lets consider a general reaction:
aA + bB ----> cC + dD
Rate = - 1/a delta A / delta t = - 1/b deltaB / delta t = - 1/c delta C / delta t = - 1/d delta D / delta t
So,the speed of any event is measured by the change that occurs in any interval of time. The speed of a reaction or reaction rate is expressed as the change in concentration of a reactant or product over a certain amount of time.
Since concentration of the reactant decreases over time, final concentration minus the initial will give a negative value . And that is why we see a negative sign above in the rate equation of the reactants A and B .
look at the rate of appearance of a product. As a product appears, its concentration increases. The rate of appearance is a positive quantity.
rate of appearance of a product is equal to the negative of rate of disappearance of a reactant
In the reaction :
2O3(g) -----> 3O2 (g)
the rate of the disaperence of O3 is the -1 / 2 delta O3 / delta t
and the rate of the appearence of the O2 is 1/3 O2/ delta t and so,
-1/2 delta O3 / delta t = 1/3 O2/ delta t