## 6.7 Gibbs Free Energy Gibbs free energy ($G$) is a fundamental thermodynamic quantity that measures the amount of *usable energy* in a system at constant temperature and pressure. It is also known as "Available energy". This concept was introduced by the American scientist Josiah Willard Gibbs in 1876. ### Definition and Equation Gibbs free energy is defined as the enthalpy of a system minus the product of its absolute temperature and entropy: $$G = H - TS$$ Where: - $G$ = Gibbs free energy (typically in Joules or kJ) - $H$ = Enthalpy of the system (typically in Joules or kJ) - $T$ = Temperature in Kelvin (absolute temperature) - $S$ = Entropy of the system (typically in $J\,K^{-1}$ or $kJ\,K^{-1}$) ### Gibbs free energy as a State Function Gibbs free energy is a *state function*, meaning its value depends only on the initial and final states of the system, not on the path taken to get there. Therefore, the change in Gibbs free energy ($\Delta G$) for a process is particularly significant. For a process occurring at standard conditions (denoted by $\circ$), the change in Gibbs free energy is given by: $$\Delta G^\circ = \Delta H^\circ - \Delta(TS)$$ When a reaction is carried out at **constant temperature** ($\Delta T = 0$), this equation simplifies significantly: $$\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ$$ This crucial equation is known as the **Gibbs-Helmholtz equation**. It is immensely useful for predicting the spontaneity and equilibrium state of a chemical reaction. ### Spontaneity and Equilibrium from $\Delta G^\circ$ The sign and magnitude of $\Delta G^\circ$ provide direct information about the spontaneity of a process and its position relative to equilibrium. There are three possibilities: - **If $\Delta G^\circ < 0$ (negative):** The process is **spontaneous** in the forward direction. This means the system releases energy and naturally proceeds towards a lower energy state. - **If $\Delta G^\circ > 0$ (positive):** The process is **non-spontaneous** in the forward direction. It would require an input of energy to proceed as written. The reverse reaction would be spontaneous. - **If $\Delta G^\circ = 0$:** The system is at **equilibrium**. There is no net change in the concentrations of reactants and products, and the forward and reverse reaction rates are equal. ### Application: Phase Transitions Gibbs free energy is also valuable for understanding phase transitions, such as melting, freezing, boiling, and condensation. Consider the melting of ice: $$\mathrm{H_2O_{(s)}} \rightarrow \mathrm{H_2O_{(l)}}$$ - At the normal melting point of ice ($0^\circ\text{C}$ or $273.15\,\text{K}$), solid ice and liquid water are in equilibrium. Thus, $\Delta G^\circ = 0$. - Below the melting point (e.g., $-5^\circ\text{C}$), $\Delta G^\circ$ for melting is positive, indicating that the solid phase is more stable (melting is non-spontaneous). - Above the melting point (e.g., $+5^\circ\text{C}$), $\Delta G^\circ$ for melting is negative, indicating that the liquid phase is more stable (melting is spontaneous). ### Enthalpy and Calorie Contents of Food The amount of energy contained within a food or drink is commonly measured in **calories**. These "food calories" represent the chemical energy stored in the food, which is then released through metabolic processes within our bodies. - The *enthalpy change* associated with these metabolic processes corresponds to the amount of energy released (exothermic) or absorbed (endothermic) during digestion and metabolism. - The energy content of food is typically measured in **Kilocalories (kcal)**, often colloquially referred to simply as "calories" (with a capital 'C' sometimes used to distinguish from thermodynamic calories). - The conversion factor between kilocalories and kilojoules (the SI unit for energy) is approximately: $$1\;\text{Kilocalorie (kcal)} \approx 4.184\;\text{Kilojoules (kJ)}$$ This conversion factor allows us to relate the energy released from food metabolism to standard thermodynamic energy units. ### Worked Examples *No numerical worked examples were provided in the source text.* ### Possible Questions and Answers - **Q:** What is Gibbs free energy a measure of? **A:** It is a measure of the *usable energy* of a system at constant temperature and pressure. - **Q:** State the Gibbs-Helmholtz equation. **A:** $$\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ$$ - **Q:** If a chemical reaction has a negative $\Delta G^\circ$ value, what does this indicate about the reaction? **A:** A negative $\Delta G^\circ$ indicates that the reaction is **spontaneous** in the forward direction. - **Q:** How do calories in food relate to enthalpy change? **A:** Calories in food represent stored chemical energy. When this food is metabolized, the energy released corresponds to an *enthalpy change* during the metabolic processes in the body. - **Q:** Convert $500\,\text{kcal}$ to kilojoules. **A:** $500\,\text{kcal} \times 4.184\,\frac{\text{kJ}}{\text{kcal}} = 2092\,\text{kJ}$