# 6.2: Thermodynamics and Equilibrium Chemistry

• • Contributed by David Harvey
• Professor (Chemistry and Biochemistry) at DePauw University

Thermodynamics is the study of thermal, electrical, chemical, and mechanical forms of energy. The study of thermodynamics crosses many disciplines, including physics, engineering, and chemistry. Of the various branches of thermodynamics, the most important to chemistry is the study of how energy changes during a chemical reaction.

Consider, for example, the general equilibrium reaction shown in equation \ref{6.1}, which involves the species A, B, C, and D, with stoichiometric coefficients of a, b, c, and d.

$a A+b B \rightleftharpoons c C+d D \label{6.1}$

By convention, we identify the species on the left side of the equilibrium arrow as reactants and those on the right side of the equilibrium arrow as products. As Berthollet discovered, writing a reaction in this fashion does not guarantee that the reaction of A and B to produce C and D is favorable. Depending on initial conditions the reaction may move to the left, it may move to the right, or it may exist in a state of equilibrium. Understanding the factors that determine the reaction’s final equilibrium position is one of the goals of chemical thermodynamics.

The direction of a reaction is that which lowers the overall free energy. At a constant temperature and pressure, which is typical of many benchtop chemical reactions, a reaction’s free energy is given by the Gibb’s free energy function

$\Delta G=\Delta H-T \Delta S \label{6.2}$

where T is the temperature in kelvin, and ∆G, ∆H, and ∆S are the differences in the Gibb's free energy, the enthalpy, and the entropy between the products and the reactants.

Enthalpy is a measure of the flow of energy, as heat, during a chemical reaction. A reaction that releases heat has a negative ∆H and is called exothermic. An endothermic reaction absorbs heat from its surroundings and has a positive ∆H. Entropy is a measure of energy that is unavailable for useful, chemical work. The entropy of an individual species is always positive and generally is larger for gases than for solids, and for more complex molecules than for simpler molecules. Reactions that produce a large number of simple, gaseous products usually have a positive ∆S.

For many students, entropy is the most difficult topic in thermodynamics to understand. For a rich resource on entropy, visit the following web site: http://entropysite.oxy.edu/.

The sign of ∆G indicates the direction in which a reaction moves to reach its equilibrium position. A reaction is thermodynamically favorable when its enthalpy, ∆H, decreases and its entropy, ∆S, increases. Substituting the inequalities ∆< 0 and ∆> 0 into equation \ref{6.2} shows that a reaction is thermodynamically favorable when ∆G is negative. When ∆G is positive the reaction is unfavorable as written (although the reverse reaction is favorable). A reaction at equilibrium has a ∆G of zero.

Equation \ref{6.2} shows that the sign of ∆G depends on the signs of ∆H and of ∆S, and the temperature, T. The following table summarizes the possibilities.

 $$\Delta H$$ $$\Delta S$$ $$\Delta G$$ $$–$$ $$+$$ $$\Delta G < 0$$ at all temperatures $$-$$ $$-$$ $$\Delta G < 0$$ at low temperatures only $$+$$ $$+$$ $$\Delta G < 0$$ at high temperatures only $$+$$ $$-$$ $$\Delta G > 0$$ at all temperatures

Note that the what constitutes "low temperatures" or "high temperatures" depends on the reaction.

As a reaction moves from its initial, non‐equilibrium condition to its equilibrium position, its value of ∆G approaches zero. At the same time, the chemical species in the reaction experience a change in their concentrations. The Gibb's free energy, therefore, must be a function of the concentrations of reactants and products.

As shown in equation \ref{6.3}, we can divide the Gibb’s free energy, ∆G, into two terms.

$\triangle G=\Delta G^{\circ}+R T \ln Q_r \label{6.3}$

The first term, ∆Go, is the change in the Gibb’s free energy when each species in the reaction is in its standard state, which we define as follows: gases with unit partial pressures, solutes with unit concentrations, and pure solids and pure liquids. The second term includes the reaction quotient, $$Q_r$$, which accounts for non‐standard state pressures and concentrations. For reaction \ref{6.1} the reaction quotient is

$Q_r = \frac{[\mathrm{C}]^{c}[\mathrm{D}]^{d}}{[\mathrm{A}]^{a}[\mathrm{B}]^{b}} \label{6.4}$

where the terms in brackets are the concentrations of the reactants and products. Note that we define the reaction quotient with the products in the numerator and the reactants in the denominator. In addition, we raise the concentration of each species to a power equivalent to its stoichiometry in the balanced chemical reaction. For a gas, we use partial pressure in place of concentration. Pure solids and pure liquids do not appear in the reaction quotient.

Although not shown here, each concentration term in equation \ref{6.4} is divided by the corresponding standard state concentration; thus, the term [C]c really means

$\left\{\frac{[\mathrm{C}]}{[\mathrm{C}]^{\circ}}\right\} \nonumber$

where [C]o is the standard state concentration for C. There are two important consequences of this: (1) the value of Q is unitless; and (2) the ratio has a value of 1 for a pure solid or a pure liquid. This is the reason that pure solids and pure liquids do not appear in the reaction quotient.

At equilibrium the Gibb’s free energy is zero, and equation \ref{6.3} simplifies to

$\triangle G^{\circ}=-R T \ln K \nonumber$

where K is an equilibrium constant that defines the reaction’s equilibrium position. The equilibrium constant is just the reaction quotient’s numerical value when we substitute equilibrium concentrations into equation \ref{6.4}.

$K = \frac{[\mathrm{C}]_{\mathrm{eq}}^{c}[\mathrm{D}]_{\mathrm{eq}}^{d}}{[\mathrm{A}]_{\mathrm{eq}}^{a}[\mathrm{B}]_{\mathrm{eq}}^{b}} \label{6.5}$

Here we include the subscript “eq” to indicate a concentration at equilibrium. Although generally we will omit the “eq” when we write an equilibrium constant expressions, it is important to remember that the value of K is determined by equilibrium concentrations.

As written, equation \ref{6.5} is a limiting law that applies only to infinitely dilute solutions where the chemical behavior of one species is unaffected by the presence of other species. Strictly speaking, equation \ref{6.5} is written in terms of activities instead of concentrations. We will return to this point in Chapter 6.9. For now, we will stick with concentrations as this convention already is familiar to you.