Lecture 21. Case studies of equilibria

Tuesday 9 April 2024

Chemical reactions and equilibrium. The equilibrium constant, K_eq. Rules for writing expressions for equilibrium constants. Calculating the equilibrium constant from measured equilibrium concentrations. The reaction quotient (Q) and its comparison with K_eq.

Reading: Tro NJ. Chemistry: Structure and Properties (3rd ed.) - Ch.11, Ch.13, Ch.15 (pp.639-656t).

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Summary

Our aim is to enumerate several general types of equilibria, and develop a qualitative and quantitative treatment of specific examples of each type. We'll see that although the specific examples vary in detail, there are some clear common principles that govern equilibria of all types.

Why do some reactions proceed to completion, attaining nearly 100% conversion of reactants to products, while others fall far short of such product-favored final compositions? We say that a reacting system that reaches its final composition and undergoes no further observable changes in reactant and product concentrations with time has reached a state of chemical equilibrium. We can restate the observations about the varying degrees of completion among chemical reactions by saying that while some reactions show a highly product-favored state of equilibrium, others show more intermediate equilibria in which neither products or reactants are highly favored. Still others display equilibria in which very little net conversion to products has occurred. Such reactions can justifiably be called reactant-favored. A deeper understanding of this state of affairs requires that we recognize the dynamic nature of chemical equilibrium, and the relation between reaction rates and equilibria. Any chemical reaction is in principle reversible, and chemical equilibrium is the result of the balance between the rates of the forward reaction (reactants → products) and the reverse reaction (reactants ← products). When these rates are equal, no change in the chemical composition of the reacting system is observed on the macroscopic level, yet on the nanoscale the forward and reverse reactions continue to interconvert reactants and products. This dynamic nature of chemical equilibrium is shared with physical processes, such as vapor-liquid equilibrium attained by a liquid in an enclosed container (with at least some volume in the container available for the gas phase to occupy).

Case studies in dynamic equilibrium

1. A liquid-vapor equilibrium. Consider the system of a closed container with a small amount of water at room temperature. If the partial pressure of water vapor pressure is initially less than its equilibrium vapor pressure (0.0231 atm at 20 °C), then H₂O(l) will evaporate until that partial pressure is reached. If the partial pressure of water vapor pressure is initially more than its equilibrium vapor pressure, then H₂O(g) will condense until that partial pressure is reached. Equilibrium in this physical process - which is a dynamic equilibrium in which the rate at which water molecules continue to evaporate from the liquid phase is equal to the rate at which vapor phase molecules "crash" back into the surface of the liquid - will be attained spontaneously, provided the total amount of water is sufficient to reach equilibrium vapor pressure given the volume of the system and still leave some small amount of water in the liquid phase. This is an example of a heterogeneous system, where there are two different phases present. In this case, the equation for this equilibrium-attaining process is

H₂O(l) ⇌ H₂O(g)

The "equals" sign used here should be interpreted as explicitly symbolizing a condition of dynamic equilibrium, in which concentrations of the species on both sides remains constant with time due to the equal forward and reverse rates. Note that the "equals" sign here is NOT meant to suggest any equivalence in the amounts of the left and right that pertain to the condition of equilibrium. We are already well aware that chemical reactions show a range of relative compositions reached at the conclusion of any spontaneous change in reactant and product concentrations.

2. Solubility equilibrium. Solutions are homogeneous mixtures of pure substances, and typically we are concerned with liquid-phase solutions in which a predominant liquid phase serves as the solvent, and smaller amounts of other substances serve as solutes; that is to say they are dissolved in the solvent phase. Of course, water is our primary example of a solvent, and solutions with solvent water are termed aqueous solutions. For example, sugar and table salt both dissolve readily in water, forming homogeneous aqueous phase solutions. However, the amount of solute that will mix into a solvent has a limit. A saturated solution results when the concentration of the solute reaches a maximum value (for a fixed temperature), and any further addition of solute results in accumulation of the undissolved substance. At this point, the rate at which undissolved solute enters into solution is balanced by the rate at which dissolved species rejoin the solid phase. The equations representing the formation of aqueous solutions of sugar and salt are

C₁₂H₂₂O₁₁(s)  ⇌  C₁₂H₂₂O₁₁(aq)

NaCl(s)  ⇌  Na⁺(aq)  +  Cl⁻(aq)

The equilibria that exist in saturated solutions of these substances is, relatively speaking, quite favorable to the products. That means that the concentrations attained by the dissolved species on the right-hand side of the above equations under saturation (equilibrium) conditions are relatively high. On the other hand, the following equilibria

AgCl(s)  ⇌  Ag⁺(aq)  +  Cl⁻(aq)        

CO₂(g)  ⇌  CO₂(aq)

The substances on the "reactant" side of these equations would be considered "insoluble" or sparingly soluble as the concentrations attained by the dissolved species on the right-hand side of the above equations under saturation (equilibrium) conditions are relatively low. One question we'll want to address, using these case studies, asks why the equilibrium conditions for some reactions or physical processes are very product favored, others intermediate - not strongly favoring reactants or products, and still others markedly favor reactants at equilibrium.

