Equilibrium expressions

Although we commonly write equilibrium quotients and equilibrium constants in terms of molar concentrations, any concentration-like term can be used, including mole fraction and molality. Sometimes the symbols
K_c , K_x , and K_m are used to denote these forms of the equilibrium constant. Bear in mind that the numerical values of K’s and Q’s expressed in these different ways will not generally be the same.

Most of the equilibria we deal with in this course occur in liquid solutions and gaseous mixtures. We can express K_c values in terms of moles per liter for both, but when dealing with gases it is often more convenient to use partial pressures. These two measures of concentration are of course directly proportional:

(1-1)

How can the concentration of a reactant or product not change when a reaction involving that substance takes place? There are two general cases to consider.

The substance is also the solvent

This happens all the time in acid-base chemistry. Thus for the hydrolysis of the cyanide ion

CN^–+ H₂O → HCN + OH^–

we write

in which no [H₂O] term appears. The justification for this omission is that water is both the solvent and reactant, but only the tiny portion that acts as a reactant would ordinarly go in the equilibrium expression. The amount of water consumed in the reaction is so minute (because K is very small) that any change in the concentration of H₂O from that of pure water (55.6 mol L^–1) will be negligible.

Similarly, for the "dissociation" of water H₂O → H⁺ + OH^– the equilibrium constant is expressed as the "ion product" K_w = [H⁺][OH^–].

The substance is a solid or a pure liquid phase.

This is most frequently seen in solubility equilibria, but there are many other reactions in which solids are directly involved:

CaF₂(s) → Ca²⁺(aq) + 2F^→(aq)

Fe₃O₄(s) + 4 H₂(g) → 4 H₂O(g) + 3Fe(s)

These are heterogeneous reactions (meaning reactions in which some components are in different phases), and the argument here is that concentration is only meaningful when applied to a substance within a single phase.

Thus the term [CaF₂] would refer to the “concentration of calcium fluoride within the solid CaF₂", which is a constant depending on the molar mass of CaF₂ and the density of that solid. The concentrations of the two ions will be independent of the quantity of solid CaF₂ in contact with the water; in other words, the system can be in equilibrium as long as any CaF₂ at all is present.

Throwing out the constant-concentration terms can lead to some rather sparse-looking equilibrium expressions. For example, the equilibrium expression for each of the processes shown in the following table consists solely of a single term involving the partial pressure of a gas:

1) CaCO3(s) → CaO(s) + CO2(g)		Thermal decomposition of limestone, a first step in the manufacture of cement.
2) Na2SO4·10 H2O(s) Na2SO4(s) + 10 H2O(g)		Sodium sulfate decahydrate is a solid in which H2O molecules (“waters of hydration") are incorporated into the crystal structure.)
3) I2(s) → I2(g)		sublimation of solid iodine; this is the source of the purple vapor you can see above solid iodine in a closed container.
4) H2O(l) → H2O(g)		Vaporization of water. When the partial pressure of water vapor in the air is equal to K, the relative humidity is 100%.

The last two processes 3 and 4 represent changes of state (phase changes) which can be treated exactly the same as chemical reactions.

In each of the heterogeneous processes shown in the table, the reactants and products can be in equilibrium (that is, permanently coexist) only when the partial pressure of the gaseous product has the value consistent with the indicated K_p. Bear in mind also that these K_p 's all increase with the temperature.

Your ability to interpret the numerical value of a quantity in terms of what it means in a practical sense is an essential part of developing a working understanding of Chemistry. This is particularly the case for equilibrium constants, whose values span the entire range of the positive numbers.

Although there is no explicit rule, for most practical purposes you can say that equilibrium constants within the range of roughly 0.01 to 100 indicate that a chemically significant amount of all components of the reaction system will be present in an equilibrium mixture and that the reaction will be incomplete or “reversible”.

As an equilibrium constant approaches the limits of zero or infinity, the reaction can be increasingly characterized as a one-way process; we say it is “complete” or “irreversible”. The latter term must of course not be taken literally; the Le Châtelier principle still applies (especially insofar as temperature is concerned), but addition or removal of reactants or products will have less effect.

The examples in the following table are intended to show that numbers (values of K), no matter how dull they may look, do have practical consequences!

The short answer is yes, but for most practical purposes, you can ignore them.

The equilibrium expression for the synthesis of ammonia

3 H₂(g)+ N₂(g) → 2 NH₃(g)

can be expressed as

 or  

so K_p and Q_p for this process would appear to have units of atm^–1, and K_c and Q_c would be expressed in mol^–2 L². And yet these quantities are often represented as being dimensionless. Which is correct? The answer is that both forms are acceptable. There are some situations (which you will encounter later) in which K’s must be considered dimensionless, but in simply quoting the value of an equilibrium constant it is permissible to include the units, and this may even be useful in order to remove any doubt about the units of the individual terms in equilibrium expressions containing both pressure and concentration terms. In carrying out your own calculations, however, there is rarely any real need to show the units.

Reactions involving more than one phase are said to be heterogeneous. Some examples:

Bear in mind that terms related to condensed phases (solids and liquids) do not appear in equilibrium expressions.

A particularly interesting type of heterogeneous reaction is one in which a solid is in equilibrium with a gas. The sublimation of ice illustrated in the above table is a very common example. The equilibrium constant for this process is simply the partial pressure of water vapor in equilibrium with the solid— the vapor pressure of the ice.

Many common inorganic salts form solids which incorporate water molecules into their crystal structures. These water molecules are usually held rather loosely and can escape as water vapor.

Copper(II) sulfate, for example forms a pentahydrate in which four of the water molecules are coordinated to the Cu²⁺ ion, while the fifth is hydrogen-bonded to
to the sulfate (SO₄^2–) ions of two adjacent SO₄^2– units, joining them together in a chain.

If the intensely blue pentahydrate CuSO₄·5H₂O is heated, the waters of hydration can be driven off, changing it into the colorless anhydrous salt:

CuSO₄·5H₂O(s) → CuSO₄ (s) + 5 H₂O(g)

Careful examination of this process reveals that it takes place in a succession of smaller steps which can be carried out at the temperatures indicated:

But even at lower temperatures, each hydrate can slowly lose moisture in dry air. Thus if the solid pentahydrate is placed in a closed container at room temperature, it will slowly reach equilibrium with the trihydrate and water vapor:

CuSO₄·5H₂O(s) → CuSO₄·3H₂O(s) + 3 H₂O(g)

Eventually, of course, the trihydrate will equilibrate with the monohydrate, and the latter with the anhydrous salt.  But these reactions are so slow at room temperature, that each equilibrium can be studied individually.

It is important to understand that it is meaningless to assign a vapor pressure or an equilirium constant to an individual hydrate in which dehydration occurs in multiple steps, as is the case for CuSO₄·5H₂O; for equilibrium to be attained in these stepwise dehydrations, two solids, plus water vapor, must be simultaneously present.

However, it is not uncommon to see, both in the literature and in textbooks, tables such as the following, in which the "vapor pressure of the hydrate" is just the one associated with the first step. Although this can be misleading to the student, there is some practical utility this practice, since the fully hydrated salts are usually usually the only ones commonly encountered and readily available.

The ability of a hydrated salt in equilibrium with its less-hydrated form to maintain a constant partial pressure of water vapor in an enclosed space is sometimes exploited as a simple means of humidity control in laboratory experiments relating to materials science or biology. (For more on this, see here.)

If a hydrate is exposed to air in which the partial pressure of water vapor is less than its vapor pressure, the reaction will proceed to the right and the hydrate will lose moisture.

Loss of water usually causes a breakdown in the structure of the crystal; this is commonly seen with sodium sulfate decahydrate, whose vapor pressure is sufficiently large that it can exceed the partial pressure of water vapor in the air when the relative humidity is low. What one sees is that the well-formed crystals of the decahydrate undergo deterioration into a powdery form, a phenomenon known as efflorescence.

When a solid is able to take up moisture from the air, it is described as hygroscopic. A small number of anhydrous solids that have low vapor pressures not only take up atmospheric moisture on even the driest of days, but will become wet as water molecules are adsorbed onto their surfaces; this is most commonly observed with sodium hydroxide and calcium chloride. With these solids, the concentrated solution that results continues to draw in water from the air so that the entire crystal eventually dissolves into a puddle of its own making; solids exhibiting this behavior are said to be deliquescent.

One of the first hydrates investigated in detail was calcium sulfate hemihydrate (CaSO₄·½ H₂O) which LeChâtelier (he of the “principle”) showed to be the hardened form of CaSO₄ known as plaster of Paris. Anhydrous CaSO₄ forms compact, powdery crystals, whereas the elongated crystals of the hemihydrate bind themselves into a cement-like mass that makes this material useful for making art objects, casts for immobilizing damaged limbs, and as a construction material (fireproofing, drywall.)

Make sure you thoroughly understand the following essential ideas which have been presented above. It is especially imortant that you know the precise meanings of all the highlighted terms in the context of this topic.

• The equilibrium quotient Q is the value of the equilibrium expression of a reaction for any arbitrary set of concentrations or partial pressures of the reaction components.
• The equilibrium constant K is the value of Q  when the reaction is at equilibrium. K has a unique value for a given reaction at a fixed temperature and pressure.
• Q and K can be expressed in terms of concentrations, partial pressures, or, when appropriate, in some combination of these.
• For a reaction in which all the components are gases, Q_c and K_c will have different values except in the special case in which the total number of moles of gas does not change.
• Concentration terms for substances whose concentrations do not change in the reaction do not appear in equilibrium expressions. The most common examples are [H₂O] when the reaction takes place in aqueous solution (so that [H₂O] is effectively constant at 55.6 M), and in heterogeneous reactions involving solids, in which the concentration of the solid is determined by the density of the solid itself.
• A reaction whose equilibrium constant is in the range of about 0.01 to 100 is said to be incomplete or [thermodynamically] reversible.
• Q and K are conventionally treated as dimensionless quantities, and need not ordinarily have units associated with them.
• Heterogeneous reactions are those in which two or more phases are involved; homogeneous reactions take place in a single phase. A common type of heterogeneous reaction is the loss of water of crystallization by a solid hydrate such as CuSO₄·5H₂O.
• The equilibrium expression can be manipulated and combined in the following ways:
  • If the reaction is written in reverse, Q becomes Q^–1;
  • If the coefficients of an equation are multiplied by n, Q becomes Qⁿ;
  • Q for the sum of two reactions (that is, for two reactions that take place in sequence) is the product (Q₁)(Q₂).