## 7.7 Gibbs Free Energy Change and Feasibility of a Reaction This section explores Gibbs Free Energy, a thermodynamic quantity used to predict the spontaneity or feasibility of a chemical reaction under constant temperature and pressure. ### What is Gibbs Free Energy? Gibbs free energy ($G$) is a measure of the amount of useable energy (or work-potential) in a system at constant temperature and pressure. It is also known as "Available Energy." It was conceptualized by the American scientist Josiah Willard Gibbs in 1876. The Gibbs free energy of a system is defined by the equation: $$G = H - TS$$ Where: - $G$ = Gibbs Free Energy - $H$ = Enthalpy of the system - $T$ = Absolute temperature (in Kelvin) - $S$ = Entropy of the system Gibbs free energy is a **state function**, meaning its value depends only on the current state of the system, not on the path taken to reach that state. ### Gibbs Free Energy Change ($\Delta G$) For a chemical process, we are interested in the *change* in Gibbs free energy ($\Delta G$). The change is given by the **Gibbs-Helmholtz equation** for a process at constant temperature: $$\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ$$ Where: - $\Delta G^\circ$ = Standard Gibbs free energy change - $\Delta H^\circ$ = Standard enthalpy change - $\Delta S^\circ$ = Standard entropy change - $T$ = Absolute temperature ### Predicting Reaction Spontaneity and Feasibility The sign and magnitude of $\Delta G$ determine whether a reaction is spontaneous (feasible), non-spontaneous, or at equilibrium. A **feasible reaction** is one that, once started, will proceed to completion without a continuous supply of external energy. The three possible conditions are: | Value of $\Delta G^\circ$ | Description | Characteristics | | :--- | :--- | :--- | | $\Delta G^\circ < 0$ (negative) | **Spontaneous / Feasible** | The reaction proceeds in the forward direction. The system releases free energy. | | $\Delta G^\circ > 0$ (positive) | **Non-spontaneous / Not Feasible** | The reaction does not proceed in the forward direction. Energy must be supplied for it to occur. The reverse reaction is spontaneous. | | $\Delta G^\circ = 0$ | **At Equilibrium** | The rates of the forward and reverse reactions are equal. There is no net change in the system. | #### Temperature Dependence of Spontaneity The feasibility of a reaction depends on the balance between enthalpy ($\Delta H$) and entropy ($T\Delta S$). 1. **Exothermic reactions with increasing entropy ($\Delta H < 0, \Delta S > 0$):** $\Delta G$ is always negative. Spontaneous at all temperatures. 2. **Endothermic reactions with decreasing entropy ($\Delta H > 0, \Delta S < 0$):** $\Delta G$ is always positive. Non-spontaneous at all temperatures. 3. **Endothermic reactions with increasing entropy ($\Delta H > 0, \Delta S > 0$):** Spontaneous only at high temperatures where $T\Delta S > \Delta H$. 4. **Exothermic reactions with decreasing entropy ($\Delta H < 0, \Delta S < 0$):** Spontaneous only at low temperatures where $|T\Delta S| < |\Delta H|$. ### Gibbs Free Energy and Phase Transitions $\Delta G$ is also useful for understanding phase transitions, such as the melting of ice into liquid water. **Reaction:** $$\mathrm{H_2O_{(s)}} \rightarrow \mathrm{H_2O_{(l)}}$$ - **At the melting point (0°C or 273.15 K):** The solid and liquid phases are in equilibrium. The rate of melting equals the rate of freezing. Here, $\Delta G = 0$. - **Above the melting point (> 0°C):** Melting is a spontaneous process. The liquid phase is more stable. Here, $\Delta G < 0$. - **Below the melting point (< 0°C):** Melting is a non-spontaneous process. The solid phase is more stable, and the reverse reaction (freezing) is spontaneous. Here, $\Delta G > 0$. ### Relation between $\Delta G^\circ$ and Equilibrium Constant ($K$) The standard free energy change is related to the equilibrium constant of a reaction by the following expression: $$\Delta G^\circ = -RT \ln K$$ Or in terms of base 10 logarithm: $$\Delta G^\circ = -2.303 RT \log K$$