Question

In a certain gaseous reaction between X and Y, \[X + 3Y \to X{Y_3}\], the initial rates are reported as follows-

[X]	[Y]	Rate
\[0.1M\]	\[0.1M\]	\[0.002M{s^{ - 1}}\]
\[0.2M\]	\[0.1M\]	\[0.002M{s^{ - 1}}\]
\[0.3M\]	\[0.2M\]	\[0.008M{s^{ - 1}}\]
\[0.4M\]	\[0.3M\]	\[0.018M{s^{ - 1}}\]

The rate law is:
A.\[r = K[X]{[Y]^3}\]
B.\[r = K{[X]^0}{[Y]^2}\]
C.\[r = K[X][Y]\]
D.\[r = K{[X]^0}{[Y]^3}\]

Answer 1

Hint: The order of the reaction depends on the power of the rate of concentration of each reactant which is present in a reaction. In the case of first order reaction, the rate depends only on the concentration of one species. And the rate of the reaction which represents the speed of chemical reaction and it is proportional to the rising the concentration of the product and reducing the concentration of reactant.

Complete answer:
According to the given table, the order of Y is not equal to three. Thus,\[r = K[X]{[Y]^3}\]is not the rate law of the reaction. Hence, option (A) is incorrect.
The order of the reaction with respect to X and Y is equal to a and b.
From the given table, we get
\[0.002 = K{\left[ {0.1} \right]^a}{\left[ {0.1} \right]^b} \ldots \left( 1 \right)\]
\[0.002 = K{\left[ {0.2} \right]^a}{\left[ {0.1} \right]^b} \cdots \left( 2 \right)\]
And dividing the equation one with equation two, will get
\[1 = {\left[ {\dfrac{1}{2}} \right]^a}\]Here, a is equal to zero
Consider next two reactions,
\[0.008 = K{\left[ {0.3} \right]^a}{\left[ {0.2} \right]^b} \ldots \left( 3 \right)\]
\[0.018 = K{\left[ {0.3} \right]^a}{\left[ {0.3} \right]^b} \cdots \left( 4 \right)\]
Divide the equation three with equation four, will get
\[\dfrac{8}{{18}} = {\left[ {\dfrac{2}{3}} \right]^b}\]Here, b is equal to two
So, the order with respect to X is equal to zero and Y is equal to two. Therefore, the rate law is ,\[r = K{[X]^0}{[Y]^2}\]. Hence, option (B) is correct.
The order of X and Y is not one. Thus, the rate law is not equal to \[r = K[X][Y]\]. Hence, option (C) is incorrect.
According to the given values of X and Y, the order of both X and Y is not equal to zero and three. Thus, the rate law of the reaction will not be equal to \[r = K{[X]^0}{[Y]^3}\]. Hence, the option (D) is incorrect.

Hence, option (B) is correct.

Note:
We know that the rate law indicates the rate of the reaction to rate constant and the reactant concentrations. And the rate constant is represented as ‘k’ which is known as proportionality constant. The rate law mainly relates the reactant concentrations. And the rate mainly depends on the concentration of the reactants. But sometimes, the rate does not depend on the concentration of the reactant.