Equilibrium constant (K) Knowledge-Based Study

Study notes

1. Dynamic Chemical Equilibrium

   In a reversible chemical reaction occurring in a closed system at constant temperature, reactants convert to products while products simultaneously react to reform reactants. When the rate of the forward reaction equals the rate of the reverse reaction, the system reaches a state of dynamic equilibrium. At this point, macroscopic properties such as concentration, color, and pressure remain constant, although microscopic chemical processes continue unabated.

   The Law of Mass Action

   For a general reversible reaction at equilibrium:

   $$a\text{A} + b\text{B} \rightleftharpoons c\text{C} + d\text{D}$$

   the relationship between the concentrations of reactants and products is defined by the equilibrium constant expression:

   $$K_c = \frac{[\text{C}]^c [\text{D}]^d}{[\text{A}]^a [\text{B}]^b}$$

   In this expression:

   • Square brackets represent molar concentrations (mol/L) at equilibrium.
   • The exponents $a, b, c,$ and $d$ correspond to the stoichiometric coefficients from the balanced chemical equation.
   • $K_c$ is the equilibrium constant expressed in terms of concentration, which remains constant at a specified temperature.

   Homogeneous and Heterogeneous Equilibria

   Equilibrium systems are classified based on the physical states of the participating substances:

   • Homogeneous equilibrium: All reactants and products are in the same phase (e.g., all gases or all aqueous species).
   • Heterogeneous equilibrium: Reactants and products are present in more than one phase.

   A crucial rule in formulating equilibrium constant expressions is that pure solids (s) and pure liquids (l) are omitted because their molar concentrations (densities divided by molar mass) remain essentially constant throughout the reaction.

   For example, in the thermal decomposition of calcium carbonate:

   $$\text{CaCO}_3(s) \rightleftharpoons \text{CaO}(s) + \text{CO}_2(g)$$

   The equilibrium expression simplifies to:

   $$K_c = [\text{CO}_2]$$

   Equilibrium Constant for Gaseous Systems ($K_p$)

   For reactions involving gases, the equilibrium constant can be expressed in terms of partial pressures (usually in atmospheres or bar):

   $$K_p = \frac{(P_\text{C})^c (P_\text{D})^d}{(P_\text{A})^a (P_\text{B})^b}$$

   The relationship between $K_p$ and $K_c$ is derived from the ideal gas law ($P = \frac{n}{V}RT = CRT$):

   $$K_p = K_c (RT)^{\Delta n}$$

   where:

   • $R$ is the universal gas constant (0.0821 L·atm/(mol·K)).
   • $T$ is absolute temperature in Kelvin.
   • $\Delta n = (c + d) - (a + b)$, the change in the number of moles of gas (moles of gaseous products minus moles of gaseous reactants).

   Interpreting the Magnitude of $K$

   The numerical value of $K$ provides vital information regarding the extent of a chemical reaction at equilibrium:

   • $K \gg 1$ (e.g., $K > 10^3$): Products predominate over reactants at equilibrium; the reaction proceeds almost to completion.
   • $K \ll 1$ (e.g., $K < 10^{-3}$): Reactants predominate over products at equilibrium; very little product is formed.
   • $K \approx 1$: Significant and comparable concentrations of both reactants and products are present at equilibrium.

   Predicting Reaction Direction: The Reaction Quotient ($Q$)

   When a reaction is not necessarily at equilibrium, the reaction quotient ($Q$) is calculated using the same mathematical formula as $K$, but using non-equilibrium concentrations or pressures:

   • $Q = K$: The system is already at dynamic equilibrium. No net change occurs.
   • $Q < K$: The concentration of products is lower than at equilibrium. The forward reaction proceeds faster than the reverse reaction, shifting the system to the right.
   • $Q > K$: The concentration of products is higher than at equilibrium. The reverse reaction predominates, shifting the system to the left.

   Mathematical Operations on Chemical Equations

   Altering the stoichiometry of a chemical equation changes the corresponding equilibrium constant in predictable ways:

   • Reversing the reaction: If a reaction is reversed, its new equilibrium constant is the reciprocal of the original: $$K' = \frac{1}{K}$$
   • Multiplying by a factor $n$: If the coefficients of a reaction are multiplied by a factor $n$, the new equilibrium constant is raised to the power of $n$: $$K' = K^n$$
   • Adding reactions: When two or more chemical equations are combined to yield a net equation, the equilibrium constant of the overall reaction is the product of the individual constants: $$K_\text{net} = K_1 \times K_2$$

   Factors Affecting the Value of $K$

   It is essential to distinguish between factors that shift the position of equilibrium and factors that alter the equilibrium constant itself:

   • Temperature: Temperature is the only factor that changes the numerical value of $K$. For endothermic reactions ($\Delta H > 0$), increasing temperature increases $K$. For exothermic reactions ($\Delta H < 0$), increasing temperature decreases $K$.
   • Concentration, Pressure, and Volume: Changes in concentration or pressure temporarily alter $Q$, causing the system to shift until $Q = K$ again, but the value of $K$ remains constant.
   • Catalysts: Catalysts increase the rates of both forward and reverse reactions equally. They shorten the time required to reach equilibrium, but they do not alter the position of equilibrium or the value of $K$.