Question

Total order of the reaction $\text{ X + Y }\to \text{ XY }$ is 3. The order of reaction with respect to X is 2. State the differential rate equation for the reaction.
A) $\text{ }-\dfrac{\text{d}\left[ \text{X} \right]}{\text{dt}}\text{ = k }{{\left[ \text{X} \right]}^{3}}{{\left[ \text{Y} \right]}^{0}}\text{ }$
B) $\text{ }-\dfrac{\text{d}\left[ \text{X} \right]}{\text{dt}}\text{ = k }{{\left[ \text{X} \right]}^{0}}{{\left[ \text{Y} \right]}^{3}}\text{ }$
C) $\text{ }-\dfrac{\text{d}\left[ \text{X} \right]}{\text{dt}}\text{ = k }{{\left[ \text{X} \right]}^{2}}{{\left[ \text{Y} \right]}^{1}}\text{ }$
D) $\text{ }-\dfrac{\text{d}\left[ \text{X} \right]}{\text{dt}}\text{ = k }{{\left[ \text{X} \right]}^{1}}{{\left[ \text{Y} \right]}^{2}}\text{ }$

Answer 1

Hint: The chemical reactions are generally classified based on the order of the reaction. The order of the reaction is the sum of the powers to which the concentration or pressure terms are raised in the rate law expression. For general reaction, $\text{ xA + yB }\to \text{ L + M }$ the rate of reaction is expressed as,
$\text{ }\dfrac{\text{d}\left[ \text{A} \right]}{\text{dt}}\text{ = rate = k }{{\left[ \text{A} \right]}^{\text{x}}}{{\left[ \text{B} \right]}^{\text{y}}}\text{ }$
Where x and y are the order of reaction with respect to A and B respectively and the order of the reaction is equal to $\text{ x + y }$ .

Complete answer:
We have given the following reaction. X and Y react with each other to form a product$\text{ XY }$ . The order of the reaction is 3 and the reaction with respect to the X is 2.
We have to determine the differential equation for the reaction,
$\text{ X + Y }\to \text{ XY }$
The rate of reaction depends on the concentration of reactant X and Y.As the reaction proceeds the concentration of X keeps on falling with the time .the rate of the reaction at any given time is given by the expression
$\text{ r = }-\dfrac{d{{C}_{\text{X}}}}{dt}\text{ = }-\dfrac{d{{C}_{\text{Y}}}}{dt}\text{ }$ ……………….. (1)
Where $\text{ }d{{C}_{\text{X}}}\text{ }$ and $\text{ }d{{C}_{\text{Y}}}\text{ }$is the infinitesimally small decrease in the concentration of X and Y in an infinitesimally small; interval of time $\text{ }dt\text{ }$ , ${{C}_{\text{X}}}$and $\text{ }{{C}_{\text{Y}}}\text{ }$ gives the concentration of reactant X and Y at a given instant.
The concentration of products $\text{ XY }$ goes on increasing with time. Hence , the rate of reaction can be also expressed in terms of the increase in the concentration of the product $\text{ XY }$in an infinitesimally small interval of time $\text{ }dt\text{ }$.
$\text{ r = }\dfrac{d{{C}_{\text{XY}}}}{dt}\text{ }$ ……………….. (2)
Now, the differential rate equation for the reaction is determined from the equation (1) and (2) is,
$\text{ r = }-\dfrac{d{{C}_{\text{X}}}}{dt}\text{ = }-\dfrac{d{{C}_{\text{Y}}}}{dt}\text{ = }\dfrac{d{{C}_{\text{XY}}}}{dt}\text{ }$
Thus the rate equation depends on the concentration of the reactant as,
$\text{ }-\dfrac{\text{d}\left[ \text{X} \right]}{\text{dt}}\text{ = k }{{\left[ \text{X} \right]}^{\text{m}}}{{\left[ \text{Y} \right]}^{\text{n}}}\text{ }$
Here, n and m are the order of reaction with respect to the reactant. The order of the reaction is the sum of the powers of the concentration terms raised in the rate law expression. for reaction,
$\text{ X + Y }\to \text{ XY }$
The order of a reaction is 3.Thus,
$\text{ m + n = 3 }$
But we know that the order of reaction with respect to X is 2.Thus, the order of reaction with respect to Y is,
$\begin{align}
  & \text{ 2 + n = 3} \\
 & \Rightarrow \text{n = 3 }-2\text{ = 1 } \\
\end{align}$
Thus the order of reaction with respect to Y is 1. Thus the differential equation for the reaction is, $\text{ }-\dfrac{\text{d}\left[ \text{X} \right]}{\text{dt}}\text{ = k }{{\left[ \text{X} \right]}^{2}}{{\left[ \text{Y} \right]}^{1}}\text{ }$

Hence, (C) is the correct option.

Note:
It may be noted that the order of the reaction is essentially an experimental quantity. The order of reaction for the general reaction is usually equal to the sum of the powers of concentration or the pressure term.
There is one more term along with the order of the reaction: molecularity. It is the number of molecules taking part in the reaction.