✓ SECTION 5.6 CHECKLIST

📖 READING PORTION

Gibbs Free Energy

One of the challenges of using the Second Law of Thermodynamics to determine if a process is spontaneous is that we must determine 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

[latex]G = H\;-\;TS[/latex]

Free energy is a state function, and at constant temperature and pressure, the free energy change (ΔG) may be expressed as

We can understand the relationship between this system property and the spontaneity of a process by recalling the previously derived Second Law expression

[latex]{\Delta}S_{\text{univ}} = {\Delta}S\;+\;\frac{q_{\text{surr}}}{T}[/latex]

The First Law requires that q_surr = −q_sys, and at constant pressure q_sys = ΔH, and so this expression may be rewritten as

[latex]{\Delta}S_{\text{univ}} = {\Delta}S\;-\;\frac{{\Delta}H}{T}[/latex]

ΔH is the enthalpy change of the system. Multiplying both sides of this equation by −T, and rearranging yields

[latex]-T{\Delta}S_{\text{univ}} = {\Delta}H\;-\;T{\Delta}S[/latex]

Comparing this equation to the previous one for free energy change shows

[latex]{\Delta}G = -T{\Delta}S_{\text{univ}}[/latex]

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 1 summarizes the relation between the spontaneity of a process and the arithmetic signs of these indicators.

Calculating Free Energy Change

Gibbs free energy is a state function, so its value depends only on the conditions of the initial and final states of the system that have undergone some change. 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 according to Equation 1.

The symbol ° on ΔG, ΔH, and ΔS refers to standard state, a special condition for the species involved in a reaction:

• the partial pressure of gases is 1 atm
• the concentration of dissolved solutes is 1 M
• gases and solutes behave ideally
• substances are in their most stable physical forms

Temperature is not part of the definition of standard state, but 298 K is a temperature at which most data are collected.

Free energy changes may also use the standard free energy of formation ([latex]{\Delta}G_{\text{f}}^{\circ}[/latex]), for each of the reactants and products involved in the reaction. The standard free energy of formation is the free energy change that accompanies the formation of one mole of a substance from its elements in their standard states. Similar to the standard enthalpies of formation, [latex]{\Delta}G_{\text{f}}^{\circ}[/latex] is by definition zero for elemental substances under standard state conditions. The approach to computing the free energy change for a reaction using this approach is the same as that demonstrated previously for enthalpy and entropy changes. At a given temperature and pressure,

For the reaction

[latex]a\text{A}\;+\;b\text{B}\;{\longrightarrow}\;c\text{C}\;+\;d\text{D}[/latex],

the standard free energy change at 298 K may be calculated as

[latex]{\Delta}G_{298}^{\circ} = {\Delta}G^{\circ} = {\sum}n\times {\Delta}G_{298}^{\circ}(\text{products})\;-\;{\sum}n\times {\Delta}G_{298}^{\circ}(\text{reactants}) = [c{\Delta}G_{\text{f, C}}^{\circ}\;+\;d{\Delta}G_{\text{f, D}}^{\circ}]\;-\;[a{\Delta}G_{\text{f, A}}^{\circ}\;+\;b{\Delta}G_{\text{f, B}}^{\circ}][/latex]

Temperature-Dependence of Spontaneity

In Section 5.5, we saw that 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. Likewise, some chemical reactions can also exhibit temperature dependent spontaneities. To illustrate this concept, the equation relating free energy change to the enthalpy and entropy changes for the process is considered

[latex]{\Delta}G = {\Delta}H\;-\;T{\Delta}S[/latex]

The spontaneity of a process, as reflected in the arithmetic sign of its 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 ΔH and ΔS are positive. This condition describes an endothermic process that involves an increase in system entropy. In this case, ΔG will be negative if the magnitude of the TΔS term is greater than ΔH. If the TΔS term is less than ΔH, the free energy change will be positive. Such a process is spontaneous at high temperatures and nonspontaneous at low temperatures.
• Both ΔH and ΔS are negative. This condition describes an exothermic process that involves a decrease in system entropy. In this case, ΔG will be negative if the magnitude of the TΔS term is less than ΔH. If the TΔS term’s magnitude is greater than ΔH, the free energy change will be positive. Such a process is spontaneous at low temperatures and nonspontaneous at high temperatures.
• ΔH is positive and ΔS is negative. This condition describes an endothermic process that involves a decrease in system entropy. In this case, ΔG will be positive regardless of the temperature. Such a process is nonspontaneous at all temperatures.
• ΔH is negative and ΔS is positive. This condition describes an exothermic process that involves an increase in system entropy. In this case, ΔG will be negative regardless of the temperature. Such a process is spontaneous at all temperatures.

These four scenarios are summarized in Figure 1.

Free Energy and Equilibrium

The free energy change for a process may be viewed as a measure of its driving force. A negative value for ΔG represents a finite driving force for the process in the forward direction, while a positive value represents a driving force for the process in the reverse direction. When ΔG is zero, the forward and reverse driving forces are equal, and so the process occurs in both directions at the same rate (the system is at equilibrium).

In Section 3.3, the reaction quotient, Q, was introduced as a convenient measure of the status of an equilibrium system. Recall that Q is the numerical value of the mass action expression for the system, and that we may use its value to identify the direction in which a reaction will proceed in order to achieve equilibrium. When Q is lesser than the equilibrium constant, K, the reaction will proceed in the forward direction until equilibrium is reached and Q = K. Conversely, if Q < K, the process will proceed in the reverse direction until equilibrium is achieved.

The free energy change for a process taking place with reactants and products present under nonstandard conditions, ΔG_r, is related to the standard free energy change, ΔG_r°, according to this equation:

Previously, we defined the reaction quotients and equilibrium constants as purely in terms of concentrations (Q_c or K_c) or partial pressures (Q_p or K_p).  Mixed reaction quotients and mixed equilibrium constants have gaseous species in partial pressures (atm or bar) and aqueous solutes in molar concentrations.

For a system at equilibrium, Q = K and ΔG_r = 0, and Equation 3 may be written as

[latex]0 = {\Delta}G_{\text{r}}^{\circ}\;+\;RT\;\text{ln}\;K\;\;\;\;\;\;\;(\text{at equilibrium})[/latex]

and rearranged into

This form of the equation provides a useful link between these two essential thermodynamic properties, and it can be used to derive equilibrium constants from standard free energy changes and vice versa. The relations between standard free energy changes and equilibrium constants are summarized in Table 2.

To further illustrate the relation between these two essential thermodynamic concepts, consider the observation that reactions spontaneously proceed in a direction that ultimately establishes equilibrium. As may be shown by plotting the free energy change versus the extent of the reaction (for example, as reflected in the value of Q), equilibrium is established when the system’s free energy is minimized, Figure 2. If a system is present with reactants and products present in nonequilibrium amounts (Q ≠ K), the reaction will proceed spontaneously in the direction necessary to establish equilibrium.

The nonzero slopes of tangent lines of the curves in Figure 2 represent nonstandard ΔG values. The slope begins with a very large negative value (extent of reaction = 0, Q = 0, ΔG is very negative). At equilibrium, slope = 0 and thus, ΔG = 0 (free energy has been minimized). Near the end, the slope has a very large positive value (extent of reaction approaches 1, Q is very large, ΔG is very positive, nonspontaneous forward reaction).