytic reactions than can show such a behaviour. E.g. a reaction $\ce{A + B \rightarrow 2\ B}$ might show the rate $k[\ce{A}] [\ce{B}]$.
The reaction of permanganate with oxalic acid is a well known autocatalytic reaction. $$ \mathrm{ 5\ (COOH)_2 + 2\ MnO_4^- + 6\ H^+ \rightarrow 10\ CO_2 + 2\ Mn^{2+} + 8\ H_2O} $$ Here the generated manganese(II) ions catalyze the reaction. The reaction starts slowly if no initial manganese(II) ions are present, and then speeds up.
No, because the reaction order is defined by how the rate of reaction changes when the concentrations of one reactant is changed (though rate of product formation can be considered as well), and can be thought of as a number the sum of the possible successful collisions that the atoms can make. For example, in the reaction $$\ce{A +B->products}$$ There order of the reaction can be:
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If $\ce{B}$ is not a reactant, then it has no place in a rate equation with $\ce{A}$ because the two aren't colliding in order to react.
You have given a reaction for an irreversible reaction: $$\ce{A -> B}$$ so the products would not really come into consideration. For a reversible reaction: $$\ce{A <=> B}$$ the relative rates of reaction of the forward and backward reactions would yield the equilibrium constant, $\ce{k_{eq}}$, which is given by: $$\ce{k_{eq} = \dfrac{[B]}{[A]}}$$. The forward reaction would be $$\ce{ r_{A} = k_{A} [A]^{n_{A}}}$$ and the reverse reaction or B would be $$\ce{ r_{B} = k_{B} [B]^{n_{B}}}$$ The quirk here is that $\ce{n_{A}}$ and $\ce{n_{B}}$ do not need to be integers for the rate equation, but in the equilibrium equation the exponents for the concentrations of A and B would be the same as the integers in the stoichiometry of the chemical equation.
