Learning Objectives By the end of this section, you will be able to: Define Gibbs free energy, and describe its relation to spontaneity Explain how temperature affects the spontaneity of some processes One of the challenges of using the second law of thermodynamics to determine if a process is spontaneous is that it requires measurements of the entropy change for the system and the entropy change for the surroundings. An alternative approach involving a new thermodynamic property defined in terms of system properties only was introduced in the late nineteenth century by American mathematician Josiah Willard Gibbs . This new property is called the Gibbs free energy (\(G\)) (or simply the free energy ), and it is defined in terms of a system’s enthalpy and entropy as the following: \[G=H-T S \nonumber \] Free energy is a state function, and at constant temperature and pressure, the free energy ch ange (\(ΔG\)) may be expressed as the following: \[\Delta G=\Delta H-T \Delta S \nonumber \] (For simplicity’s sake, the subscript “sys” will be omitted henceforth.) The relationship between this system property and the spontaneity of a process may be understood by recalling the previously derived second law expression: \[\Delta S_{\text {univ }}=\Delta S+\frac{q_{\text {surr }}}{T} \nonumber \] The first law requires that \(q_{surr} = −q_{sys}\), and at constant pressure \(q_{sys} = ΔH\), so this expression may be rewritten as: \[\Delta S_{\text {univ }}=\Delta S-\frac{\Delta H}{T} \nonumber \] Multiplying both sides of this equation by \(−T\), and rearranging yields the following: \[-T \Delta S_{\text {univ }}=\Delta H-T \Delta S \nonumber \] Comparing this equation to the previous one for free energy change shows the following relation: \[\Delta G=-T \Delta S_{\text {univ }} \nonumber \] The free energy change is therefore a reliable indicator of the spontaneity of a process, being directly related to the previously identified spontaneity indicator, \(ΔS_{univ}\). Table \(\PageIndex{1}\) summarizes the relation between the spontaneity of a process and the arithmetic signs of these indicators. 0 1 2 \(ΔS_{univ} > 0\) ΔG < 0 moves spontaneously in the forward direction, as written, to reach equilibrium \(ΔS_{univ} < 0\) ΔG > 0 nonspontaneous in the forward direction, as written, but moves spontaneously in the reverse direction, as written, to reach equilibrium \(ΔS_{univ} = 0\) ΔG = 0 reversible (at equilibrium) What’s “Free” about Δ G ? In addition to indicating spontaneity, the free energy change also provides information regarding the amount of useful work (\(w\)) that may be accomplished by a spontaneous process. Although a rigorous treatment of this subject is beyond the scope of an introductory chemistry text, a brief discussion is helpful for gaining a better perspective on this important thermodynamic property. For this purpose, consider a spontaneous, exothermic process that involves a decrease in entropy. The free energy, as defined by \[\Delta G=\Delta H-T \Delta S \nonumber \] may be interpreted as representing the difference between the energy produced by the process, Δ H , and the energy lost to the surroundings, T Δ S . The difference between the energy produced and the energy lost is the energy available (or “free”) to do useful work by the process, Δ G . If the process somehow could be made to take place under conditions of thermodynamic reversibility, the amount of work that could be done would be maximal: \[\Delta G=w_{max } \nonumber \] where \(w_{max}\) refers to all types of work except expansion (pressure-volume) work. However, as noted previously in this chapter, such conditions are not realistic. In addition, the technologies used to extract work from a spontaneous process (e.g., batteries) are never 100% efficient, and so the work done by these processes is always less than the theoretical maximum. Similar reasoning may be applied to a nonspontaneous process, for which the free energy change represents the minimum amount of work that must be done on the system to carry out the process. Calculating Free Energy Change Free energy is a state function, so its value depends only on the conditions of the initial and final states of the system. A convenient and common approach to the calculation of free energy changes for physical and chemical reactions is by use of widely available compilations of standard state thermodynamic data. One method involves the use of standard enthalpies and entropies to compute standard free energy changes , \(ΔG^o\), according to the following relation. \[\Delta G^{\circ}=\Delta H^{\circ}-T \Delta S^{\circ} \nonumber \] It is important to understand that for phase changes, \(\Delta G^º\) tells you if the phase change is spontaneous or not; will it happen, or not happen. For chemical reactions, \(\Delta G^º\) tells you the extent of a reaction. In other words, \(\Delta G^º\) for a reaction tells you how much product will be present at equilibrium. A reaction with \(\Delta G^º\) < 0 is considered product-favored at equilibrium ; there will be more products than reactants when the reaction reaches equilibrium. A reaction with \(\Delta G^º\) > 0 is considered reactant-favored at equilibrium ; there will be more reactants than products when the reaction reaches equilibrium. Temperature Dependence of Spontaneity and Extent of Reaction As was previously demonstrated in this chapter’s section on entropy, the spontaneity of a process may depend upon the temperature of the system. Phase transitions, for example, will proceed spontaneously in one direction or the other depending upon the temperature of the substance