GENERAL CHEMISTRY TOPICS
Quantitative treatment of equilibria
The reaction quotient expression, Q. A consistent value of Q is achieved at equilibrium - Q = Keq. Rules for writing the reaction quotient expression from the equation for a given chemical reaction. Heterogeneous systems. Two forms of the equilibrium constant: KP and Kc.
Here is a brief review of the essentials of a quantitative treatment of chemical systems, including the specification of conditions at equilibrium. An expression for what is called the reaction quotient (Q) can be written for a given balanced chemical equation that represents a chemical reaction or other process (e.g. phase changes). The rules for writing the expression for Q are presented, along with examples. For any given system of reactants and products that can interconvert according to a corresponding chemical equation, the expression for Q is evaluated by substitution of the molar concentrations for each reactant and product species appearing in the expression. The equilibrium constant Keq is the specific, characteristic value that the expression for Q evaluates as when the molar concentrations of reactants and products at equilibrium are substituted.
For the generalized chemical reaction,
aA + bB + cC + ···· ⇌ xX + yY + zZ + ····
the reaction quotient expression, Q is given as
Q = [A]−a[B]−b[C]−c···· [X]x [Y]y [Z]z····
It is important that we be specific about the phases of all reactants and products in the reaction equation, for it affects the how the quantities represented by the terms [A], …, [Z] appear in this expression. For now, let us adopt the simplifying assumption that A, …, Z are all solutes in a dilute aqueous solution unless otherwise specified. As a practical consequence, this means we treat the terms [A],…, [Z] as numerically equivalent to molar concentrations of the species A(aq),…, Z(aq).
It is also crucial to distinguish clearly between Q and the equilibrium constant Keq, which we define below. For Q is an arbitrary quantity in the sense that the concentrations at any one time may be consistent with an equilibrium or a nonequilibrium state. Thus, in following the changes of concentrations with time associated with a chemical reaction occurs, the value of Q changes - either becoming larger (as products increase) or smaller (the reaction is proceeding in a reverse direction as written). But when the reacting system reaches a state of equilibrium, the concentrations of the reactants and products no longer change with time, so the value of the variable expression Q is constant with time. Remarkably, any nonequilibrium starting conditions at a fixed temperature lead to the same value for Q, a result which leads to give this particular value for the reaction quotient a special name. The equilibrium constant (Keq) is defined as
Keq = [A]eq−a [B]eq−b [C]eq−c … [X]eqx [Y]eqy [Z]eqz…
where [A]eq, …, [Z]eq are a set of values of [A],…, [Z] occurring when a state of equilibrium pertains. The magnitude of the equilibrium constant (Keq) determines the extent to which a reaction favors products (Keq large, >>1) or reactants (Keq small, <<1) at its equilibrium.
Bear in mind that Keq is dependent on temperature, so for a given value of it we either implicitly or explicitly are specifying a temperature. When comparing Keq and Q, the terms refer in principle to a chemical reaction or other process occurring at a fixed temperature.
Heterogeneous systems
In general, for any chemical reaction or physical process, we must be aware of the states of all reactants and products that comprise the system. This is necessary not only to specify the nature of the equilibrium that the system will spontaneously approach (if it is not already there), but also to write the correct expressions for Q and Keq.
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)
and we would probably write
Q = [H2O(g)] / [H2O(l)] Keq = [H2O(g)]eq / [H2O(l)]eq
but this can be simplified, and writing such an expression in this case is often considered incorrect. To understand why, consider that in our system, two phases are present under various nonequilibrium conditions as well as when the system is at equilibrium. Note however, that whether or not the system is at equilibrium, as long as some liquid water is present, the term [H2O(l)] is unchanged. The concentration of liquid is set by the temperature and the liquid density at that temperature. Only the [H2O(g)] term changes in the process of approach to equilibrium from nonequilibrium starting conditions. All that is required is that some liquid water be present at equilibrium. Therefore, in writing the Q and Keq expressions, by convention we drop the constant concentration of the liquid. In this case, we will also prefer to use partial pressure of water vapor; thus we replace the above incorrect forms with the following correct forms:
QP = PH2O Keq = PH2O, eq
The equilibrium constant is simply the equilibrium vapor pressure.
The general rule for various types of heterogeneous systems is that terms drop out of Q and Keq expressions when they correspond to system components whose concentrations do not change. This applies to solids as well as liquids, and to solvents (e.g. water) in sufficiently dilute solutions when the solvent happens to be a participant in the reaction.
Example: (from Ref. 2, pp.29-30) Consider the reaction
BrO3– + 2Cr3+ + 4H2O ⇌ Br– + Cr2O72– + 8H+
(i) Write the expression corresponding to Q for this reaction.
The phases are not specified, so we assume all species are in dilute aqueous solution. Note that the solvent, water, is also a participant in the reaction. Following the general definition, but also recognizing the concentration of water is essentially constant, we write
Q = [BrO3–]−1[Cr3+]−2[Br–][Cr2O72–][H+]8
(ii) Given a value of Keq of 1 × 1011 at 25 °C, and concentrations at this temperature of the solutes as follows,
[BrO3–] = 0.043 M; [Cr3+] = 0.0030 M; [Br–] = 1.0 M; [Cr2O72–] = 0.10 M; [H+] = 5.0 M,
is the system at equilibrium?
(iii) Predict the effect of changing the above conditions by doubling the dichromate (Cr2O72–) concentration from 0.10 to 0.20 M. Compare predictions using Le Châtelier's Principle and a Q vs. Keq analysis.
Two forms of the equilibrium constant: KP and Kc
As we progress in our study of chemistry and chemical equilibria, we will see that equilibrium constants appear in many different guises or "flavors"; however in every case they represent the same fundamental concept. Initially, we will define and distinguish between a form of the reaction quotient expression useful for gas phase reactions denoted QP, and one useful for reactions occurring in solution called Qc. This leads to two corresponding definitions of the equilibrium constant, KP and Kc. The form of the reaction quotient expression QP is expressed in partial pressures of the reactants and products in a gas phase reaction. Thus, for a generalized gas-phase reaction,
aA(g) + bB(g) ⇌ cC(g) + dD(g)
the reaction quotient expression QP is given as
QP = PA−a PB−b PCc PDd
where PA, PB, etc., are the partial pressures of their subscripted components of the gas phase mixture*, and the corresponding version of the equilibrium constant becomes
KP = (PA, eq)−a(PB, eq)−b(PC, eq)c(PD, eq)d
In case of reactions involving solute species in a solution, the formulation above is not appropriate, so we define for the following generalized reaction in solution (taking the example of an aqueous solution)
aA(aq) + bB(aq) ⇌ cC(aq) + dD(aq)
the reaction quotient expression Qc as
Qc = [A]−a[B]−b[C]c[D]d
where [A], [B],... are the molar concentrations of the solute species*, which leads to the following definition of the equilibrium constant Kc as
Kc = ([A]eq)−a ([B]eq)−b ([C]eq)c ([D]eq)d
* Strictly speaking, each of these quantities is divided by a reference pressure or concentration, respectively. The reference value for pressure is taken as exactly 1 atm (or 1 bar), and for concentration it is exactly 1 M. Thus, reaction quotients and equilibrium constants are technically unitless quantities, and the partial pressures or concentrations we plug into these expressions are only their numerical values.
Examples
A more quantitative approach to equilibria uses weak acids and weak bases as important examples.