Question

Express the rate of the reaction $$2 \mathrm{C}_{2} \mathrm{H}_{6}(g)+7 \mathrm{O}_{2}(g) \longrightarrow 4 \mathrm{CO}_{2}(g)+6 \mathrm{H}_{2} \mathrm{O}(g)$$ in terms of (a) \(\Delta\left[\mathrm{C}_{2} \mathrm{H}_{6}\right]\) (b) \(\Delta\left[\mathrm{CO}_{2}\right]\)

Short answer

Question: Express the rate of the given reaction in terms of the concentration changes of C2H6 and CO2. Answer: (a) The rate of reaction in terms of the concentration change of C2H6 is given by: $$\text{rate} = -\frac{1}{2} \times \frac{d[\mathrm{C}_{2} \mathrm{H}_{6}]}{dt}$$. (b) The rate of reaction in terms of the concentration change of CO2 is given by: $$\text{rate} = \frac{1}{4} \times \frac{d[\mathrm{CO}_{2}]}{dt}$$.

Step-by-step solution

Recall the definition of reaction rate

The rate of a chemical reaction is defined as the change in the concentration of a reactant or product over time. Mathematically, the rate of reaction for a species A can be given by: $$\text{rate}_{A} = \frac{1}{\text{stoichiometric coefficient}} \times \frac{d[\text{A}]}{dt}$$. Note that if A is a reactant, the stoichiometric coefficient is negative, and if A is a product, the stoichiometric coefficient is positive.

Apply the formula to express the rate of reaction in terms of \(\Delta[\mathrm{C}_{2} \mathrm{H}_{6}]\)

For part (a), we are to express the rate of reaction in terms of the change in concentration of C2H6, which is a reactant with a stoichiometric coefficient of 2. Using the formula from step 1, we can write: $$\text{rate} = -\frac{1}{2} \times \frac{d[\mathrm{C}_{2} \mathrm{H}_{6}]}{dt}$$

Apply the formula to express the rate of reaction in terms of \(\Delta[\mathrm{CO}_{2}]\)

Similarly, for part (b), we are to express the rate of reaction in terms of the change in concentration of CO2, which is a product with a stoichiometric coefficient of 4. Using the formula from step 1, we can write: $$\text{rate} = \frac{1}{4} \times \frac{d[\mathrm{CO}_{2}]}{dt}$$ In summary, the rate of the given reaction can be expressed in terms of the concentration changes of C2H6 and CO2 as follows: (a) $$\text{rate} = -\frac{1}{2} \times \frac{d[\mathrm{C}_{2} \mathrm{H}_{6}]}{dt}$$ (b) $$\text{rate} = \frac{1}{4} \times \frac{d[\mathrm{CO}_{2}]}{dt}$$

Key concepts

Chemical Kinetics

Understanding the speed of a chemical reaction, or its reaction rate, is a fundamental part of chemical kinetics. In essence, chemical kinetics studies how rapidly chemical reactions occur and what factors—such as temperature, pressure, and concentration—affect this speed. For example, a reaction that produces products quickly is said to have a high reaction rate. On the other hand, a reaction that takes a long time to convert reactants to products has a low reaction rate.

This concept is crucial in various fields, from industrial synthesis of chemicals, where maximizing efficiency is key, to environmental chemistry, where the rates at which pollutants are broken down can have significant implications. One of the most important aspects to consider in chemical kinetics is the rate law, which is an equation that links the rate of reaction to the concentration of reactants. This understanding helps to predict how changes in conditions can alter the speed of a chemical reaction.

Stoichiometry

Stoichiometry can be thought of as the 'recipe' for a chemical reaction. It refers to the quantitative relationship between the amounts of reactants and products involved in a chemical reaction based on the balanced chemical equation. In the context of reaction rates, stoichiometry dictates how the changes in the amount of one substance relate to the changes in another.

Understanding Reaction Coefficients

In our exercise, the stoichiometric coefficients (the numbers in front of the chemical formulas in the balanced equation) tell us that 2 moles of C2H6 react with 7 moles of O2 to produce 4 moles of CO2 and 6 moles of H2O. In terms of reaction rates, these coefficients are used to relate the rates of consumption of reactants to the rates of formation of products, making sure the conservation of mass is respected in the reaction. This insight is essential for everything from laboratory experiments to engineering applications where precise calculations are necessary for successful outcomes.

Concentration Changes

To put it simply, concentration is a measure of how much of a given substance is present in a mixture. In chemistry, changes in concentration over time typically indicate that a reaction is taking place. The exercise on hand exemplifies how the concentrations of reactants decrease while those of products increase as the reaction proceeds.

Rate Expressions

The changes in concentration are crucial for calculating the reaction rate. When answering the exercise, we used differential rate expressions which involve the derivative of concentration with respect to time. This represents the instantaneous rate of change, giving us a clear picture of how fast the reaction is occurring at any given moment. In practice, understanding concentration changes allows chemists and engineers to design reaction conditions that optimize product yields and minimize waste, which is key for sustainability and economic viability in chemical manufacturing.