Question

In a reaction,$xA + yB \to zC$ if $ - \dfrac{{d[A]}}{{dt}} = - \dfrac{{d[B]}}{{dt}} = 1.5\dfrac{{d[C]}}{{dt}}$. Then x, y and z are:
A. 1, 1, 1
B. 3, 2, 3
C. 3, 3, 2
D. 2, 3, 3

Answer 1

Hint: Rate of a reaction can be defined as the change in the concentration of a reactant or product in unit time that is the rate of decrease in concentration of any of the reactants or the rate of increase in concentration of any one of the products.

Complete step by step solution:
We know that the rate of a reaction can be the average rate or the instantaneous rate, the average rate depends upon the change in concentration of reactants or products and the time taken for that change to occur. For example, for $A \to B$
${r_{av}} = \dfrac{{\Delta [B]}}{{\Delta t}} = \dfrac{{{{[B]}_2} - {{[B]}_1}}}{{{t_2} - {t_1}}} = \dfrac{{{{[A]}_2} - {{[A]}_1}}}{{{t_2} - {t_1}}}$
However, the average rate cannot be used to predict the rate of reaction at a particular instant as it would be constant for the time interval for which it is calculated. So, to express the rate at a particular moment of time we determine the instantaneous rate.
${r_{inst}} = - \dfrac{{d[R]}}{{dt}} = \dfrac{{d[P]}}{{dt}}$
For expressing the rate of such a reaction like above where stoichiometric coefficients of reactants or products are not equal to one, rate of disappearance of any of the reactants or the rate of appearance of the products is divided by their respective stoichiometric coefficients. Therefore the rate of above reaction is:
$r = - \dfrac{1}{x}\dfrac{{d[A]}}{{dt}} = - \dfrac{1}{y}\dfrac{{d[B]}}{{dt}} = \dfrac{1}{z}\dfrac{{d[C]}}{{dt}}$
Comparing this equation with the equation in the question we get the value of x, y and z as:
$
  x = y = \dfrac{z}{{1.5}} \\
  x = y = \dfrac{{2z}}{3}(multiply\,by\, 3) \\
  x = 3,y = 3,z = 2
 $

Hence the correct option is C.

Note:
Kinetic studies not only help us to determine the speed or rate of the reaction but also describes the conditions by which the reaction rates can be altered. The factors such as concentration, temperature, pressure and catalyst affect the rate of a reaction.