THE CONCEPT OF CHEMICAL EQUILIBRIUM

Chemical Equilibrium

In a chemical reaction, chemical equilibrium is the state in which both reactants and products are present in concentrations which have no further tendency to change with time.

The chemical equilibrium is achieved when the rate of forward reaction is same as the reverse reaction. Since the rates are equal, there are no net changes in the concentrations of the reactant(s) and product(s). This state is known as dynamic equilibrium.

The entire process can be graphically represented by this diagram.

Law of Chemical Equilibrium & Equilibrium Constant K_c

• Chemical Equilibrium can easily be understood by understanding Chemical Kinetics. It is the study of the rate of the reactions under various conditions.
• Another concept that forms the basis is the Law of mass action.
• Law of mass action states that the Rate of a chemical reaction is directly proportional to the product of the concentrations of the reactants raised to their respective stoichiometric coefficients.
• So, Given a reaction:
  aA(g) + bB(g) ⇌ cC(g) + dD(g)
  Using the Law of mass action,
  Forward reaction rate   = k₊ [A]^a[B]^b
  Backward reaction rate = k_– [C]^c[D]^d 

where, [A], [B], [C] and [D] being the active masses and k₊ and k₋ are rate constants.

• At equilibrium forward and backward rates are equal and the ratio of the rate constants is a constant and is known as an equilibrium constant.
• In a reaction mixture at equilibrium, the concentrations of the reactants and products are related by an equilibrium constant. For the reaction
  aA(g) + bB(g) ⇌ cC(g) + dD(g)
  
  Forward reaction rate   = Backward reaction rate
  k₊ [A]^a[B]^b = k_– [C]^c[D]^d
  We have,
  K_c = k₊ / k_–
  K_c = [ C ]^c·[ D ]^d / [ A ]^a·[ B ]^b

• Equilibrium Constant & Gibbs Free energy:
  • The Equilibrium constant is related to the standard Gibbs free energy change for the reaction.
  • The relation is given by the equation:
  • In the above equation,
    • R is the universal gas constant.
    • T is the temperature.
    • K_eq is the equilibrium constant
• Temperature Dependency: Equilibrium constant K_c depends on thetemperature of the reaction and the relation is given by Van’t Hoff equation.
  • In the above equation,
    • K₁ is the equilibrium constant at absolute temperature T₁,
    • K₂ is the equilibrium constant at absolute temperature T_2,
    • R is the ideal gas constant,
    • ΔH reaction enthalpy assumed to be constant over the temperature range.
  • This equation can be used to estimate a new equilibrium constant at a new absolute temperature assuming a constant standard enthalpy change over the temperature range.
  • Now, Let’s understand the change in the equilibrium constant for two different kinds of reactions: Endothermic & Exothermic. Using the definition of Gibbs free energy & Gibbs free energy Isotherm equation, we have:
  • In the above equation,
    • K_eq is the equilibrium constant at temperature T.
    • ΔH^⊖ and ΔS^⊖ are constants and are enthalpy and entropy of the system respectively.
  • This graph of this equation is called the Van ‘t Hoff plot. The plot is used to estimate the enthalpy and entropy of a chemical reaction.
  • From the plot between “ln K_eq” & “1/T“, we have  −ΔH/R as the slope, and ΔS/R as the intercept of the linear fit.
• Exothermic reactions
  • For an exothermic reaction, heat is released, making the net enthalpy change negative.
  • ΔH < 0, so slope (−ΔH/R) is positive.
  •  Van ‘t Hoff plot has a positive slope. 

• Endothermic reactions
  • For an endothermic reaction, heat is absorbed, making the net enthalpy change positive.
  • ΔH > 0, so slope (−ΔH/R) is negative.
  •  Van ‘t Hoff plot has a negative slope.

Types of Chemical Equilibria

There are two types of chemical equilibria:

• Homogeneous equilibria
• Heterogeneous equilibria

Homogeneous Equilibria

• The equilibrium reactions in which all the reactants and the products are in the same phase are known as homogeneous equilibrium reactions. These are divided into two categories:
  • The number of product molecules is equal to the number of reactant molecules. For example:
    • N₂ (g) + O₂ (g) ⇌ 2NO (g)
    • H₂ (g) I₂ ⇌ 2HI (g)
  • The number of product molecules is not equal to the number of reactant molecules.
    • COCl₂ (g) ⇌ CO (g) + Cl₂ (g)
    • 2SO₂ (g) + O₂ (g) ⇌ 2SO₃ (g)
• In gaseous phase,

• In solution phase,

• The equilibrium constant for the homogeneous reaction in gaseous systems:
• Ideal gas equation is given by, pV = nRT
  If concentration C is in mol L⁻¹ or mol dm⁻³ and p is in bar, then we can write p = c RTOr, p = [gas] RT …..(i)Where R = 0.0831 bar L mol⁻¹ K⁻¹For a general reaction,

• In the above equation,
  • Δn = (number of moles of gaseous products)− (number of moles of gaseous reactants) in the balanced chemical equation
  • While calculating K_p, pressure should be expressed in bar.
  • 1 bar = 10⁵ Pa = 10⁵ Nm⁻²

Heterogeneous Equilibria

• The equilibrium reactions in which the reactants and the products are present in different phases are known as Heterogeneous equilibrium reactions. For Example:
• The dissociation of solid calcium carbonate to give solid calcium oxide and gaseous carbon dioxide:
  • Here as CaO and CaCO₃ are pure solids, [CaO] and [CaCO₃] are constants.
    So,

• The concentrations of pure solids & pure liquids assumed to be constant and these do not appear in equilibrium concentration expression. By convention [solid] = 1 and [liquid] = 1