3. Chemical equilibria: Gas phase reactions.

Chemical reactions that occur among species in a homogeneous gas phase mixture, such as

H₂(g)  +  I₂(g)  ⇌  2 HI(g)

N₂(g)  +  3 H₂(g)  ⇌  2 NH₃(g)

are good case studies since we can treat them as taking place and reaching equilibrium within closed systems containing a mixture of ideal gases. We then have a simple equation of state - namely, the ideal gas law - that we can apply to such systems. This is generally best done with systems at high temperatures and low total pressure. The partial pressures of each reactant and product will vary while the reaction proceeds; upon establishment of an equilibrium state, the partial pressures of all components remain constant with time. The examples of gas phase reactions we consider here will provide good opportunities to look at the effects changes in state variables have on the composition of the equilibrium state.

4. Chemical equilibria in solution.

Reactions between ions in solution

Fe³⁺(aq)  +  SCN⁻(aq)  ⇌  FeSCN²⁺(aq)

This reaction is observable as a color change of a solution containing iron (III) cations (Fe³⁺), thiocyanate anions (SCN⁻), and ferricyanate anions (FeSCN²⁺). Given some arbitrary starting amounts of reactants - or reactants and products - the initial color changes until equilibrium is established, as indicated by the lack of any further color change in the solution. Investigation of this reaction is discussed in Reference 2, pp.589-591.

Weak acid dissociation. A particularly important example of a chemical equilibrium in solution is the result of the dissociation of a weak acid (such as acetic acid) in water:

CH₃COOH(aq)  ⇌  H⁺(aq)  +  CH₃COO⁻(aq)

You may see the formula for acetic acid written in several ways: C₂H₄O₂, HC₂H₃O₂, CH₃CO₂H, CH₃COOH. The latter three formulae signify that one of the hydrogens is different. Three of the hydrogens are covalently bonded to carbon, forming what is called a methyl group. The fourth hydrogen is covalently bonded to oxygen, and this is the acidic hydrogen, meaning the hydrogen has some tendency to be transferred as an ion to a group of one or more water molecules when acetic acid is in solution. The weak acid dissociation in water is an example of a Brønsted-Lowry acid-base reaction, with water acting as a base. This is explicitly represented in the following equivalent form of the acid dissociation equation

CH₃COOH(aq)  +  H₂O(l)  ⇌  H₃O⁺(aq)  +  CH₃COO⁻(aq)

which serves as a reminder that H⁺(aq) is actually an abbreviation for H₃O⁺(aq).

General principles of equilibria

In each case, the equilibrium state is characterized by

• a minimum value for the potential energy of the system
• an unchanged or fixed composition of the system at the macroscale
• an equality of rates of opposing processes at the molecular scale

Further exploration of equilibria

Where do we go from here? We will endeavor to answer some general types of questions about equilibria :

• When equilibrium has been attained, do reactants or products predominate?
• What factors intrinsically favor a particular direction of a reaction, thus determining its equilibrium position?
• Given initial concentrations of reactants and products, in which direction will the reaction go?
• Given initial concentrations, what concentrations of reactants and products are present at equilibrium?

Complete answers to such questions will require a more quantitative description of equilibrium as well as a further development of applicable qualitative reasoning.

Equilibrium and chemical potential energy

In macroscopic mechanical systems, there is a characteristic feature noted for spontaneous processes, those occurring without any outside input of energy. Such processes always lower the potential energy of the system. Water always flows downhill, and a stretched spring always relaxes when released. Does a similar principle govern systems in which processes such as phase transitions and chemical reactions are occurring? At the nanoscale, the dynamic equilibrium between reactants and products is a result of equal rates for the forward and reverse reactions. In going from reactants to products, both the change in energy (enthalpy) and the change in entropy combine to determine the position of equilibrium, which is also temperature-dependent. For constant pressure processes, what we can refer to as "chemical" potential energy is a combination of enthalpy and entropy. We will not fully address the quantitative nature of this relationship, although we have some familiarity with the meaning of entropy from previous discussions, and Tro brings up the matter in his treatment of solutions in Ch.14 (p.511).

The liquid-vapor equilibrium introduced earlier provides a good illustration of the role of enthalpy and entropy in establishing equilibrium, and is in particular a case where the two contributions oppose one another. The temperature dependence of equilibrium is striking in such cases. The evaporation of water is an energy-requiring (endothermic) process, as reflected by the positive value for ΔH. The enthalpy contribution thus favors the lower energy liquid state. On the other hand, the kinetic energy of motion of water molecules is more broadly distributed when they are in the gas phase, so that entropy increases upon evaporation. An increase in entropy is favorable, so the contribution of entropy in this case favors the gas phase. Furthermore, the entropy contribution becomes more important as temperature increases. Thus, below the boiling point of water, equilibrium is favorable to liquid, while above its boiling point equilibrium becomes favorable to the gas phase.