Problem 15 - A Complete Resource Book in Chemistry for JEE Main

Question

The unit of second-order reaction rate constant is (a) \(\mathrm{L}^{-1} \cdot \mathrm{mol}^{-1} \mathrm{ds}^{-1}\) (b) \(\mathrm{L}^{2} \mathrm{~mol}^{-2} \mathrm{~s}^{-1}\) (c) L.mol \(^{-1} \mathrm{~s}^{-1}\) (d) \(s^{-t}\)

Step-by-Step Solution

Answer

The unit of the second-order reaction rate constant is (c) L.mol \(^{-1}\) \mathrm{s}^{-1\).

Step 1: Understand the Rate Law for Second-Order Reactions

In a second-order reaction, the rate of the reaction is proportional to the square of the concentration of one reactant or the product of the concentrations of two reactants. The rate law can be expressed as \( r = k[A]^2 \) or \( r = k[A][B] \), where \( k \) is the rate constant.

Step 2: Determine the Units of Concentration and Rate

The concentration [A] or [B] is measured in moles per liter (mol/L). The rate \( r \), which has the units of concentration change per unit time, is expressed in mol L\(^{-1}\) s\(^{-1}\).

Step 3: Derive the Units of the Rate Constant k

For the rate law \( r = k[A]^2 \), rearrange to get \( k = \frac{r}{[A]^2} \). Substitute the rate and concentration units: \( k = \frac{\text{mol L}^{-1} \text{s}^{-1}}{(\text{mol L}^{-1})^2} = \text{L mol}^{-1} \text{s}^{-1} \).

Step 4: Select the Correct Option Based on Units

Based on the derivation, the correct unit for the second-order reaction rate constant \( k \) is L mol\(^{-1}\) s\(^{-1}\). Check the options for this unit. Option (c) matches exactly.

Key Concepts

Reaction Rate Constant

The reaction rate constant, commonly denoted as \( k \), is a crucial factor in understanding the speed of a chemical reaction. It is a proportionality factor in the rate law, which shows how the rate of reaction depends on the concentration of reactants. In a second-order reaction, the rate is directly proportional to either

• the square of the concentration of a single reactant, expressed as \( [A]^2 \)
• or the product of the concentrations of two reactants, \( [A][B] \).

In such equations, the reaction rate constant \( k \) ensures that the rate equation balances. This constant is specific to each reaction and changes with temperature and catalysts. But at a given condition, it tells us how quickly the reactants are converted to products.

Rate Law

The rate law expresses the relationship between the rate of a chemical reaction and the concentration of its reactants. For second-order reactions, the rate law can appear as \( r = k[A]^2 \) or \( r = k[A][B] \). Here, \( r \) represents the reaction rate, \( k \) is the rate constant, and \( [A] \) and \( [B] \) are the molar concentrations of the reactants.

The equations tell us:

• If the concentration of a reactant is doubled, the reaction rate will increase by four times if the reaction is second-order with respect to that reactant.
• If the reaction is involving two different reactants, the mixing of both concentrations will determine the rate.

Understanding the rate law helps chemists control reaction speeds by adjusting reactant concentrations.

Units of Concentration

Concentration is often measured in terms of moles per liter, abbreviated as mol/L or M (molarity). This is a standard unit in chemistry used to describe how much solute is present in a given volume of solution.

In the context of rate laws:

• [A] or [B] signifies the concentration of reactants involved in the reaction.
• The units of concentration play a significant role when calculating the units for other parameters like the rate of reaction and the reaction rate constant.

For second-order reactions, these concentrations become squared (\( [A]^2 \) or \( [A][B] \)), meaning we consider interactions on more than a linear level, pointing out their significance in the rate equation.

Units of Rate

The units of rate in a reaction refer to how the concentration of a reactant or product changes over time. Typically, it is expressed in mol/L/s, which reads as moles per liter per second. This format shows:

• The change in molarity (concentration) as the reaction progresses.
• The time frame in seconds over which this change happens.

In a second-order reaction, the rate is proportional to either the square of a reactant's concentration or the product of two concentrations. This must be taken into account when deducing the rate constant. By understanding these units, one can predict the speed or progress of a chemical reaction, providing valuable insight into how the reaction might proceed in varied conditions.