As you work through this section, if you find that you need a bit more background material to help you understand the topics at hand, you can consult “Chemistry: The Molecular Science” (5th ed. Moore and Stanitski) Chapter 16-6, 12-1 and 12-2, and/or Chapter 12.6-12.8 and Chapter 13.1-13.3 in the Additional Reading Materials section.

D28.1 Gibbs Free Energy

One of the challenges of using the second law of thermodynamics to determine if a process proceeds forward of its own accord is that we must determine ΔS_system and ΔS_surroundings. An alternative approach involving a new thermodynamic property was introduced in the late 19^th century by American mathematician Josiah Willard Gibbs. The property is called the Gibbs free energy (G) (or simply the free energy), and is defined in terms of a system’s enthalpy, entropy, and temperature:

Because H, T, and S are state functions, G must also be a state function. The change in Gibbs free energy (ΔG) at constant temperature may be expressed as:

Gibbs free energy change is related to whether a process proceeds forward of its own accord. This can be derived from the second law expression:

Multiplying both sides of this equation by −T, and rearranging yields:

Hence:

For a process that proceeds forward ΔS_univ must be positive. Because thermodynamic temperature is always positive, the Gibbs free energy change must be negative for a process that proceeds forward of its own accord.

D28.2 Calculating ΔG

Because Gibbs free energy is a state function, its value depends only on the conditions of the initial and final states of the system. A convenient and common approach to calculate ΔG for reactions is by using standard state thermodynamic data. One method involves using the definition of ΔG given above but restricting to standard-state conditions:

It is also possible to use the standard Gibbs free energy of formation ([latex]{\Delta}G_{\text{f}}^{\circ}[/latex]) for each of the reactants and products involved in the reaction to calculate Gibbs free energy changes. [latex]{\Delta}_{\text{f}}G^{\circ}[/latex] is the Gibbs free energy change that accompanies the formation of one mole of a substance from its elements in their standard states. [latex]{\Delta}G_{\text{f}}^{\circ}[/latex] is by definition zero for elemental substances under standard state conditions.

For a reaction:

the standard Gibbs free energy change at room temperature is:

D28.3 Temperature Dependence

Consider:

Whether a reaction is product-favored, as reflected in the arithmetic sign of its Δ_rG°, is determined by the signs of Δ_rH° and Δ_rS°, and, in some cases, the absolute temperature (which can only have positive values). Four possibilities therefore exist:

1. Both Δ_rH° and Δ_rS° are positive. This condition describes an endothermic process that involves an increase in system entropy. In this case, Δ_rG° will be negative if the magnitude of the TΔ_rS° term is greater than Δ_rH°, and positive if the TΔ_rS° term is less than Δ_rH°. Such a process is product-favored at high temperatures and reactant-favored at low temperatures.
2. Both Δ_rH° and Δ_rS° are negative. This condition describes an exothermic process that involves a decrease in system entropy. In this case, Δ_rG° will be negative if the magnitude of the TΔ_rS° term is less than Δ_rH° and positive if the TΔ_rS° term is greater than Δ_rH°. Such a process is product-favored at low temperatures and reactant-favored at high temperatures.
3. Δ_rH° is positive and Δ_rS° is negative. This condition describes an endothermic process that involves a decrease in system entropy. In this case, Δ_rG° will be positive regardless of the temperature. Such a process is reactant-favored at all temperatures.
4. Δ_rH° is negative and Δ_rS° is positive. This condition describes an exothermic process that involves an increase in system entropy. In this case, Δ_rG° will be negative regardless of the temperature. Such a process is product-favored at all temperatures.

These four scenarios are summarized in Figure 1.

A process that is reactant-favored at one temperature but product-favored at a different temperature must have some temperature where it changes from reactant-favored to product-favored (as reflected by the sign of its Δ_rG°). This is illustrated graphically in Figure 2, where Δ_rG° is plotted on the y axis versus T on the x axis :

A process whose Δ_rH° and Δ_rS° have the same arithmetic sign becomes more or less product-favored depending on temperature as depicted by the two yellow lines in the plot. Each line crosses from product-favored to reactant-favored at a temperature that is characteristic of the specific process. This temperature is represented by the x-intercept of the line—the value of T for which Δ_rG° is zero:

And so, saying a process is product-favored at “high” or “low” temperatures means the temperature is above or below, respectively, that temperature at which Δ_rG° for the process is zero.

D28.4 Chemical Equilibrium

A chemical reaction is usually written in a way that suggests it proceeds in one direction, the direction in which we read, but all chemical reactions are reversible: both the forward and reverse reaction occur at the same time. When reactions involve gases or solutions, where concentrations change as the reaction proceeds, the reaction eventually reaches a dynamic equilibrium. In a chemical equilibrium, the forward and reverse reactions occur at equal rates, and the concentrations of products and reactants remain constant. This implies that, if a reaction occurs in a closed system so that the products cannot escape, the reaction often does not yield 100% products. Instead, some reactants remain after the concentrations stop changing. At this point, when there is no further change in concentrations of reactants and products, we say the reaction is at equilibrium.

For example, when we place a sample of dinitrogen tetraoxide (N₂O₄, a colorless gas) in a glass tube, the color becomes darker as N₂O₄ is converted to nitrogen dioxide (NO₂, a red-brown gas) by the reaction:

The double arrow, ⇌, indicates that the products react to form reactants as well as the reactants producing products. When the system reaches equilibrium, both N₂O₄ and NO₂ are present (Figure 3).

