Understanding the Calculation of Reaction Spontaneity through the Sign of ΔG

Reaction spontaneity determines whether a chemical process occurs naturally without external input. Calculating the sign of ΔG reveals this fundamental property.

This article explores the detailed methods to calculate Gibbs free energy change (ΔG), interpret its sign, and apply it to real-world chemical reactions.

• Calculate the spontaneity of a reaction with ΔH = -50 kJ/mol and ΔS = 100 J/mol·K at 298 K.
• Determine the sign of ΔG for the reaction: N2 + 3H2 → 2NH3 at 500 K given ΔH and ΔS values.
• Find the temperature at which a reaction with ΔH = 20 kJ/mol and ΔS = 150 J/mol·K becomes spontaneous.
• Analyze the spontaneity of the combustion of methane at standard conditions using ΔG calculations.

Comprehensive Tables of Common Values for Reaction Spontaneity Calculations

Fundamental Formulas for Calculating Reaction Spontaneity and Their Variables

The spontaneity of a chemical reaction is primarily determined by the Gibbs free energy change (ΔG). The core formula is:

• ΔG: Gibbs free energy change (kJ/mol). Negative values indicate spontaneous reactions.
• ΔH: Enthalpy change (kJ/mol). Represents heat absorbed or released.
• T: Absolute temperature (Kelvin). Must be in Kelvin for accuracy.
• ΔS: Entropy change (J/mol·K). Reflects disorder change in the system.

Note that ΔS is often given in J/mol·K, so it must be converted to kJ/mol·K by dividing by 1000 to maintain unit consistency with ΔH.

Another important relationship connects ΔG to the equilibrium constant (K) of the reaction:

• ΔG°: Standard Gibbs free energy change (kJ/mol), measured under standard conditions.
• R: Universal gas constant, 8.314 J/mol·K (or 0.008314 kJ/mol·K).
• T: Temperature in Kelvin.
• Q: Reaction quotient, ratio of product and reactant concentrations at any point.

At equilibrium, ΔG = 0 and Q = K, so the equation simplifies to:

This formula allows calculation of the equilibrium constant from thermodynamic data or vice versa.

Additional Useful Formulas

• Temperature for spontaneity change: When ΔG = 0, solve for T:
  T = ΔH / ΔS
• Conversion of entropy units:
  ΔS (kJ/mol·K) = ΔS (J/mol·K) / 1000

Detailed Real-World Examples of Reaction Spontaneity Calculations

Example 1: Spontaneity of the Formation of Ammonia (Haber Process)

The Haber process synthesizes ammonia from nitrogen and hydrogen gases:

N₂(g) + 3H₂(g) → 2NH₃(g)

Given data at 500 K:

• ΔH = -92.4 kJ/mol (exothermic)
• ΔS = -198.3 J/mol·K (entropy decreases due to fewer gas molecules)

Calculate ΔG and determine if the reaction is spontaneous at 500 K.

Step 1: Convert ΔS to kJ/mol·K:

Step 2: Calculate ΔG:

Interpretation: ΔG is positive, so the reaction is non-spontaneous at 500 K under these conditions.

However, lowering the temperature favors spontaneity due to the exothermic nature and negative entropy change.

Example 2: Temperature at Which a Reaction Becomes Spontaneous

Consider a reaction with:

• ΔH = 20 kJ/mol (endothermic)
• ΔS = 150 J/mol·K (entropy increases)

Find the temperature at which the reaction becomes spontaneous (ΔG = 0).

Step 1: Convert ΔS to kJ/mol·K:

Step 2: Calculate T:

Interpretation: Above 133.33 K, ΔG becomes negative, and the reaction is spontaneous. Below this temperature, it is non-spontaneous.

Expanded Discussion on Variables and Their Impact on Reaction Spontaneity

The sign of ΔG is influenced by the interplay between enthalpy and entropy changes, modulated by temperature. Understanding each variable’s role is critical for accurate spontaneity predictions.

• Enthalpy (ΔH): Reflects heat exchange. Exothermic reactions (negative ΔH) tend to be spontaneous at lower temperatures.
• Entropy (ΔS): Measures disorder. Positive ΔS favors spontaneity, especially at higher temperatures.
• Temperature (T): Acts as a scaling factor for entropy’s contribution. High temperatures amplify the TΔS term.
• Gibbs Free Energy (ΔG): The net driving force. Negative ΔG means the reaction can proceed without external energy.

In reactions where ΔH and ΔS have the same sign, temperature determines spontaneity:

• ΔH 0: Reaction always spontaneous (ΔG negative).
• ΔH > 0 and ΔS < 0: Reaction never spontaneous (ΔG positive).
• ΔH < 0 and ΔS < 0: Spontaneous at low temperatures.
• ΔH > 0 and ΔS > 0: Spontaneous at high temperatures.

Practical Considerations and Advanced Applications

In industrial chemistry, controlling reaction conditions to optimize ΔG is essential. For example, the Haber process operates at moderate temperatures and high pressures to balance kinetics and thermodynamics.

Electrochemical cells also rely on ΔG calculations to predict cell potential and feasibility. The relationship between Gibbs free energy and electromotive force (E) is:

• n: Number of moles of electrons transferred.
• F: Faraday constant (96485 C/mol).
• E: Cell potential (Volts).

This formula links thermodynamics with electrochemistry, enabling prediction of spontaneous redox reactions.

Summary of Key Points for Optimized Calculation of Reaction Spontaneity

• Always convert entropy units to kJ/mol·K for consistency.
• Use absolute temperature in Kelvin.
• Interpret the sign of ΔG carefully: negative means spontaneous.
• Apply the relationship between ΔG, ΔH, ΔS, and T to predict temperature-dependent spontaneity.
• Utilize equilibrium constant relations to connect thermodynamics with reaction extent.
• Consider real-world constraints such as pressure, concentration, and kinetics.

Recommended Authoritative Resources for Further Study

• LibreTexts: Gibbs Free Energy and Spontaneity
• American Chemical Society: Thermodynamics and Reaction Spontaneity
• LibreTexts: Equilibrium Constants and Gibbs Free Energy
• NIST Thermochemical Databases