Question

For a reaction: $xA + yB \to zC$, if $ - \dfrac{{d\left[ A \right]}}{{dt}} = - \dfrac{{d\left[ B \right]}}{{dt}} = 1 \cdot 5\dfrac{{d\left[ C \right]}}{{dt}}$; then x, y, and z are:
A) ${\text{1,1,1}}$
B) ${\text{3,2,3}}$
C) ${\text{3,3,2}}$
D) ${\text{2,2,3}}$

Answer 1

Hint: The reaction is given and the rate equation for that reaction is given. One can compare the rate equation with the initial reaction equation and find out the stoichiometry ratio which is given in the options which fit for the given reaction.

Complete step by step answer:
1) First of all we will understand what has been given in the question. The reaction is given and the ratio of reactants and products has been asked for a given reaction.
2) Now we will write the reaction which is given,
$xA + yB \to zC$
In this equation, we need to find out the values of x, y, and z.
Now we will write the reaction which we can compare to find out the stoichiometric ratio,
$ - \dfrac{1}{x}\dfrac{{d\left[ A \right]}}{{dt}} = - \dfrac{1}{y}\dfrac{{d\left[ B \right]}}{{dt}} = \dfrac{1}{z}\dfrac{{d\left[ C \right]}}{{dt}}$
Let us consider this as equation 1
3) Now let us write the given equation in question a below,
$ - \dfrac{{d\left[ A \right]}}{{dt}} = - \dfrac{{d\left[ B \right]}}{{dt}} = 1 \cdot 5\dfrac{{d\left[ C \right]}}{{dt}}$
Let us consider this as equation 2
4) Now in order to get equation 1 we need to multiply equation 2 by $\dfrac{1}{3}$ and we get,
$ - \dfrac{1}{3}\dfrac{{d\left[ A \right]}}{{dt}} = - \dfrac{1}{3}\dfrac{{d\left[ B \right]}}{{dt}} = 1 \cdot 5 \times \dfrac{1}{3}\dfrac{{d\left[ C \right]}}{{dt}}$
Now by doing the multiplication part in the last part of the equation we will get the value as below,
$ - \dfrac{1}{3}\dfrac{{d\left[ A \right]}}{{dt}} = - \dfrac{1}{3}\dfrac{{d\left[ B \right]}}{{dt}} = \dfrac{1}{2}\dfrac{{d\left[ C \right]}}{{dt}}$
Now if we compare this calculated equation with equation 1 we will get the values of x, y, and z as 3, 3, and 2 respectively.
Therefore, the value of x, y, and z is $3,3,2$ which shows option C as the correct choice of answer.

Note: While we write the rate equation for the reaction the molecularity of a reaction is always multiplied by the denominator value and one should remember that when calculating the ratio the value of x, y, and Z is in the denominator and not in the numerator and ratio can be calculated accordingly.