Problem 40 - Introductory Chemistry Atoms First

Question

Write the definition of equilibrium in terms of the general rate laws Rate \(_{\text {forward rxn }}=k_{\text {forward rxn }}[\text { Reactants }]^{\text {order }}\) and Rate \(_{\text {reverse rxn }}=k_{\text {reverse rxn }}[\text { Products }]^{\text {order }}\)

Step-by-Step Solution

Answer

Equilibrium in terms of general rate laws can be defined when the forward and reverse reaction rates become equal, represented by the equation: \(k_{\text{forward rxn}}[\text{Reactants}]^{\text{order}} = k_{\text{reverse rxn}}[\text{Products}]^{\text{order}}\) Here, \(k_{\text{forward rxn}}\) and \(k_{\text{reverse rxn}}\) are the rate constants for forward and reverse reactions, and \(\text{order}\) represents the order of the reaction with respect to reactants and products.

Step 1: Forward Reaction Rate Law

The forward reaction rate law can be represented as: Rate\(_{\text{forward rxn}} = k_{\text{forward rxn}}[\text{Reactants}]^{\text{order}}\) Here, Rate\(_{\text{forward rxn}}\) represents the rate of the forward reaction, \(k_{\text{forward rxn}}\) is the rate constant for the forward reaction, \([\text{Reactants}]\) represents the concentration of the reactants, and \(\text{order}\) is the order of the reaction with respect to the reactants.

Step 2: Reverse Reaction Rate Law

Similarly, the reverse reaction rate law can be represented as: Rate\(_{\text{reverse rxn}} = k_{\text{reverse rxn}}[\text{Products}]^{\text{order}}\) Here, Rate\(_{\text{reverse rxn}}\) represents the rate of the reverse reaction, \(k_{\text{reverse rx_{}{n}}}\) is the rate constant for the reverse reaction, \([\text{Products}]\) represents the concentration of the products, and \(\text{order}\) is the order of the reaction with respect to the products.

Step 3: Equilibrium Condition

At equilibrium, the rates of the forward and reverse reactions become equal. This can be mathematically represented as: Rate\(_{\text{forward rxn}}\) = Rate\(_{\text{reverse rxn}}\)

Step 4: Equilibrium Definition

Now, using the rate laws for forward and reverse reactions, we can define equilibrium as: \(k_{\text{forward rxn}}[\text{Reactants}]^{\text{order}} = k_{\text{reverse rxn}}[\text{Products}]^{\text{order}}\) This equation defines the condition for equilibrium in terms of the general rate laws for forward and reverse reactions.

Key Concepts

Rate Laws

Rate laws are mathematical expressions that describe the speed of a chemical reaction. They include factors that influence the reaction rate such as the concentration of reactants and a proportionality constant known as the rate constant. For a forward reaction, the rate law is expressed by \[ \text{Rate}_{\text{forward}} = k_{\text{forward}}[\text{Reactants}]^{\text{order}} \]where:

• \(k_{\text{forward}}\) is the rate constant specific to the forward reaction.
• \([ ext{Reactants}]\) indicates the concentration of the reactants involved.
• \(\text{order}\) represents the power to which the concentration is raised, often determined experimentally.

For reverse reactions, the law is similar,\[ \text{Rate}_{\text{reverse}} = k_{\text{reverse}}[ ext{Products}]^{\text{order}} \].Understanding these laws helps in predicting how changing reactant concentrations will affect the speed of the reactions in both directions.

Reaction Rates

Reaction rates refer to how quickly a chemical reaction proceeds. They are influenced by several factors like temperature, pressure, concentration, and the presence of catalysts. The rate law expressions give a quantitative analysis of how reaction rates vary with the concentration of reactants or products. For instance, the forward reaction rate is given by \[ \text{Rate}_{\text{forward}} \] while the reverse reaction rate is expressed as:\[ \text{Rate}_{\text{reverse}} \].
At equilibrium, both the forward and reverse reaction rates are equal, meaning the rate at which products form equals the rate at which they revert to reactants. This creates a dynamic balance where reactions occur continuously, yet the concentrations remain constant over time.

Equilibrium Constant

The equilibrium constant is a critical concept used to quantify the balance between the concentrations of reactants and products at equilibrium. It is derived from the rate laws when the rates of forward and reverse reactions are equal. Given the equilibrium condition,\[ k_{\text{forward}}[\text{Reactants}]^{\text{order}} = k_{\text{reverse}}[\text{Products}]^{\text{order}} \],the equilibrium constant \(K\) can be expressed as:
\[ K = \frac{k_{\text{forward}}}{k_{\text{reverse}}} = \frac{[\text{Products}]^{\text{order}}}{[\text{Reactants}]^{\text{order}}} \].
This dimensionless constant gives insights into the extent of a reaction at equilibrium, indicating whether products or reactants are favored under certain conditions. Understanding how \(K\) functions aids in predicting reaction behavior in varied environments.

Forward and Reverse Reactions

Forward and reverse reactions are integral components of dynamic chemical equilibrium. A forward reaction involves the transformation of reactants into products, while a reverse reaction involves the conversion of products back into reactants. These reactions occur simultaneously in a closed system.

• **Forward reaction**: \( \text{Reactants} \rightarrow \text{Products} \)
• **Reverse reaction**: \( \text{Products} \rightarrow \text{Reactants} \)

Understanding these concepts is crucial when considering equilibrium; at equilibrium, the rates of forward and reverse reactions are equal, not their amounts. This balance indicates that the system has reached a state where reactions are ongoing, but no net change in concentration occurs, maintaining the balance indefinitely unless conditions change.