Question

The rate law for the reaction \(2 \mathrm{NO}_{2}(g) \longrightarrow \mathrm{N}_{2} \mathrm{O}_{4}(g)\) is rate \(=k\left[\mathrm{NO}_{2}\right]^{2}\). Which of these changes will change the value of \(k ?\) (a) The pressure of \(\mathrm{NO}_{2}\) is doubled. (b) The reaction is run in an organic solvent. (c) The volume of the container is doubled. (d) The temperature is decreased. (e) A catalyst is added to the container.

Short answer

The value of 'k' will change when the temperature is decreased (d) or a catalyst is added (e) to the reaction.

Step-by-step solution

Understanding the Rate Constant 'k'

The rate constant 'k' in a chemical reaction is a proportionality constant that depends on the temperature of the reaction medium and the presence of a catalyst. Let's go through each of the options and see if they affect the value of the rate constant.

The Pressure of NO2 is doubled

The rate constant 'k' is not affected by changes in concentration or pressure of the reactants. Thus, even if the pressure of NO2 is doubled, it will not change the value of 'k'.

The Reaction is Run in an Organic Solvent

Changing the medium in which the reaction is taking place (such as changing it to an organic solvent) can change how molecules interact, but it doesn't directly impact the value of 'k'. It could influence the reaction rate by changing the concentration of reactants, but it won't change the rate constant itself.

The Volume of the Container is Doubled

Doubling the volume of the container is equivalent to halving the concentration of the gases present. However, this change will not affect 'k', as the rate constant is not dependent on concentration.

The Temperature is Decreased

The rate constant 'k' is directly related to the temperature of the reaction medium based on the Arrhenius equation. Therefore, decreasing the temperature will indeed change the value of 'k'.

A Catalyst is Added to the Container

Adding a catalyst will affect the rate constant 'k' as it changes the rate of reaction by providing an alternative reaction pathway with a lower activation energy.

Key concepts

Understanding Rate Constant 'k'

In a chemical reaction, the rate constant 'k' plays a crucial role in determining the speed at which the reaction proceeds. It is specific to each reaction and is influenced by factors such as temperature and the presence of catalysts, but not by the concentration of reactants or the pressure of the system.

For the reaction: \(2 \text{NO}_{2}(g) \to \text{N}_{2} \text{O}_{4}(g)\), the rate constant 'k' appears in the rate law: \(rate = k[\text{NO}_{2}]^{2}\). This shows that while the reaction rate is proportional to the square of the concentration of \(\text{NO}_{2}\), the rate constant 'k' itself remains unchanged with variations in \(\text{NO}_{2}\) pressure. Therefore, doubling the pressure of \(\text{NO}_{2}\) does not alter the value of 'k'. Learning about 'k' through such examples reinforces its conceptual understanding.

Arrhenius Equation and Temperature's Effect on 'k'

The Arrhenius equation is a mathematical expression that describes how the rate constant 'k' varies with temperature. It shows that 'k' increases exponentially with an increase in temperature.

The equation is often presented as: \(k = A \text{e}^{-E_{a}/(RT)}\), where \(A\) is the frequency factor, \(E_{a}\) is the activation energy, \(R\) is the universal gas constant, and \(T\) is the absolute temperature. A decrease in temperature, as indicated in the exercise, implies an increase in the exponent's magnitude, leading to a smaller value for 'k'. This elucidation helps students grasp why the rate constant changes with temperature variations.

Impact of Reaction Medium on Reaction Rate

While the reaction medium, such as an organic solvent, doesn't directly change the rate constant 'k', it can influence the reaction rate by affecting solubility, reactant dispersion, and ionic strength. These factors can alter the frequency of effective collisions among reactant molecules. Thus, changing the medium to an organic solvent may result in subtle shifts in the reaction dynamics, potentially changing concentration over time, though without affecting the inherent value of 'k'. Discussing this clarifies the distinction between the rate constant and the actual rate of the reaction, which can confuse students.

Role of Catalysts in Chemistry

Catalysts are substances that increase the rate of a chemical reaction without being consumed in the process. They provide alternative pathways for the reaction with lower activation energies, effectively increasing the reaction rate. Nonetheless, it's important to note that while adding a catalyst to a reaction, as in the exercise, may seem to change the rate constant 'k', it typically changes the effective rate constant under the catalyzed conditions.

Catalysts do not alter the original rate constant 'k' for the uncatalyzed reaction; instead, they modify the reaction mechanism available to the reactants. Understanding this can help students comprehend why reactions proceed faster with catalysts without the initial 'k' being directly altered.