Consider the example in Figure 3. Not only can N₂O₄ decompose to form NO₂, but the NO₂ produced can react to form N₂O₄. As soon as the forward reaction produces any NO₂, the reverse reaction begins and NO₂ starts to react to form N₂O₄. As the reaction proceeds the rate of the forward reaction decreases and the rate of the reverse reaction increases. (The forward reaction is first order and as concentration of N₂O₄ decreases the forward rate decreases. The reverse reaction is second order; as concentration of NO₂ increases the rate also increases.) At equilibrium, the concentrations of N₂O₄ and NO₂ no longer change because the rate of NO₂ formation is exactly equal to the rate of NO₂ consumption, and the rate of formation of N₂O₄ is exactly equal to the rate of consumption of N₂O₄. Chemical equilibrium is a dynamic process: the numbers of reactant and product molecules remain constant, but the forward and reverse reactions do not stop.

We use an equilibrium arrow when writing an equation for a reversible reaction. Such a reaction may or may not be at equilibrium. When we wish to speak about one particular component of a reversible reaction, we use a single arrow. For example, in the equilibrium shown in Figure 3, the rate of the forward reaction

is equal to the rate of the backward reaction

An equilibrium can be established for a physical change as well as for a chemical reaction. For example:

Figure 4 shows a sample of liquid bromine at equilibrium with bromine vapor in a closed container. When we pour liquid bromine into an empty bottle in which there is no bromine vapor, some liquid evaporates: the amount of liquid decreases and the amount of vapor increases. If we seal the container so no vapor escapes, the amount of liquid and vapor will eventually stop changing and an equilibrium between the liquid and the vapor will be established. If the container were not sealed, the bromine vapor would escape and no equilibrium would be reached.

D28.5 Equilibrium Constants

An equilibrium constant (K) is a ratio of equilibrium concentrations of products and reactants that remains constant for a given reaction at a given temperature. For example, consider this general equation for a reversible reaction:

For this reaction, the concentration equilibrium constant, K_c, is equal to the molar concentrations of the products, each raised to the power of the coefficient in the equation and all multiplied together, divided by the molar concentrations of the reactants, each raised to the power of its coefficient and all multiplied together:

This mathematical expression is called the equilibrium constant expression. Square brackets indicate equilibrium concentrations of reactants and products.

Because each concentration has units mol/L, an equilibrium constant can have units depending on the coefficients in the chemical equation. For example, the equilibrium constant for the N₂O₄ reaction described earlier has units mol/L because the concentration in the numerator is squared but the denominator is not. However, the equilibrium constant for reaction of H₂ and I₂ to form 2 HI has no units because the sum of exponents in the numerator of the equilibrium constant expression for this reaction equals the sum of exponents in the denominator.

N₂O₄(g) ⇌ 2 NO₂(g)    K_c = [latex]\dfrac{[\text{NO}_2]^2}{[\text{N}_2\text{O}_4]}[/latex]     units mol/L

H₂(g)  +  I₂(g)  ⇌  2 HI(g)    K_c = [latex]\dfrac{[\text{H}_2][\text{I}_2]}{[\text{HI}]^2}[/latex]     units cancel so K_c is unitless

Units are often omitted when evaluating equilibrium constants because it is easy to tell what units are needed from the mathematical form of the equilibrium constant expression.

Reaching equilibrium requires that concentrations of reactants and products change until the rates of forward and reverse reactions are equal. Therefore, only substances whose concentrations change as a reaction occurs are included in an equilibrium constant expression. For example, consider this equilibrium:

2 HgO(s)   ⇌   2 Hg(l)  +  O₂(g)          K_c = [O₂]

Because HgO is a pure solid, the number of HgO formula units in a given volume of HgO is the same throughout the reaction; it depends only on the density of HgO at the temperature of the reaction. Similarly, the number of Hg atoms per unit volume of pure Hg(l) is constant throughout the reaction. Thus, these concentrations are not included in the equilibrium constant expression. It is necessary for some HgO(s) and some Hg(l) to be present for the equilibrium to be maintained, but the quantity of each does not matter. In general, K_c expressions do not contain terms for pure solids or pure liquids.

For dilute solutions, the concentration of solvent remains constant throughout an equilibrium reaction and is not included in the equilibrium constant expression, even though the solvent may appear in the reaction equation.

A homogeneous equilibrium is one in which all of the reactants and products are present in a single phase (are all in a solution, which, by definition, is a homogeneous mixture). Examples of homogeneous equilibria are reactions in the gas phase and reactions in liquid solutions. For example:

In these aqueous equilibrium systems, H₂O(l) is the solvent. Its concentration does not appear in the K_c expression.

A heterogeneous equilibrium is a system in which reactants and products are found in two or more phases. Some heterogeneous equilibria involve chemical changes, for example:

Other heterogeneous equilibria involve phase changes, for example:

The equilibrium constant expression is different if a reaction changes stoichiometry, is reversed, or is the sum of several steps. The value of the equilibrium constant therefore changes.

When all the coefficients in a balanced chemical equation are multiplied by some factor n, then the new equilibrium constant is the original equilibrium constant raised to the n^th power. For example:

When a reaction’s direction is reversed, the equilibrium constant for the new reaction is simply the reciprocal (inverse) of that for the original reaction. For example:

When two reactions are added together to obtain a new reaction, the equilibrium constant for the new reaction is the product of the equilibrium constants for the original reactions. For example: