Question

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

Short answer

Question: Express the rate of reaction for the given chemical reaction in terms of (a) ∆[N2O4] and (b) ∆[N2]. Answer: (a) The rate of reaction in terms of ∆[N2O4] is: Rate of reaction = -\(\dfrac{d[\mathrm{N}_{2}\mathrm{O}_{4}]}{dt}\) (b) The rate of reaction in terms of ∆[N2] is: Rate of reaction = \(\dfrac{1}{3}\dfrac{d[\mathrm{N}_{2}]}{dt}\)

Step-by-step solution

Understanding the chemical reaction

The given reaction is: $$2 \mathrm{NH}_{4}(l)+\mathrm{N}_{2} \mathrm{O}_{4}(l) \longrightarrow 3 \mathrm{N}_{2}(g)+4 \mathrm{H}_{2} \mathrm{O}(g)$$ In this reaction, two moles of N2H4 react with one mole of N2O4, resulting in the formation of three moles of N2 and four moles of H2O.

Writing the rate of reaction for each substance

To express the rate of reaction, we will denote it with "-r" (the negative sign indicates that reactants are decreasing in concentration). We can write the rate in terms of the change in concentration of each reactant and product: Rate of reaction = -\(\dfrac{1}{2}\dfrac{d[\mathrm{N}_{2}\mathrm{H}_{4}]}{dt}\) = -\(\dfrac{d[\mathrm{N}_{2}\mathrm{O}_{4}]}{dt}\) = \(\dfrac{1}{3}\dfrac{d[\mathrm{N}_{2}]}{dt}\) = \(\dfrac{1}{4}\dfrac{d[\mathrm{H}_{2}\mathrm{O}]}{dt}\)

Expressing the rate of reaction in terms of ∆[N2O4]

In order to express the rate of reaction in terms of ∆[N2O4], we can use the second part of the rate equation: Rate of reaction = -\(\dfrac{d[\mathrm{N}_{2}\mathrm{O}_{4}]}{dt}\) This means that the rate of reaction is equal to the negative change in concentration of N2O4 with respect to time.

Expressing the rate of reaction in terms of ∆[N2]

Likewise, we can express the rate of reaction in terms of ∆[N2] using the third part of the rate equation: Rate of reaction = \(\dfrac{1}{3}\dfrac{d[\mathrm{N}_{2}]}{dt}\) This means that the rate of reaction is one-third of the change in concentration of N2 with respect to time. In conclusion: (a) The rate of reaction in terms of \(\Delta\left[\mathrm{N}_{2} \mathrm{O}_{4}\right]\) is given by: Rate of reaction = -\(\dfrac{d[\mathrm{N}_{2}\mathrm{O}_{4}]}{dt}\) (b) The rate of reaction in terms of \(\Delta\left[\mathrm{N}_{2}\right]\) is given by: Rate of reaction = \(\dfrac{1}{3}\dfrac{d[\mathrm{N}_{2}]}{dt}\)

Key concepts

Rate of Reaction Expression

The rate of a chemical reaction expresses how quickly reactants turn into products over time. For the reaction
\(2 \text{N}_2\text{H}_4(l) + \text{N}_2\text{O}_4(l) \rightarrow 3 \text{N}_2(g) + 4 \text{H}_2\text{O}(g)\),
the rate can be expressed in terms of the disappearance of reactants or the appearance of products. It's a mathematical representation of the speed at which a chemical process occurs. Consider the variable 'r' to represent this rate. Since the concentration of reactants decreases over time, the rate in terms of reactants is negative, indicating a decline. For products, the rate is positive, highlighting an increase in concentration. For instance, the rate expression \(-\dfrac{d[\text{N}_2\text{O}_4]}{dt}\) denotes the rate of disappearance of \(\text{N}_2\text{O}_4\), meaning for every unit of time, how much the concentration of \(\text{N}_2\text{O}_4\) goes down. The balancing coefficients from the chemical equation are crucial to correctly represent the rate for each substance, reflecting their stoichiometry.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that studies the rates of chemical reactions and the factors that affect these rates. It involves understanding not just how quickly reactions occur, but also the steps (mechanisms) by which they proceed. The reaction rate is influenced by several factors, including

• Concentration of reactants
• Temperature
• Presence of a catalyst
• Surface area of reactants

One of the key elements of kinetics is the rate law, which provides a quantitative relationship between the concentration of reactants and the rate of reaction. The law is derived from experimental data and often takes the form \(r = k[A]^m[B]^n\), where \(k\) is the rate constant, and \(m\) and \(n\) are orders of the reaction with respect to reactants \(A\) and \(B\), respectively. Kinetic studies can also help in deducing reaction mechanisms, predicting the effects of changing conditions, and controlling how fast products are created in industrial processes.

Concentration Change Over Time

The concentration change over time is central to understanding chemical reactions. It's tied directly to the rate at which reactants are used up and products are formed. When expressing the rate of a chemical reaction, differential equations are used to relate the rate to changes in concentration over small time intervals. In the given equation, for example, the change in concentration of \(\text{N}_2\text{O}_4\) over time can tell us the pace at which it's being consumed. This change is described by the derivative \(-\dfrac{d[\text{N}_2\text{O}_4]}{dt}\). A steeper slope indicates a faster reaction rate, pointing to a more rapid consumption of \(\text{N}_2\text{O}_4\). Similarly, changes in the concentration of products relate directly to how quickly they appear. Studying these changes offers insights into the kinetics of the reaction, allowing us to predict how long it will take for a reaction to go to completion or reach a certain point under given conditions.