Problem 38 - A Complete Resource Book in Chemistry for JEE Main

Question

If the rate law of a reaction \(\mathrm{nA} \longrightarrow \mathrm{B}\) is expressed as Rate \(=-\frac{1}{n} \frac{d[A]}{d t}=+\frac{d[B]}{d t}=k[A]^{x}\) The unit of the rate constant will be (a) \(\mathrm{mol}^{\mathrm{x}} / \mathrm{L}^{\mathrm{x}} / \mathrm{s}\) (b) \(\mathrm{L}^{\mathrm{x}} / \mathrm{mol}^{\mathrm{t}} \mathrm{s}\) (c) \(m o l^{(1-x)} / L^{(x-1)} \cdot S^{-1}\) (d) \(\operatorname{mol}^{(x-1)} / L^{(1-x)} \cdot S^{-1}\)

Step-by-Step Solution

Answer

The unit of the rate constant is \(\operatorname{mol}^{(x-1)} / L^{(1-x)} \cdot \text{s}^{-1}\) (option d).

Step 1: Understand the Given Rate Law

The rate law is given as \(\text{Rate} = k[A]^x\). Here, \(k\) is the rate constant, \([A]\) is the concentration of reactant \(A\), and \(x\) is the order with respect to \(A\). The rate of reaction has units of concentration/time, typically \(\frac{\text{mol}}{\text{L}\cdot\text{s}}\).

Step 2: Determine the Units of Rate

The units of reaction rate are \(\frac{\text{mol}}{\text{L}\cdot\text{s}}\). This is consistent with the units for \(\frac{-1}{n} \frac{d[A]}{dt}\) and \(\frac{d[B]}{dt}\). These represent the change in concentration over time.

Step 3: Identify Units of Concentration Term

The expression \([A]^x\) involves \([A]\), which has units of \(\frac{\text{mol}}{\text{L}}\). Raising it to the power \(x\), the units become \(\left(\frac{\text{mol}}{\text{L}}\right)^x = \frac{\text{mol}^x}{\text{L}^x}\).

Step 4: Solve for Rate Constant Units

In the rate law \(\text{Rate} = k[A]^x\), substituting the units:\[\frac{\text{mol}}{\text{L} \cdot \text{s}} = k \cdot \frac{\text{mol}^x}{\text{L}^x}\]To solve for \(k\), divide by \([A]^x\):\[k = \frac{\text{mol}^{1-x}}{\text{L}^{1-x} \cdot \text{s}}\]

Step 5: Select Matching Option

Compare the derived unit for \(k\), which is \(\frac{\text{mol}^{1-x}}{\text{L}^{1-x} \cdot \text{s}}\), to the options given. The correct match is option (d): \(\text{mol}^{(x-1)} / \text{L}^{(1-x)} \cdot \text{s}^{-1}\).

Key Concepts

Rate Law

In chemical kinetics, understanding the rate law is crucial for predicting how fast reactions proceed. The rate law is an equation expressing the relationship between the concentration of reactants and the reaction rate. For a general reaction, \[-nA \rightarrow B\] the rate can be defined as \[\text{Rate} = -\frac{1}{n} \frac{d[A]}{dt} = +\frac{d[B]}{dt} = k[A]^x\] where \

• \( k \) is the rate constant, a multiplier for how quickly a reaction occurs.
• [A] is the concentration of reactant A.
• \( x \) indicates the reaction order for A.

By examining the rate law, scientists can gain insights into the pathway the reaction takes and estimate how altering parameters, like concentration, affects the speed of the reaction.

Rate Constant

The rate constant, often denoted by \( k \), is a crucial component of the rate law. It provides specifics on how fast a reaction progresses at constant conditions. However, it is critical to understand that while rate constants are typically constant at a given temperature, they are influenced by factors such as temperature and pressure.

• The rate constant has different units depending on the reaction order, providing a clue about the influence of various reactants.
• In this example, where the rate law is \( \text{Rate} = k[A]^x \), the unit for \( k \) is determined to balance the rate equation, yielding \( \frac{\text{mol}^{1-x}}{\text{L}^{1-x} \cdot \text{s}} \) for this specific case.

This means the units of \( k \) change based on the value of \( x \), the reaction order of A. Understanding these units helps predict how reactions will respond to changes in conditions.

Reaction Order

Reaction order, symbolized as \( x \) in the rate law \( \text{Rate} = k[A]^x \), is a key player in chemical kinetics. It tells us how the rate depends on the concentration of reactants.

• If \( x = 1 \), the reaction is first-order with respect to A, meaning the rate is directly proportional to [A].
• If \( x = 2 \), the reaction is second-order with respect to A, indicating that the rate depends on the concentration of A squared.

Reactions can have mixed orders (e.g., non-integer values) or be zero-order concerning a reactant if changes in its concentration have no effect on the rate. Knowing the order aids in setting up experiments and devising strategies for process optimization or scaling up chemical reactions in industrial settings.

Units of Measurement

In reaction kinetics, paying attention to units of measurement is essential for clarity and correctness in calculations. Each component of the rate law, from reactants to rate constants, is expressed in specific units.

• The rate of reaction is usually expressed as \( \frac{\text{mol}}{\text{L} \cdot \text{s}} \), indicating how the concentration of a reactant changes over time.
• The concentration of a reactant, like [A], is expressed in \( \text{mol/L} \).
• Combining these, the units of the rate constant \( k \) must adjust to ensure dimension consistency in the rate law. For \( \text{Rate} = k[A]^x \), the correct unit of \( k \) was determined to be \( \frac{\text{mol}^{1-x}}{\text{L}^{1-x} \cdot \text{s}} \).

Being meticulous with units is not only foundational to solving kinetic problems but also to ensure reliable and interpretable results across various conditions.