Applications of Equilibrium Constant

• Important features of the equilibrium constant:
  • Applicable only in equilibrium state
  • Independent of initial concentration of reactants and products
  • Depends on temperature; has a unique value for a reaction at a given temperature
  • The equilibrium constant for the forward direction is the inverse of the Equilibrium constant for the reverse direction.
  • The equilibrium constant for a reaction is related to the equilibrium constant of the corresponding reaction whose equation is obtained by multiplying or dividing the equation for the original reaction by a small integer.
• Predicting the extent of a reaction:
  • If K_C> 10³, then the products predominate over the reactants.
  • If K_C< 10⁻³, then the reactants predominate over the products.
  • If 10⁻³< K_C< 10³, then appreciable concentrations of both reactants and products are present.
• Predicting the direction of a reaction: Reaction quotient, Q (Q_C with molar concentration and Q_p with partial pressure)
  For a general reaction,
  The concentrations are not necessarily equilibrium values.
  • If Q_C> K_C, then the reaction will proceed in the reverse direction.
  • If Q_C< K_C, then the reaction will proceed in the forward direction.
  • If Q_C= K_C, then the reaction is at equilibrium.

Relation between K, Q, and G (Gibbs Energy)

• There is a very important relationship between Standard Gibbs free energy (G^o) and the equilibrium constant (K_eq).
• We know that the reaction quotient (Q) measures the relative amounts of products and reactants present during a reaction at a particular point in time.
• The free energy change at any point in of the chemical reaction is given by:

• At equilibrium,  ΔG = 0 (zero) and Q = K_eq. This condition gives the relation,
•  The equilibrium constant (K) is another way we can tell if a reaction is spontaneous.
• Here,  ΔG^o is equal to negative R, the gas constant, multiplied by the temperature in Kelvin, multiplied by the natural log of the equilibrium constant.
• This relationship allows us to directly relate the standard free energy change to the equilibrium constant. It also tells us about the extent of the reaction.
• If, ΔG^o < 0, ln K_eq = positive, Hence K_eq > 1 implying that it is a spontaneous reaction.
• If, ΔG^o < 0, ln K_eq = negative, Hence K_eq < 1 implying that it is a non-spontaneous reaction.

Factors Affecting Chemical Equilibrium

• The factors that can influence chemical equilibrium are:
  • Change in concentration
  • Change in pressure (or volume)
  • Change in temperature.
• The effect of the change in one of these factors on the extent of reaction can be qualitatively explained by using Le Chatelier’s principle.
• Le Chatelier’s principle
  According to this principle, if a system is in equilibrium and it is subjected to any change in any of the factors that determine the equilibrium conditions of the system, then it will shift the equilibrium in such a way to reduce or to counteract the effect of the change.

To help you understand the effects of various changes better, here is the summary:

Chemical equilibrium is a dynamic process that consists of a forward reaction, in which reactants are converted to products, and a reverse reaction, in which products are converted to reactants. At equilibrium, the forward and reverse reactions proceed at equal rates. Consider, for example, a simple system that contains only one reactant and one product, the reversible dissociation of dinitrogen tetroxide (N₂O₄) to nitrogen dioxide (NO₂). You may recall from Chapter 14 “Chemical Kinetics” that NO₂ is responsible for the brown color we associate with smog. When a sealed tube containing solid N₂O₄ (mp = −9.3°C; bp = 21.2°C) is heated from −78.4°C to 25°C, the red-brown color of NO₂appears (Figure 15.1 “The “). The reaction can be followed visually because the product (NO₂) is colored, whereas the reactant (N₂O₄) is colorless:

The double arrow indicates that both the forward and reverse reactions are occurring simultaneously; it is read “is in equilibrium with.”

Figure 15.2 “The Composition of N” shows how the composition of this system would vary as a function of time at a constant temperature. If the initial concentration of NO₂ were zero, then it increases as the concentration of N₂O₄ decreases. Eventually the composition of the system stops changing with time, and chemical equilibrium is achieved. Conversely, if we start with a sample that contains no N₂O₄ but an initial NO₂ concentration twice the initial concentration of N₂O₄ in part (a) in Figure 15.2 “The Composition of N”, in accordance with the stoichiometry of the reaction, we reach exactly the same equilibrium composition, as shown in part (b) in Figure 15.2 “The Composition of N”. Thus equilibrium can be approached from either direction in a chemical reaction.

Figure 15.3 “The Forward and Reverse Reaction Rates as a Function of Time for the ” shows the forward and reverse reaction rates for a sample that initially contains pure NO₂. Because the initial concentration of N₂O₄ is zero, the forward reaction rate (dissociation of N₂O₄) is initially zero as well. In contrast, the reverse reaction rate (dimerization of NO₂) is initially very high (2.0 × 10⁶ M/s), but it decreases rapidly as the concentration of NO₂ decreases. (Recall from Chapter 14 “Chemical Kinetics” that the reaction rate of the dimerization reaction is expected to decrease rapidly because the reaction is second order in NO₂: rate = k_r[NO₂]², where k_r is the rate constant for the reverse reaction shown in Equation 15.1.) As the concentration of N₂O₄ increases, the rate of dissociation of N₂O₄ increases—but more slowly than the dimerization of NO₂—because the reaction is only first order in N₂O₄ (rate = k_f[N₂O₄], where k_f is the rate constant for the forward reaction in Equation 15.1). Eventually, the forward and reverse reaction rates become identical, k_F = k_r, and the system has reached chemical equilibrium. If the forward and reverse reactions occur at different rates, then the system is not at equilibrium.