Read the Passage carefully and answer the Questions.

Kinetic studies not only help us to determine the rate of a chemical reaction but also describe 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. The speed of a reaction or the rate of a reaction can be defined as the change in concentration of a reactant or product in unit time.

Rate = $k [A]^x [B]^y$

Here k is the rate constant

$x + y$ gives the overall order of a reaction, whereas x and y represent the order with respect to the reactants A and B, respectively. 

Table: Initial rate of formation of $NO_2$

Based on the above observations, the rate of formation of $NO_2$ can be represented as

Rate of reaction = $k[NO]^2 [O_2]$

The correct answer is Option (1) → Rate of reaction = $k[NO]^2 [O_2]$

The rate of formation of $\text{NO}_2$ can be represented by the general rate law:

$\text{Rate} = k[\text{NO}]^x[\text{O}_2]^y$

where $x$ and $y$ are the reaction orders with respect to $\text{NO}$ and $\text{O}_2$, respectively.

We determine $x$ and $y$ using the method of initial rates.

1. Determine the Order with Respect to $\text{NO}$ ($x$)

We compare experiments where the concentration of $\text{O}_2$ is held constant (Experiments 1 and 2).

$\frac{\text{Rate}_2}{\text{Rate}_1} = \frac{k[\text{NO}]_2^x[\text{O}_2]_2^y}{k[\text{NO}]_1^x[\text{O}_2]_1^y}$

$\frac{0.384}{0.096} = \left(\frac{0.60}{0.30}\right)^x \left(\frac{0.30}{0.30}\right)^y$

$4 = (2)^x$

$\mathbf{x = 2}$

The reaction is second order with respect to $\text{NO}$.

2. Determine the Order with Respect to $\text{O}_2$ ($y$)

We compare experiments where the concentration of $\text{NO}$ is held constant (Experiments 1 and 3).

$\frac{\text{Rate}_3}{\text{Rate}_1} = \frac{k[\text{NO}]_3^x[\text{O}_2]_3^y}{k[\text{NO}]_1^x[\text{O}_2]_1^y}$

$\frac{0.192}{0.096} = \left(\frac{0.30}{0.30}\right)^x \left(\frac{0.60}{0.30}\right)^y$

$2 = (2)^y$

$\mathbf{y = 1}$

The reaction is first order with respect to $\text{O}_2$.

3. Write the Rate Law

Substituting the orders ($x=2$ and $y=1$) into the general rate law:

$\text{Rate of reaction} = k[\text{NO}]^2[\text{O}_2]$

The correct option is: $\text{Rate of reaction} = k[\text{NO}]^2[\text{O}_2]$