A
At equilibrium, the concentrations of A and B are constant (though not necessarily equal), and the reactions A
Equilibrium can be thought of as a balance between the two reactions (forward and reverse). Better still, equilibrium should be understood on the basis of entropy, which is the measure of the distribution of energy throughout a system or between a system and its environment. For a reversible reaction at a given temperature, the reaction will reach equilibrium when the system’s entropy—or energy distribution—is at a maximum and the Gibbs free energy of the system is at a minimum.
LAW OF MASS ACTION
For a generic reversible reaction aA + bB
The law of mass action is actually related to the expressions for the rates of the forward and reverse reactions. Consider the following one-step reversible reaction:
2A
Because the reaction occurs in one step, the rates of the forward and reverse reactions are given by
ratef = kf[A]2, and rater = kr[B][C]
When ratef = rater, the system is in equilibrium. Because the rates are equal, we can set the rate expressions for the forward and reverse reactions equal to each other:
Because kf and kr are both constants, we can define a new constant Kc, where Kc is called the equilibrium constant and the subscript c indicates that it is in terms of concentration. (When dealing with gases, the equilibrium constant is referred to a Kp, and the subscript p indicates that it is in terms of pressure.) For dilute solutions, Kc and Keq are used interchangeably. The new equation can thus be written
Key Concept
For most purposes, you will not need to distinguish between different K values. For dilute solutions, Keq
While the forward and the reverse reaction rates are equal at equilibrium, the concentrations of the reactants and products are not usually equal. This means that the forward and reverse reaction rate constants, kf and kr, are not usually equal. The ratio of kf to kr is Kc (Keq).
When a reaction occurs by more than one step, the equilibrium constant for the overall reaction is found by multiplying together the equilibrium constants for each step of the reaction. When you do this, the equilibrium constant for the overall reaction is equal to the concentrations of the products divided by the concentrations of the reactants in the overall reaction, each concentration term raised to the stoichiometric coefficient for the respective species. The forward and reverse rate constants for the nth step are designated kn and k-n, respectively. For example, if the reaction
aA + bB
occurs in three steps, then
Example: What is the expression for the equilibrium constant for the following reaction?
3 H2 (g) + N2 (g)
Solution: Kc =
MCAT Expertise
Remember our earlier warning about confusion between reaction coefficients and rate laws? Well, here the coefficients are equal to the exponents in the equilibrium expression. On Test Day, if the reaction is balanced, then the equilibrium expression should just about write itself on your scratch paper.
REACTION QUOTIENT
The law of mass action defines the position of equilibrium by stating that the ratio of the product of the concentrations of the products, each raised to their respective stoichiometric coefficients, to the product of the concentrations of the reactants, each raised to their respective stoichiometric coefficients, is constant. However, equilibrium is a state that is achieved only through time. Depending on the actual rates of the forward and reverse reactions, equilibrium might be achieved in minutes or years. What can we use as a kind of “timer” that tells us how far along in the process toward equilibrium the reaction has reached? We can take another measurement of concentrations called the reaction quotient, Qc. At any point in time of a reaction, we can measure the concentrations of all the reactants and products and calculate the reaction quotient according to the following equation:
You should be struck by how similar this looks to the equation for Keq. It’s the same form, but the information it provides is quite different. While the concentrations used for the law of mass action are equilibrium (constant) concentrations, when calculating a value of Qc for a reaction, the concentrations of the reactants and products may not be constant. In fact, if Qc changes over time because the concentrations of reaction species are changing, the reaction by definition is not at the equilibrium state. Thus, the utility of Qc is not the value itself but rather the comparison that can be made of the calculated Qc at any given moment in time of the reaction to the known Keq for the reaction at a given temperature. For any reaction, if
• Qc < Keq, then the reaction has not yet reached equilibrium.
• Qc > Keq, then the reaction has exceeded equilibrium.
• Qc = Keq, then the reaction is in dynamic equilibrium.
Any reaction that has not yet reached the equilibrium state, as indicated by Qc < Keq , will continue spontaneously in the forward direction (that is, consuming reactants to form products) until the equilibrium ratio of reactants and products is reached. Any reaction in the equilibrium state will continue to react in the forward and reverse direction, but the reaction rates for the forward and reverse reactions will be equal and the concentrations of the reactants and products will be constant, such that Qc = Keq. Once a reaction is at equilibrium, any further “movement” either in the forward direction (resulting in an increase in products) or in the reverse direction (resulting in the re-formation of reactants) will be nonspontaneous. In Chapter 6, Thermochemistry, we’ll review methods of introducing changes in systems at equilibrium and discuss how those systems respond in terms of enthalpy, spontaneity, and entropy.