Problem 116 - A Complete Resource Book in Chemistry for JEE Main

Question

For a gaseous reaction \(2 \mathrm{~A}+\mathrm{B} \longrightarrow 2 \mathrm{AB}\) this following rate data were obtained at \(300 \mathrm{~K}\). $$ \begin{array}{llll} \hline \text { Expt } & \text { Concentration } & \text { Rate of disappearance } \\ & {[\mathbf{A}]} & {\left[\mathrm{B}_{2}\right]} & \text { of } \mathrm{B}_{2}\left(\mathrm{~mol} \mathrm{~L} \mathbf{~ m i n}^{-1}\right) \\ \hline 1 . & 0.015 & 0.15 & 1.8 \times 10^{-3} \\ 2 . & 0.09 & 0.15 & 1.08 \times 10^{-2} \\ 3 . & 0.015 & 0.45 & 5.4 \times 10^{-3} \\ \hline \end{array} $$ What is the rate law? (a) \(\mathrm{r}=k[\mathrm{~A}]\left[\mathrm{B}_{2}\right]\) (b) \(r=[\mathrm{A}]^{2}\left[\mathrm{~B}_{2}\right]^{1}\) (c) \(\mathrm{r}=k[\mathrm{~A}]\left[\mathrm{B}_{2}\right]^{2}\) (d) \(\mathrm{r}=k\left[\mathrm{~B}_{2}\right]\)

Step-by-Step Solution

Answer

The rate law is \( r = k[A][B] \), which corresponds to option (a).

Step 1: Determine Reaction Order with Respect to A

To find the order of reaction with respect to A, compare experiments 1 and 2 where the concentration of B remains constant. Divide the rate of reaction of experiment 2 by that of experiment 1, which gives \( \frac{1.08 \times 10^{-2}}{1.8 \times 10^{-3}} = 6.00 \). Similarly, divide the concentration of A in experiment 2 by that of experiment 1, which gives \( \frac{0.09}{0.015} = 6.00 \). This implies the reaction is first order with respect to A, as \( 6^1 = 6 \).

Step 2: Determine Reaction Order with Respect to B

Next, compare experiments 1 and 3, keeping A constant. The rate in experiment 3 divided by the rate in experiment 1 is \( \frac{5.4 \times 10^{-3}}{1.8 \times 10^{-3}} = 3.00 \). The concentration of B in experiment 3 divided by that in experiment 1 is \( \frac{0.45}{0.15} = 3.00 \). Hence, the reaction is first order with respect to B, because \( 3^1 = 3 \).

Step 3: Write the Rate Law

Since the reaction is first order with respect to both A and B, the rate law is \( r = k[A]^1[B]^1 \), which simplifies to \( r = k[A][B] \).

Step 4: Match the Rate Law to the Given Options

Among the provided options, compare the form of the derived rate law to the options: (a) \( r = k[A][B] \) is a match. The other options either include different powers or exclude terms. Thus, the answer is option (a).

Key Concepts

Chemical Kinetics

Chemical kinetics is the branch of chemistry that studies the speed at which a chemical reaction proceeds. It focuses on understanding the factors that affect reaction rates, such as concentration, temperature, and catalysts. This area of chemistry helps to answer questions like how fast a reaction occurs or to predict the change in concentration of reactants/products over time.

If we focus on the role of concentration, we often turn to rate laws and reaction orders to describe how the concentration of reactants impacts the speed of a reaction. These factors illuminate the underlying mechanisms that dictate how molecules interact to form products. For many reactions, we can observe trends or derive the reaction mechanism from the kinetics, providing insights into molecular behavior.

In practice, chemical kinetics is applied in various fields, from developing new pharmaceuticals, ensuring safety in chemical engineering processes, to enhancing the efficiency of food storage.

Reaction Order

Reaction order is a vital concept in chemical kinetics that tells us how the concentration of a reactant affects the rate of reaction. It is determined experimentally and not directly from the stoichiometry of the reaction equation. The order with respect to a particular reactant is an exponent in the rate law, showing the extent to which concentration changes impact reaction rate.

For example, in the given solution, the reaction is found to be first order with respect to both reactant A and B. This was determined by observing that changes in rate matched proportional changes in the concentration of each reactant independently. Here is how it works:

• If reaction order is 1 with respect to a reactant, doubling its concentration doubles the reaction rate.
• If reaction order is 2, then doubling its concentration quadruples the rate.
• Zero order implies changing the concentration does not affect the rate.

The overall reaction order is the sum of the orders with respect to all reactants, and it provides insight into the complexity and steps involved in the mechanism of the reaction.

Rate of Reaction

The rate of reaction is a measure of how quickly reactants are converted to products in a chemical reaction. It is usually expressed as the change in concentration of a reactant or product per unit time. In this context, it lets us understand how a reaction progresses and guide optimal conditions for various processes.

The rate of reaction can be affected by several factors:

• Concentration: As we see in the exercise, changes in reactant concentrations directly influence the reaction rate according to the reaction order.
• Temperature: Increasing the temperature usually increases the reaction rate, as molecules move faster and collide more frequently.
• Catalysts: These substances increase the reaction rate without being consumed, by providing an alternative pathway with a lower activation energy.
• Surface Area: For reactions involving solids, increased surface area can lead to faster rates due to more exposed particles available for collisions.

By understanding these factors, we can control reaction speeds, making chemical processes more efficient and safer, depending on practical needs.