12.2 Equilibrium constants and their temperature dependence

Chemical reactions reach a balance between reactants and products called equilibrium. Equilibrium constants measure this balance, showing how much product forms compared to reactants. The constant's value depends on temperature, which affects whether reactions favor products or reactants.

Understanding equilibrium constants helps predict how reactions behave under different conditions. Temperature changes can shift the balance, influencing product yields. This knowledge is key for optimizing chemical processes in industry and understanding natural chemical systems.

Equilibrium Constant and Law of Mass Action

Defining Equilibrium Constant

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• Equilibrium constant ($K$) quantifies the relationship between reactants and products at equilibrium
• Determined by the ratio of product concentrations to reactant concentrations, each raised to their stoichiometric coefficients
• For a general reaction $aA + bB \rightleftharpoons cC + dD$, the equilibrium constant is expressed as: $K = \frac{[C]^c[D]^d}{[A]^a[B]^b}$
• Square brackets denote molar concentrations (molarity) of the respective species at equilibrium
• Equilibrium constants are dimensionless quantities, as the units of concentration cancel out

Law of Mass Action

• The law of mass action states that at equilibrium, the ratio of the product of the concentrations of the products to the product of the concentrations of the reactants, each raised to a power equal to its stoichiometric coefficient, is a constant at a given temperature
• Provides a mathematical relationship between the concentrations of reactants and products at equilibrium
• Applies to any chemical reaction at equilibrium, regardless of the complexity or number of reactants and products involved
• The law of mass action is the basis for deriving the equilibrium constant expression for a given reaction

Reaction Quotient and Equilibrium

• The reaction quotient ($Q$) has the same mathematical form as the equilibrium constant expression but uses instantaneous concentrations instead of equilibrium concentrations
• Comparing $Q$ to $K$ helps determine the direction in which a reaction will proceed to reach equilibrium:
  • If $Q < K$, the reaction will proceed in the forward direction (towards products) to reach equilibrium
  • If $Q > K$, the reaction will proceed in the reverse direction (towards reactants) to reach equilibrium
  • If $Q = K$, the reaction is at equilibrium, and no net change in concentrations occurs
• Monitoring the reaction quotient allows for predicting the direction of a reaction and determining when equilibrium is reached (when $Q$ becomes equal to $K$)

Temperature Dependence of Equilibrium Constants

Effect of Temperature on Equilibrium

• Temperature changes affect the equilibrium constant and the position of equilibrium in a chemical reaction
• Increasing temperature favors the endothermic direction of a reaction, while decreasing temperature favors the exothermic direction
• The magnitude of the equilibrium constant changes with temperature, but the direction of change depends on whether the reaction is exothermic or endothermic
• For exothermic reactions, $K$ decreases with increasing temperature, while for endothermic reactions, $K$ increases with increasing temperature

Van 't Hoff Equation

• The van 't Hoff equation describes the relationship between the equilibrium constant and temperature: $\ln \frac{K_2}{K_1} = -\frac{\Delta H^{\circ}}{R} \left(\frac{1}{T_2} - \frac{1}{T_1}\right)$
• $K_1$ and $K_2$ are the equilibrium constants at temperatures $T_1$ and $T_2$, respectively
• $\Delta H^{\circ}$ is the standard enthalpy change of the reaction
• $R$ is the universal gas constant (8.314 J/mol·K)
• The van 't Hoff equation allows for calculating the equilibrium constant at a different temperature, given the equilibrium constant at a known temperature and the standard enthalpy change of the reaction

Exothermic and Endothermic Reactions

• In exothermic reactions ($\Delta H^{\circ} < 0$), heat is released from the system to the surroundings
  • Examples: combustion reactions, neutralization reactions (acid-base reactions)
  • Increasing temperature shifts the equilibrium towards the reactants side, decreasing the equilibrium constant
• In endothermic reactions ($\Delta H^{\circ} > 0$), heat is absorbed by the system from the surroundings
  • Examples: thermal decomposition of calcium carbonate (limestone), melting of ice
  • Increasing temperature shifts the equilibrium towards the products side, increasing the equilibrium constant
• Understanding the temperature dependence of equilibrium constants is crucial for predicting the effect of temperature changes on the position of equilibrium and the yield of products in chemical reactions

Thermodynamic Basis of Equilibrium Constants

Relationship between Equilibrium Constant and Standard Gibbs Free Energy Change

• The equilibrium constant is related to the standard Gibbs free energy change ($\Delta G^{\circ}$) of a reaction by the following equation: $\Delta G^{\circ} = -RT \ln K$
• $R$ is the universal gas constant (8.314 J/mol·K), and $T$ is the absolute temperature in Kelvin
• The standard Gibbs free energy change represents the driving force for a chemical reaction at standard conditions (1 atm pressure, 298 K, and 1 M concentrations)
• A negative $\Delta G^{\circ}$ indicates a spontaneous reaction (favoring products), while a positive $\Delta G^{\circ}$ indicates a non-spontaneous reaction (favoring reactants)

Calculating Equilibrium Constants from Standard Gibbs Free Energy Change

• The equation relating $\Delta G^{\circ}$ and $K$ can be rearranged to solve for the equilibrium constant: $K = e^{-\Delta G^{\circ}/RT}$
• By knowing the standard Gibbs free energy change of a reaction, the equilibrium constant can be calculated at a given temperature
• Conversely, if the equilibrium constant is known, the standard Gibbs free energy change can be determined using the same equation
• This relationship highlights the thermodynamic basis of equilibrium constants and provides a link between the thermodynamic favorability of a reaction and the concentrations of reactants and products at equilibrium

Factors Influencing Standard Gibbs Free Energy Change

• The standard Gibbs free energy change depends on the standard enthalpy change ($\Delta H^{\circ}$) and the standard entropy change ($\Delta S^{\circ}$) of a reaction: $\Delta G^{\circ} = \Delta H^{\circ} - T\Delta S^{\circ}$
• $\Delta H^{\circ}$ represents the heat absorbed or released by the reaction at constant pressure, while $\Delta S^{\circ}$ represents the change in the system's disorder or randomness
• Reactions with a negative $\Delta H^{\circ}$ (exothermic) and a positive $\Delta S^{\circ}$ (increase in disorder) are more likely to have a negative $\Delta G^{\circ}$ and thus be spontaneous and have a larger equilibrium constant
• Temperature affects the relative contributions of $\Delta H^{\circ}$ and $\Delta S^{\circ}$ to $\Delta G^{\circ}$, which in turn influences the equilibrium constant
• Understanding the thermodynamic basis of equilibrium constants allows for predicting the favorability of reactions and the magnitude of equilibrium constants based on the enthalpy and entropy changes of the system