Problem 17 - Chemistry The Molecular Science

Question

Express the rate of the reaction $$ 2 \mathrm{~N}_{2} \mathrm{H}_{4}(\ell)+\mathrm{N}_{2} \mathrm{O}_{4}(\ell) \longrightarrow 3 \mathrm{~N}_{2}(\mathrm{~g})+4 \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) $$ in terms of (a) \(\Delta\left[\mathrm{N}_{2} \mathrm{O}_{4}\right]\). (b) \(\Delta\left[\mathrm{N}_{2}\right]\)

Step-by-Step Solution

Answer

(a) Rate = \\( -\frac{d[\mathrm{N_2O_4}]}{dt} \\ (b) Rate = \\( \frac{1}{3}\frac{d[\mathrm{N_2}]}{dt} \\)

Step 1: Understanding Rate of Reaction

The rate of a chemical reaction is usually expressed as the change in concentration of a reactant or a product over time. For the given reaction, we can write the rate of reaction in terms of the change in concentration of any reactant or product.

Step 2: Write General Rate Expression

For the given reaction, the general expression for the rate can be written as:\[-\frac{1}{2}\frac{d[\mathrm{N_2H_4}]}{dt} = -\frac{1}{1}\frac{d[\mathrm{N_2O_4}]}{dt} = \frac{1}{3}\frac{d[\mathrm{N_2}]}{dt} = \frac{1}{4}\frac{d[\mathrm{H_2O}]}{dt}\]This equation relates the rate of disappearance of reactants to the rate of formation of products.

Step 3: Rate in Terms of Δ[N2O4]

To express the rate in terms of \( \Delta[\mathrm{N_2O_4}] \), use the portion of the rate expression involving \( \mathrm{N_2O_4} \):\[\text{Rate} = -\frac{d[\mathrm{N_2O_4}]}{dt}\]This indicates that the rate of the reaction is equal to the negative change in concentration of \( \mathrm{N_2O_4} \) over time. Since it's a 1:1 relation with the reaction rate, there's no coefficient adjustment.

Step 4: Rate in Terms of Δ[N2]

To express the rate in terms of \( \Delta[\mathrm{N_2}] \), use the portion of the rate expression involving \( \mathrm{N_2} \):\[\text{Rate} = \frac{1}{3}\frac{d[\mathrm{N_2}]}{dt}\]This means the rate of the reaction is one-third of the change in concentration of \( \mathrm{N_2} \) over time because 3 moles of \( \mathrm{N_2} \) are produced per reaction.

Key Concepts

Chemical Kinetics

Chemical kinetics is all about understanding how quickly reactions happen. Scientists study the rate at which reactants turn into products. This is crucial because it helps predict how long a reaction takes to complete. Let's imagine a simple everyday scenario, like baking a cake. Just as we would measure how long it takes a cake to rise, in chemical kinetics, we measure how fast chemicals react.

A few key factors affect reaction rates:

• Concentration of reactants: Higher concentrations usually increase reaction rates because more molecules are available to collide and react.
• Temperature: Generally, increasing the temperature speeds up reactions. Heat provides energy, making molecules move faster and collide more.
• Catalysts: These are substances that increase the rate without being consumed in the reaction themselves. They work by lowering the energy barrier needed for the reaction to proceed.
• Surface area: More surface area can lead to faster reactions, especially in solids, as it exposes more reactant particles to the reaction.

By studying chemical kinetics, learners understand the dynamics of reactions which can contribute to innovations in fields like medicine and energy.

Rate Expression

The rate expression is a mathematical representation that links the rate of a reaction to the concentrations of reactants. It's like a recipe for how the rate depends on the reactants involved.

For the reaction \[ 2 \mathrm{~N}_{2} \mathrm{H}_{4}(\ell)+\mathrm{N}_{2} \mathrm{O}_{4}(\ell) \longrightarrow 3 \mathrm{~N}_{2}(\mathrm{~g})+4 \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \]the rate expression can show how the disappearance of hydrazine \( \mathrm{N}_{2}\mathrm{H}_{4} \) or \( \mathrm{N}_{2}\mathrm{O}_{4} \) relates to the appearance of \( \mathrm{N}_2 \) and \( \mathrm{H}_2\mathrm{O} \). The general rate equation could be written as:\[-\frac{1}{2}\frac{d[\mathrm{N_2H_4}]}{dt} = -\frac{1}{1}\frac{d[\mathrm{N_2O_4}]}{dt} = \frac{1}{3}\frac{d[\mathrm{N_2}]}{dt} = \frac{1}{4}\frac{d[\mathrm{H_2O}]}{dt}\] This rate expression is like a balanced scale, indicating how the concentration changes for each substance are related to each other. The negative signs for reactants indicate that their concentrations decrease over time, while the positive rates for products indicate increases.

Concentration Change

Concentration change is central to understanding reaction rates. It's not about quantity at any one time, but how much that quantity changes as the reaction progresses. Imagine you start with a full glass of water and then drink from it. The concentration change is like the amount of water that decreases over time.

In terms of chemical reactions, this change is often represented by the symbol \( \Delta \), which stands for the change in concentration of a reactant or product. In the provided exercise, we looked at \( \Delta[\mathrm{N}_2\mathrm{O}_4] \) and \( \Delta[\mathrm{N}_2] \).

For \( [\mathrm{N}_2\mathrm{O}_4] \), the rate expression involved its reduction over time:\[ \text{Rate} = -\frac{d[\mathrm{N}_2\mathrm{O}_4]}{dt}\]This is because it's a reactant getting used up. Conversely, for \( [\mathrm{N}_2] \), since it's being produced, its expression is:\[ \text{Rate} = \frac{1}{3}\frac{d[\mathrm{N}_2]}{dt}\] Understanding how concentration changes can help predict how fast reactions will go to completion and is key to controlling reactions in industrial and laboratory settings.