Relationship Between ΔH, ΔS, and ΔG

Introduction

Key Concepts

Thermodynamic Quantities: ΔH, ΔS, and ΔG

In thermodynamics, three primary quantities describe the energy and disorder changes in a system: enthalpy change ($\Delta H$), entropy change ($\Delta S$), and Gibbs free energy change ($\Delta G$). These parameters are crucial for determining whether a process or reaction will occur spontaneously.

Enthalpy Change ($\Delta H$)

Enthalpy ($H$) is a measure of the total heat content of a system at constant pressure. The change in enthalpy ($\Delta H$) during a reaction signifies whether the process is exothermic or endothermic.

• Exothermic Reactions: Release heat, resulting in a negative $\Delta H$.
• Endothermic Reactions: Absorb heat, leading to a positive $\Delta H$.

For instance, the combustion of methane can be represented as: $$\mathrm{CH_4(g) + 2O_2(g) \rightarrow CO_2(g) + 2H_2O(l)}$$ This reaction releases heat, indicating a negative $\Delta H$.

Entropy Change ($\Delta S$)

Entropy ($S$) measures the degree of disorder or randomness in a system. The change in entropy ($\Delta S$) during a reaction indicates whether the disorder increases or decreases.

• Positive $\Delta S$: Indicates an increase in disorder.
• Negative $\Delta S$: Signifies a decrease in disorder.

For example, the dissolution of ammonium nitrate in water increases disorder, resulting in a positive $\Delta S$.

Gibbs Free Energy Change ($\Delta G$)

Gibbs free energy ($G$) combines enthalpy and entropy to determine the spontaneity of a process. The change in Gibbs free energy ($\Delta G$) is given by: $$\Delta G = \Delta H - T\Delta S$$ Where:

• $\Delta G$: Gibbs free energy change
• $\Delta H$: Enthalpy change
• $T$: Absolute temperature in Kelvin
• $\Delta S$: Entropy change

Spontaneity and Thermodynamic Favorability

A reaction is considered spontaneous if it can occur without external intervention. The sign of $\Delta G$ determines spontaneity:

• Negative $\Delta G$: Spontaneous process.
• Positive $\Delta G$: Non-spontaneous process.
• $\Delta G = 0$: System is at equilibrium.

Interrelationship Between ΔH, ΔS, and ΔG

The interplay between $\Delta H$ and $\Delta S$ dictates the value of $\Delta G$, and consequently, the spontaneity of a reaction. There are four possible scenarios:

1. ΔH < 0 and ΔS > 0: $\Delta G$ is always negative; the reaction is spontaneous at all temperatures.
2. ΔH > 0 and ΔS > 0: $\Delta G$ can be negative at high temperatures, making the reaction spontaneous above a certain temperature.
3. ΔH < 0 and ΔS < 0: $\Delta G$ can be negative at low temperatures, making the reaction spontaneous below a certain temperature.
4. ΔH > 0 and ΔS < 0: $\Delta G$ is always positive; the reaction is non-spontaneous at all temperatures.

Temperature Dependence

Temperature plays a crucial role in the relationship between $\Delta H$, $\Delta S$, and $\Delta G$. The term $T\Delta S$ can either enhance or oppose $\Delta H$, influencing the sign of $\Delta G$. As temperature increases, the entropy term becomes more significant.

Applications in Chemical Reactions

Understanding the relationship between $\Delta H$, $\Delta S$, and $\Delta G$ allows chemists to predict the feasibility of reactions under varying conditions. For example, the formation of diamond from graphite has positive $\Delta H$ and negative $\Delta S$, making it non-spontaneous under standard conditions.

Calculating Gibbs Free Energy

To calculate $\Delta G$, the following formula is used: $$\Delta G = \Delta H - T\Delta S$$ Where:

• Ensure that $\Delta H$ and $\Delta S$ are in compatible units, typically kJ/mol and J/mol.K respectively.
• Temperature ($T$) must be in Kelvin.