in question. In a similar, but not identical fashion, some chemical reactions can switch from being product-favored at equilibrium, to being reactant-favored at equilibrium, depending on the temperature. Note The numerical value of \(\Delta G^º\) is always dependent on the temperature. In this section we are determining whether or not the sign of \(\Delta G^º\) is dependent on the temperature. To illustrate this concept, the equation relating free energy change to the enthalpy and entropy changes for the process is considered: \[ ΔG^º=ΔH^º−TΔS^º \nonumber \] The extent of a process, as reflected in the arithmetic sign of its standard free energy change, is then determined by the signs of the enthalpy and entropy changes and, in some cases, the absolute temperature. Since T is the absolute (Kelvin) temperature, it can only have positive values. Four possibilities therefore exist with regard to the signs of the enthalpy and entropy changes: Both Δ H º and Δ S º are positive. This condition describes an endothermic process that involves an increase in system entropy. In this case, Δ G º will be negative if the magnitude of the T Δ S º term is greater than Δ H º . If the T Δ S º term is less than Δ H º , the free energy change will be positive. Such a process is product-favored at equilibrium at high temperatures and reactant-favored at equilibrium at low temperatures. Both Δ H º and Δ S º are negative. This condition describes an exothermic process that involves a decrease in system entropy. In this case, Δ G º will be negative if the magnitude of the T Δ S º term is less than Δ H º . If the T Δ S º term’s magnitude is greater than Δ H º , the free energy change will be positive. Such a process is product-favored at equilibrium at low temperatures and reactant-favored at equilibrium at high temperatures. Δ H º is positive and Δ S º is negative. This condition describes an endothermic process that involves a decrease in system entropy. In this case, Δ G º will be positive regardless of the temperature. Such a process is reactant-favored at equilibrium at all temperatures. Δ H º is negative and Δ S º is positive. This condition describes an exothermic process that involves an increase in system entropy. In this case, Δ G º will be negative regardless of the temperature. Such a process is product-favored at equilibrium at all temperatures. These four scenarios are summarized in Table \(\PageIndex{1}\) Sign of \(\Delta H^o\) Sign of \(\Delta S^o\) Sign of \(\Delta G^o\) Temperature Dependence of \(\Delta G^o\) - + - The sign of \(\Delta G^o\) does not depend on the temperature.The reaction is product-favored at equilibrium at all temperatures. + - + The sign of \(\Delta G^o\) does not depend on the temperature.The reaction is reactant-favored at equilibrium at all temperatures. - - - or + The sign of \(\Delta G^o\) does depend on the temperature. The reaction will be product-favored at equilibrium at lower temperatures. + + - or + The sign of \(\Delta G^o\) does depend on the temperature. The reaction will be product-favored at equilibrium at lower temperatures. Note In this discussion, we have used two different descriptions for the meaning of the sign of Δ G º . You should be aware of the meaning of each description. a) Extent of Reaction: This description is used to predict the ratio of the product and reactant concentrations at equilibrium. In this description, we use the thermodynamic term Δ G º to tell us the same information as the equilibrium constant, K. When Δ G º < 0, K > 1, and the reaction will be product-favored at equilibrium. When Δ G º > 0, K< 1, and the reaction is reactant-favored at equilibrium. When Δ G º = 0, K =1, and the reaction will have roughly equal amounts of products and reactants at equilibrium. In all cases, the reaction will form a mixture of products and reactants at equilibrium. We use the sign and magnitude of Δ G º to tell us how much product will be made if the reaction is allowed to reach equilibrium. b) Spontaneity: This description is much more complicated because it involves two different interpretations of how a reaction at standard state occurs. One interpretation involves the hypothetical process in which the reaction proceeds from a starting point of pure reactants to a finishing point of pure products, with all substances isolated in their own containers under standard state conditions. In the second, more realistic interpretation, the reaction starts with all reactants and all products in their standard state in one container. We then allow this specific mixture to react an infinitesimally small amount so that we can obtain a rate of change in free energy with respect to the extent of reaction when all reactants and products are mixed and (essentially) in their standard states. Although each interpretation describes a different reaction scenario, the value of the difference in free energy and the value of the rate of change in free energy are the same number. If Δ G º < 0, we say that the reaction is spontaneous, meaning that the reaction would proceed in the forward direction, as written, to form pure products in their standard state. If Δ G º > 0, we say that the reaction is nonspontaneous, meaning that the reaction would proceed in the reverse direction, as written, to form pure reactants in their standard state. If Δ G º = 0, we say that the neither the reactants nor the products are favored to be formed. A detailed treatment of the meaning of ΔGº can be found in the paper, "Free Energy versus Extent of Reaction" by Richard S. Treptow, Journal of Chemical Education, 1996 , Volume 73 (1), 51-54.