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It is because it's easier to evaluate the rate at the initial point when there are an insignificant amount of products present, so you only have to take into account the reactants. The rate law can be found at other points, but it is much more complicated as there will be both reactants and products present.
The advantage of examining the initial rate is that the presence of products later in the reaction may affect the rate; the interpretation of the rate is then quite complicated.
The concentration of products can affect the rate and direction of reactions so it is much easier to evaluate considering only the forward direction.
It can't be in terms of products because if you evaluate the course of the reaction later on, you won't be able to evaluate the presence of products that well because they are constantly changing. Therefore, we examine the reactants at the beginning of the reaction because it's much easier to look at one pattern (the reactants) when there are no products present.
It is definitely a lot harder to find the rate of a reaction when using products because the progress of a reaction occurs at an exponential decay, this, the reaction rate towards the end of a reaction occurs a lot slower than at the beginning of a reaction. Therefore, we using reactant concentrations and calculate as close as we can to the initial concentrations to get a better answer.
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