Understanding the Calculation of Equilibrium Temperature When ΔG = 0

The calculation of equilibrium temperature determines the point where Gibbs free energy change equals zero. This critical temperature defines reaction spontaneity and equilibrium.

This article explores the fundamental formulas, common values, and real-world applications for calculating equilibrium temperature in chemical systems. Detailed examples and tables enhance comprehension.

• Calculate equilibrium temperature for the reaction N2 + 3H2 ⇌ 2NH3 given ΔH° and ΔS°.
• Determine equilibrium temperature when ΔG = 0 for the decomposition of CaCO3.
• Find the temperature at which the reaction CO + H2O ⇌ CO2 + H2 is at equilibrium.
• Compute equilibrium temperature for the synthesis of methane from CO2 and H2.

Comprehensive Tables of Common Values for Equilibrium Temperature Calculations

To accurately calculate the equilibrium temperature where ΔG = 0, it is essential to have reliable thermodynamic data. The following tables provide standard enthalpy changes (ΔH°), standard entropy changes (ΔS°), and other relevant parameters for common reactions and substances.

These values are typically measured at standard conditions (298 K, 1 atm) and can be adjusted for temperature dependence using heat capacity data if necessary.

Fundamental Formulas for Calculating Equilibrium Temperature (ΔG = 0)

The equilibrium temperature is the temperature at which the Gibbs free energy change (ΔG) for a reaction equals zero, indicating no net driving force for the reaction to proceed in either direction.

The fundamental thermodynamic relationship is:

Where:

• ΔG = Gibbs free energy change (J/mol or kJ/mol)
• ΔH = Enthalpy change of the reaction (J/mol or kJ/mol)
• T = Absolute temperature (Kelvin, K)
• ΔS = Entropy change of the reaction (J/mol·K)

At equilibrium, ΔG = 0, so:

Rearranging to solve for the equilibrium temperature T_eq:

It is critical to ensure that units are consistent: ΔH should be in J/mol if ΔS is in J/mol·K, or both in kJ/mol and kJ/mol·K respectively.

Explanation of Variables and Typical Values

• ΔH (Enthalpy Change): Represents the heat absorbed or released during the reaction at constant pressure. Exothermic reactions have negative ΔH, endothermic positive.
• ΔS (Entropy Change): Represents the change in disorder or randomness. Positive ΔS indicates increased disorder.
• T (Temperature): Absolute temperature in Kelvin. The equilibrium temperature is the specific T where ΔG = 0.

Note that if ΔS is negative, the equilibrium temperature calculation may yield a negative or non-physical value, indicating the reaction is spontaneous only below or above certain temperatures.

Additional Relevant Formulas

In some cases, the temperature dependence of ΔH and ΔS must be considered. Using heat capacities (C_p), the enthalpy and entropy at temperature T can be estimated:

Where:

• ΔH° and ΔS° are standard enthalpy and entropy changes at reference temperature T° (usually 298 K).
• ΔC_p is the difference in heat capacities between products and reactants.

These integrals can be approximated if ΔC_p is constant over the temperature range:

Incorporating these into the equilibrium temperature calculation yields a transcendental equation that may require numerical methods to solve.

Real-World Applications and Detailed Examples

Example 1: Ammonia Synthesis Reaction

The Haber-Bosch process synthesizes ammonia from nitrogen and hydrogen gases:

N₂(g) + 3H₂(g) ⇌ 2NH₃(g)

Given standard thermodynamic data at 298 K:

• ΔH° = -92.4 kJ/mol
• ΔS° = -198.7 J/mol·K

Calculate the equilibrium temperature where ΔG = 0.

Step 1: Convert units for consistency:

• ΔH° = -92,400 J/mol
• ΔS° = -198.7 J/mol·K

Step 2: Apply the formula:

Interpretation: The equilibrium temperature is approximately 465 K (192 °C). Below this temperature, the reaction is spontaneous (ΔG < 0), favoring ammonia formation. Above it, the reaction becomes non-spontaneous.

Note: Industrial ammonia synthesis operates at higher temperatures (~700 K) to increase reaction rates, despite thermodynamic favorability at lower temperatures.

Example 2: Thermal Decomposition of Calcium Carbonate

The decomposition reaction is:

CaCO₃(s) ⇌ CaO(s) + CO₂(g)

Given data at 298 K:

• ΔH° = 178.3 kJ/mol
• ΔS° = 160.5 J/mol·K

Calculate the equilibrium temperature where ΔG = 0.

Step 1: Convert units:

• ΔH° = 178,300 J/mol
• ΔS° = 160.5 J/mol·K

Step 2: Calculate T_eq:

Interpretation: The equilibrium temperature is approximately 1,111 K (838 °C). Above this temperature, the decomposition of CaCO₃ to CaO and CO₂ is spontaneous.

This temperature is critical in cement manufacturing and lime production industries.

Additional Considerations for Accurate Equilibrium Temperature Calculations

While the simplified formula T_eq = ΔH / ΔS provides a first approximation, several factors can influence the accuracy of equilibrium temperature calculations:

• Temperature Dependence of ΔH and ΔS: Both enthalpy and entropy vary with temperature. Incorporating heat capacity corrections improves precision.
• Pressure Effects: For reactions involving gases, equilibrium constants and ΔG depend on partial pressures. The standard Gibbs free energy assumes 1 atm pressure.
• Non-ideal Behavior: Real gases and solutions may deviate from ideality, requiring activity coefficients or fugacity corrections.
• Phase Changes: Reactions involving phase transitions may have discontinuities in thermodynamic properties.

Advanced calculations often use Gibbs-Helmholtz and van’t Hoff equations to relate temperature, equilibrium constants, and thermodynamic parameters.

Summary of Key Equations for Reference

Further Reading and Authoritative Resources

• NIST Chemistry WebBook – Comprehensive thermodynamic data for chemical species.
• LibreTexts: Gibbs Free Energy – Detailed explanations of thermodynamic principles.
• ChemEurope: Gibbs Free Energy – Overview of Gibbs free energy and applications.
• ACS Publications: Thermodynamics and Equilibrium – Peer-reviewed articles on thermodynamics.