**Example Calculation:** Consider a reaction with $\Delta H = -100 \text{ kJ/mol}$ and $\Delta S = -200 \text{ J/mol.K}$ at $T = 300 \text{ K}$. First, convert $\Delta S$ to kJ: $$\Delta S = -200 \text{ J/mol.K} = -0.200 \text{ kJ/mol.K}$$ Then, $$\Delta G = -100 \text{ kJ/mol} - (300 \text{ K})(-0.200 \text{ kJ/mol.K})$$ $$\Delta G = -100 + 60 = -40 \text{ kJ/mol}$$ Since $\Delta G$ is negative, the reaction is spontaneous at 300 K.

Standard Gibbs Free Energy Change

The standard Gibbs free energy change ($\Delta G^\circ$) refers to the change in free energy when reactants and products are in their standard states (1 atm pressure and specified temperature, usually 25°C). It provides a reference point for calculating free energy changes under non-standard conditions using the reaction quotient.

Relation to Equilibrium Constant

There's a direct relationship between $\Delta G^\circ$ and the equilibrium constant ($K$) of a reaction: $$\Delta G^\circ = -RT \ln K$$ Where:

• $R$: Gas constant ($8.314 \text{ J/mol.K}$)
• $T$: Temperature in Kelvin
• $K$: Equilibrium constant

A negative $\Delta G^\circ$ implies $K > 1$, favoring product formation, while a positive $\Delta G^\circ$ indicates $K < 1$, favoring reactants.

Hess's Law and Gibbs Free Energy

Hess's Law states that the total enthalpy change for a reaction is the sum of the enthalpy changes for each step of the reaction. Similarly, Gibbs free energy is a state function, meaning: $$\Delta G_{\text{total}} = \sum \Delta G_{\text{steps}}$$ This allows for the calculation of $\Delta G$ for complex reactions by summing the $\Delta G$ values of individual steps.

Le Chatelier's Principle and Gibbs Free Energy

Le Chatelier's Principle describes how a system at equilibrium responds to stress. Changes in temperature, pressure, or concentration can shift the equilibrium position. Understanding $\Delta H$ and $\Delta S$ helps predict the direction of this shift by analyzing the resulting changes in $\Delta G$.

Standard State Conditions

Standard state conditions are essential for defining $\Delta H^\circ$ and $\Delta G^\circ$. These conditions ensure consistency in thermodynamic calculations and comparisons. Deviations from standard states require adjustments using activity coefficients or fugacity.

Spontaneity vs. Kinetics

It's important to distinguish between thermodynamic spontaneity and reaction kinetics. A reaction may be thermodynamically spontaneous ($\Delta G < 0$) but still proceed slowly due to a high activation energy barrier. Conversely, a non-spontaneous reaction can be driven by coupling with other processes.

Entropy and the Second Law of Thermodynamics

The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time. This law underpins the concept of entropy change in reactions, influencing the calculation of $\Delta G$ and the assessment of reaction spontaneity.

Practical Examples

Applying the relationship between $\Delta H$, $\Delta S$, and $\Delta G$ can predict the behavior of real-world systems:

• Melting of Ice: At temperatures above 0°C, ice melts spontaneously because the increase in entropy ($\Delta S > 0$) drives $\Delta G$ negative despite the positive $\Delta H$.
• Formation of Ammonia (Haber Process): This exothermic process has a negative $\Delta H$ and the entropy change depends on the reaction conditions, allowing optimization for industrial synthesis.

Limitations and Considerations

While the relationship between $\Delta H$, $\Delta S$, and $\Delta G$ is powerful, it assumes ideal conditions. Real systems may experience non-ideal behavior due to interactions between molecules, changes in volume, or non-constant temperature and pressure, necessitating more complex models for accurate predictions.

Comparison Table

Summary and Key Takeaways