Calculations Using ΔG = ΔH – TΔS

Introduction

Key Concepts

Understanding Gibbs Free Energy ($\Delta G$)

Gibbs Free Energy ($\Delta G$) quantifies the maximum reversible work a thermodynamic system can perform at constant temperature ($T$) and pressure. It determines the direction of chemical processes and their spontaneity. A negative $\Delta G$ indicates a spontaneous process, while a positive $\Delta G$ suggests non-spontaneity under the given conditions.

Enthalpy Change ($\Delta H$)

Enthalpy Change ($\Delta H$) represents the heat absorbed or released during a reaction at constant pressure. It encompasses the strength of chemical bonds and the energy required to break them. Exothermic reactions have negative $\Delta H$, releasing heat, whereas endothermic reactions have positive $\Delta H$, absorbing heat from the surroundings.

Entropy Change ($\Delta S$)

Entropy Change ($\Delta S$) measures the disorder or randomness within a system. An increase in entropy ($\Delta S > 0$) signifies greater disorder, which favors spontaneity, while a decrease ($\Delta S < 0$) suggests increased order, opposing spontaneity.

Temperature ($T$) in Thermodynamic Calculations

Temperature ($T$) plays a pivotal role in the relationship between enthalpy and entropy. It acts as a balancing factor that determines the influence of $\Delta S$ on $\Delta G$. The temperature must be in Kelvin ($K$) for accurate calculations.

The Gibbs Free Energy Equation

The Gibbs Free Energy equation is given by: $$ \Delta G = \Delta H - T\Delta S $$ This equation integrates enthalpy and entropy changes to predict the spontaneity of reactions. By calculating $\Delta G$, one can assess whether a reaction will proceed forward or reverse under constant temperature and pressure conditions.

Spontaneity of Reactions

The sign of $\Delta G$ determines the spontaneity:

• Negative $\Delta G$: Reaction is spontaneous.
• Positive $\Delta G$: Reaction is non-spontaneous.
• Zero $\Delta G$: System is at equilibrium.

Example Calculation

Consider the reaction: $$ \text{N}_2(g) + 3\text{H}_2(g) \rightarrow 2\text{NH}_3(g) $$ Given:

• $\Delta H = -92.4 \, \text{kJ/mol}$
• $\Delta S = -198.4 \, \text{J/mol.K}$
• Temperature, $T = 298 \, K$

Advanced Concepts

Temperature Dependence of Spontaneity

The Gibbs Free Energy equation illustrates how temperature influences reaction spontaneity. By rearranging the equation: $$ \Delta G = \Delta H - T\Delta S $$ we observe that increasing temperature can shift the balance between $\Delta H$ and $T\Delta S$. For instance:

• If $\Delta H$ and $\Delta S$ are both negative, higher temperatures may render $\Delta G$ positive, making the reaction non-spontaneous.
• If $\Delta H$ is positive and $\Delta S$ is positive, higher temperatures favor spontaneity.

Derivation from the Second Law of Thermodynamics

The Gibbs Free Energy equation is derived from the Second Law of Thermodynamics, which states that for any spontaneous process, the total entropy of the universe increases. By isolating the system's entropy change and considering energy exchanges, Gibbs formulated the equation to predict spontaneity: $$ \Delta G = \Delta H - T\Delta S $$ This derivation integrates system and surroundings, providing a comprehensive measure of spontaneity.

Standard Gibbs Free Energy Change ($\Delta G^\circ$)

Standard Gibbs Free Energy Change ($\Delta G^\circ$) refers to $\Delta G$ under standard conditions (1 atm pressure and 298 K temperature). It facilitates the comparison of different reactions and can be used to calculate equilibrium constants ($K$) through the relationship: $$ \Delta G^\circ = -RT \ln K $$ where $R$ is the gas constant and $T$ is the temperature in Kelvin.

Equilibrium Constant and Gibbs Free Energy

At equilibrium ($\Delta G = 0$), the relationship between $\Delta G^\circ$ and the equilibrium constant ($K$) is established: $$ 0 = \Delta H - T\Delta S - RT \ln K \\ \Delta G^\circ = -RT \ln K $$ This equation links thermodynamic properties with chemical equilibrium, allowing the prediction of product and reactant concentrations at equilibrium.

Non-Standard Conditions

While $\Delta G^\circ$ provides valuable insights, real-world reactions often occur under non-standard conditions. The actual Gibbs Free Energy Change ($\Delta G$) can be calculated using: $$ \Delta G = \Delta G^\circ + RT \ln Q $$ where $Q$ is the reaction quotient. This equation adjusts $\Delta G^\circ$ based on the current concentrations of reactants and products, offering a dynamic understanding of reaction spontaneity.

Applications in Biological Systems

Gibbs Free Energy plays a crucial role in biological processes. For example, in cellular respiration, the conversion of glucose to carbon dioxide and water releases energy, making it a spontaneous process ($\Delta G < 0$). Understanding $\Delta G$ helps elucidate metabolic pathways and energy transfer mechanisms within living organisms.

Interdisciplinary Connections

The concept of Gibbs Free Energy extends beyond chemistry into fields like materials science, engineering, and environmental science. In materials science, $\Delta G$ assists in predicting phase changes and material stability. In engineering, it informs the design of energy-efficient processes. Environmental science leverages $\Delta G$ to understand and mitigate energy-related challenges, such as climate change.

Complex Problem-Solving

Advanced calculations involve multiple steps and integrating various thermodynamic principles. Consider the reaction: $$ \text{C}_6\text{H}_{12}\text{O}_6(s) + 6\text{O}_2(g) \rightarrow 6\text{CO}_2(g) + 6\text{H}_2\text{O}(l) $$ Given:

• $\Delta H^\circ = -2820 \, \text{kJ/mol}$
• $\Delta S^\circ = -310 \, \text{J/mol.K}$
• Calculate $\Delta G^\circ$ at 350 K.

Comparison Table

Summary and Key Takeaways