Problem 81 - A Complete Resource Book in Chemistry for JEE Main

Question

The rate of a certain hypothetical reaction \(\mathrm{A}+\mathrm{B}+\mathrm{C} \longrightarrow\) Products is given by \(\mathrm{r}=-\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}}=k[\mathrm{~A}]^{1 / 2}[\mathrm{~B}]^{1 / 3}[\mathrm{C}]^{1 / 4}\) The order of the reaction is (a) \(13 / 12\) (b) \(13 / 14\) (c) \(12 / 13\) (d) \(13 / 11\)

Step-by-Step Solution

Answer

(a) \(\frac{13}{12}\)

Step 1: Understand Reaction Order

The reaction order is found by adding the exponents of the concentration terms in the rate equation.

Step 2: Identify Exponents

In the given rate equation \(r = k[\mathrm{A}]^{1 / 2}[\mathrm{B}]^{1 / 3}[\mathrm{C}]^{1 / 4}\), the exponents on \([\mathrm{A}]\), \([\mathrm{B}]\), and \([\mathrm{C}]\) are \(\frac{1}{2}\), \(\frac{1}{3}\), and \(\frac{1}{4}\) respectively.

Step 3: Calculate Total Reaction Order

Add the exponents together to find the total reaction order: \(\frac{1}{2} + \frac{1}{3} + \frac{1}{4}\). To add these fractions, find a common denominator.

Step 4: Find a Common Denominator

The lowest common multiple of 2, 3, and 4 is 12. Rewrite each fraction with a denominator of 12: \(\frac{1}{2} = \frac{6}{12}\), \(\frac{1}{3} = \frac{4}{12}\), \(\frac{1}{4} = \frac{3}{12}\).

Step 5: Sum the Fractions

Add the fractions: \(\frac{6}{12} + \frac{4}{12} + \frac{3}{12} = \frac{13}{12}\).

Step 6: Conclusion

The total order of the reaction is \(\frac{13}{12}\).

Key Concepts

Rate Equation

In the world of chemical kinetics, the rate equation is a powerful tool that helps predict how the concentration of reactants influences the speed of a chemical reaction. It is expressed in the form: \[ r = k[A]^{m}[B]^{n}[C]^{o} \]where "r" represents the rate of the reaction, "k" stands for the rate constant, and [A], [B], and [C] indicate the concentrations of the reactants. The exponents "m," "n," and "o" are the powers to which the concentrations are raised in the equation. These exponents are crucial as they depict how the rate of reaction depends on each reactant.

• If the exponent is 1, the reaction is said to be first order with respect to that reactant.
• If the exponent is 2, the reaction is second order with respect to that reactant.
• Exponents of zero mean the reactant does not affect the rate.

Interestingly, the overall order of the reaction is calculated by summing up all these exponents, giving a clear indication of how each reactant influences the rate.

Exponents in Reactions

Exponents in reactions are fundamental when dealing with reaction kinetics. They help chemists determine how sensitive a reaction is to changes in concentration. In the given reaction rate equation, we observe that the exponents for each reactant are fractional:

• For \(A\), the exponent is \( \frac{1}{2} \).
• For \(B\), it is \( \frac{1}{3} \).
• For \(C\), it is \( \frac{1}{4} \).

These fractional exponents suggest that the sensitivity of the reaction rate to each reactant is not straightforward and implies a more complex interplay.
To fully understand the influence of reactants, calculate the overall reaction order by summing these values, which provides insight into the reaction mechanism and potential pathways. It is essential for determining how each reactant impacts the reaction speed and enabling targeted control of reaction conditions.

Adding Fractions

Adding fractions is a crucial skill needed to determine the overall order of a reaction once the exponents are identified. It involves finding a common denominator, which simplifies the process of summing fractions smoothly.
With the exponents of \( \frac{1}{2} \), \( \frac{1}{3} \), and \( \frac{1}{4} \), we look for a common denominator. This can be done by checking the multiples of 2, 3, and 4 until finding the smallest one they all share, which is 12.

• Convert \( \frac{1}{2} \) to \( \frac{6}{12} \).
• Change \( \frac{1}{3} \) to \( \frac{4}{12} \).
• Modify \( \frac{1}{4} \) to \( \frac{3}{12} \).

Adding these newly equivalent fractions gives us \( \frac{13}{12} \), which represents the total order of reaction. This approach is central in chemical kinetics to resolve how combined effects of different concentrations influence the rate.

Chemical Kinetics

Chemical kinetics involves studying how reactions occur, the speed at which they proceed, and the factors affecting their rates. It focuses on reaction mechanisms, which depict the step-by-step sequence of elementary reactions leading to the formation of products.

• One core aspect is understanding the influence of concentration changes on rate, which is described by the rate equation.
• Temperature is another major factor, often increasing reaction rates by providing more energy to reactant molecules.
• Catalysts can also accelerate reactions without being consumed, offering alternative pathways with lower activation energy.

Studying chemical kinetics is important for industries and research, optimizing reaction conditions for better yields and efficiencies. Thus, mastering these concepts gives you the power to predict and control reaction outcomes effectively.