Dependence of rate on the nature of reactants and other factors

Question

Consider a reaction $$\ce{aA + bB -> cC + dD}$$ whose reactants are given by A and B and the products are C and D.

The rate of this reaction is given by $$\frac{dx}{dt} = -\frac{d[\ce{A}]}{adt}= -\frac{d[\ce{B}]}{bdt} = \frac{d[\ce{C}]}{cdt} = \frac{d[\ce{D}]}{ddt}$$

The rate of the reaction also depends on the surface area of reactants, nature of reactants, stirring, temperature. Does the rate equation account for all the factors (I could find that by changing these factors the change in concentration for a time interval would vary)? If in a lab we are given different samples of reactants and measure the rate of the reaction (but same concentration), wouldn't each get a different rate value? How is the rate uniquely defined?

Answer 1

The rate is given by the differentials you present and these have values depending on concentration, temperature etc. as you mention. What defines a reaction is the rate constant. In your example $$ rate = \cdots etc. =\frac{d[D]}{ddt}=k[A]^{\alpha}[B]^{\beta}\cdots$$ where $k$ is the rate constant and $\alpha+\beta\cdots$ define the reaction order. The rate constant and exponent powers have to be determined by experiment.