Question

Is there a connection between the free-energy change for a reaction and its equilibrium constant? If there is a connection, what is it?

Short answer

ΔG and K are connected by ΔG = -RT ln K, indicating spontaneity and equilibrium position.

Step-by-step solution

Understand Free-Energy Change

The free-energy change (ΔG) for a reaction indicates the amount of work that can be done by the system or the amount of energy needed for a reaction to proceed. It can be used to determine whether a reaction is spontaneous (ΔG < 0), at equilibrium (ΔG = 0), or non-spontaneous (ΔG > 0).

Define the Equilibrium Constant

The equilibrium constant (K) expresses the ratio of product concentrations to reactant concentrations at equilibrium for a given reaction. It provides insight into the position of equilibrium, indicating whether products or reactants are favored.

Connect Free-Energy Change to Equilibrium Constant

The connection between the free-energy change (ΔG) and the equilibrium constant (K) is given by the equation: \[ \Delta G = -RT \, \ln K \]where R is the universal gas constant (8.314 J/(mol·K)), T is the temperature in Kelvin, and K is the equilibrium constant.

Interpret the Equation

This equation shows that if a reaction has a large negative ΔG, it will have a large equilibrium constant (favoring products). Conversely, a positive ΔG indicates a smaller K (favoring reactants). At ΔG = 0, K equals 1, indicating the reaction is at equilibrium.

Summary of the Connection

The free-energy change for a reaction and its equilibrium constant are directly related through the equation ΔG = -RT ln K. This relationship shows how the spontaneity of a reaction is reflected in its equilibrium position.

Key concepts

Free-Energy Change

The free-energy change, represented as \(\text{ΔG}\), is a vital concept in understanding chemical reactions. It tells us whether a reaction can occur without outside assistance, meaning its spontaneity. If \(\text{ΔG}\) is less than zero \((\text{ΔG} < 0)\), the reaction is spontaneous and can proceed on its own. If \(\text{ΔG}\) equals zero \((\text{ΔG} = 0)\), the reaction is at equilibrium, meaning the forward and backward reactions happen at the same rate. On the other hand, if \(\text{ΔG}\) is greater than zero \((\text{ΔG} > 0)\), the reaction is non-spontaneous, so it requires energy input to proceed. This energy term is critical in predicting how and if a reaction will take place under certain conditions.

Equilibrium Constant

The equilibrium constant \(K\) provides a snapshot of a reaction's status at equilibrium. It is expressed as the ratio of product concentrations to reactant concentrations when the rates of the forward and reverse reactions are equal. A large \(K\) indicates that the products are favored at equilibrium. Conversely, a small \(K\) signifies that the reactants are favored. This constant helps chemists understand the balance between reactants and products, as well as the favorability of either side in a chemical equation.

Reaction Spontaneity

Reaction spontaneity shows if a reaction can occur without external energy. Spontaneous reactions \((\text{ΔG} < 0)\) proceed on their own, while non-spontaneous reactions \((\text{ΔG} > 0)\) do not. The spontaneity is judged by the sign of \(\text{ΔG}\). When \(\text{ΔG} < 0\), the reaction releases energy and is favorable. Spontaneity helps predict whether a reaction will occur naturally or need some external energy input.

Universal Gas Constant

The universal gas constant, denoted as \(R\), is a fundamental value in many chemical equations, including those involving free energy and equilibrium constants. It has a value of 8.314 J/(mol·K). This constant links the free-energy change with the equilibrium constant through the equation \[ \text{ΔG} = -RT \ln K \] where \(T\) is the temperature in Kelvin. R is crucial when calculating changes in energy and understanding the behavior of gases and reactions under different conditions.

Equilibrium Position

The equilibrium position of a reaction indicates whether the reactants or products are favored at equilibrium. This position is closely linked to the equilibrium constant \(K\). If \(K\) is large, it means the reaction favors the formation of products; if \(K\) is small, it favors the reactants. The relationship between \(\text{ΔG}\) and \(K\) helps us understand where the equilibrium lies and the feasibility of shifting the equilibrium position under different conditions.