Question

If two equilibrium reactions are added together, what is the relationship between the equilibrium constants for the individual reactions and that for the overall reaction?

Short answer

The overall equilibrium constant is the product of the individual equilibrium constants: \( K_{ ext{overall}} = K_1 \times K_2 \).

Step-by-step solution

Understand Equilibrium Reactions

Equilibrium reactions are those where the rate of the forward reaction equals the rate of the reverse reaction. For any reaction \( aA + bB \leftrightarrow cC + dD \), the equilibrium constant \( K \) is defined as \( K = \frac{[C]^c[D]^d}{[A]^a[B]^b} \), where \([X]\) denotes the concentration of substance \( X \) at equilibrium.

Consider Two Reactions

Suppose we have two reactions: \( R_1: aA + bB \leftrightarrow cC + dD \) with equilibrium constant \( K_1 \), and \( R_2: eE + fF \leftrightarrow gG + hH \) with equilibrium constant \( K_2 \).

Add Reactions Together

Adding these two reactions together results in a new overall reaction: \( R_{ ext{overall}}: aA + bB + eE + fF \leftrightarrow cC + dD + gG + hH \).

Calculate Overall Equilibrium Constant

The equilibrium constant for the overall reaction \( R_{ ext{overall}} \) is the product of the equilibrium constants of the individual reactions: \( K_{ ext{overall}} = K_1 \times K_2 \). This is because the concentrations of products and reactants multiply when reactions are added, resulting in a new expression which is the product of the original expressions for \( K_1 \) and \( K_2 \).

Key concepts

Equilibrium Reactions

In a chemical system at equilibrium, the forward and backward reactions occur at the same rate. This balance creates a state where the concentrations of reactants and products remain constant over time. To mathematically describe this state, we use the equilibrium constant, denoted by the symbol \( K \). Given a general equilibrium reaction \( aA + bB \leftrightarrow cC + dD \), the equilibrium constant \( K \) can be calculated using the formula:\[ K = \frac{[C]^c[D]^d}{[A]^a[B]^b} \]where \([X]\) is the concentration of species \( X \) at equilibrium. Understanding equilibrium reactions helps us predict how changes in conditions (like concentration or temperature) might shift the balance between reactants and products. This concept is crucial when we consider more complex reactions or when reactions are added together, affecting the overall equilibrium.

Overall Reaction

When dealing with multiple reactions, it’s often necessary to consider what happens when they are combined into an overall reaction. This overall reaction is essentially a sum of individual reactions. Imagine we have two reactions: - Reaction 1: \( aA + bB \leftrightarrow cC + dD \) with equilibrium constant \( K_1 \)- Reaction 2: \( eE + fF \leftrightarrow gG + hH \) with equilibrium constant \( K_2 \)The overall reaction simply combines all reactants and products from these steps into one equation: \( aA + bB + eE + fF \leftrightarrow cC + dD + gG + hH \).Understanding how to calculate and interpret the equilibrium constant for this overall process is essential, as it tells us how the combined reaction system behaves in equilibrium. It reflects the equilibrium positions of the entire reaction system as opposed to the individual steps.

Adding Reactions Together

Adding reactions in chemistry involves combining two or more reactions to form a new overall reaction. This process affects the equilibrium constant of the system. When reactions are added:- Each individual reaction has its own equilibrium constant (\( K_1 \), \( K_2 \), etc.).- For the new combined or overall reaction, the equilibrium constant \( K_{\text{overall}} \) is the product of the equilibrium constants from all individual reactions combined. Thus:\[ K_{\text{overall}} = K_1 \times K_2 \]This rule works because in equilibrium states, the effect of adding reactions is multiplicative in terms of concentration. When calculating \( K_{\text{overall}} \), the concentrations from each individual equilibrium expression multiply together, creating a new balance for the added reaction sequence. This method ensures that the equilibrium constant accurately reflects all interactions present in the new, combined system.