Gibbs Free Energy Calculations and Spontaneity in Chemical Reactions

Gibbs Free Energy Calculations

Concept of Gibbs Free Energy

The Gibbs Free Energy formula allows chemists to determine the spontaneity of a chemical reaction by using enthalpy ($\Delta H$), entropy ($\Delta S$), and temperature (T) values. The sign of $\Delta G$ indicates whether a reaction is spontaneous under given conditions.

• Formula:

$\Delta G = \Delta H - T\Delta S$

• $\Delta G < 0$: Reaction is spontaneous.

• $\Delta G > 0$: Reaction is nonspontaneous.

• $\Delta G = 0$: System is at equilibrium.

Example Calculation

For a reaction with $\Delta H = 111.4$ kJ and $\Delta S = 20.8$ J/K, calculate $\Delta G$ at 298 K:

• Convert $\Delta S$ to kJ: $20.8$ J/K $= 0.0208$ kJ/K

• Apply the formula:

$\Delta G = 111.4\ \text{kJ} - (298\ \text{K})(0.0208\ \text{kJ/K}) = 111.4\ \text{kJ} - 6.1984\ \text{kJ} = 105.2\ \text{kJ}$

• Since $\Delta G > 0$, the reaction is nonspontaneous at 298 K.

Temperature Conditions and Spontaneity

Effect of Temperature on Spontaneity

The spontaneity of a reaction can depend on temperature, especially when $\Delta H$ and $\Delta S$ have the same sign. The temperature at which a reaction becomes spontaneous can be calculated by setting $\Delta G = 0$:

$0 = \Delta H - T\Delta S \implies T = \frac{\Delta H}{\Delta S}$

• If $\Delta H > 0$ and $\Delta S > 0$, reaction is spontaneous at high temperatures.

• If $\Delta H < 0$ and $\Delta S < 0$, reaction is spontaneous at low temperatures.

Example: Temperature Dependence

Given $\Delta H = 31$ kJ, $\Delta S = 82$ J/K, calculate the temperature above which the reaction is spontaneous:

• Convert $\Delta S$ to kJ: $82$J/K$= 0.082$ kJ/K

• Set $\Delta G = 0$:

$T = \frac{31\ \text{kJ}}{0.082\ \text{kJ/K}} = 378.0\ \text{K}$

• Reaction is spontaneous above 378 K.

Gibbs Free Energy of Reaction

Calculating $\Delta G^\circ$ from Standard Free Energies of Formation

The standard Gibbs free energy change of a reaction ($\Delta G^\circ_{rxn}$) can be calculated using the standard free energies of formation ($\Delta G^\circ_f$) of reactants and products:

$\Delta G^\circ_{rxn} = \sum n \Delta G^\circ_f(\text{products}) - \sum n \Delta G^\circ_f(\text{reactants})$

• Elements in their standard state have $\Delta G^\circ_f = 0$.

Example Calculation

For the reaction: $\text{HNO}_3(g) + \text{NH}_3(g) \rightarrow \text{NH}_4\text{NO}_3(s)$

• Given $\Delta G^\circ_f$ values:

  • $\text{HNO}_3(g): -78.9$ kJ/mol

  • $\text{NH}_3(g): -16.4$ kJ/mol

  • $\text{NH}_4\text{NO}_3(s): -183.8$ kJ/mol

• Calculation:

$\Delta G^\circ_{rxn} = [-183.8] - [-78.9 + (-16.4)] = -183.8 + 95.3 = -88.5$ kJ

• Negative $\Delta G^\circ_{rxn}$ indicates a spontaneous reaction under standard conditions.

Practice Problems

Sample Calculations

• Calculate $\Delta G$ for $\text{P}_4(s) + 5\text{O}_2(g) \rightarrow \text{P}_4\text{O}_{10}(s)$ at 25°C, given $\Delta H^\circ = -2984$ kJ/mol and $\Delta S^\circ = 284.5$ J/K.

• Determine if a reaction is spontaneous under standard conditions, and at what temperature it becomes spontaneous.

• Given $\Delta H_{fus}$ and $\Delta S_{fus}$ for nickel, predict if it will melt at a given temperature and pressure.

HTML Table: Standard Gibbs Free Energies of Formation

The following table summarizes standard Gibbs free energies of formation for selected substances (values in kJ/mol):

Substance	$\Delta G^\circ_f$ (kJ/mol)
HNO3(g)	-78.9
NH3(g)	-16.4
NH4NO3(s)	-183.8

Key Terms

• Gibbs Free Energy ($\Delta G$): Thermodynamic quantity indicating the maximum reversible work obtainable from a system at constant temperature and pressure.

• Spontaneous Reaction: A reaction that occurs without external input; $\Delta G < 0$.

• Standard State: The most stable form of a substance at 1 atm and 25°C.

Additional info: The notes include practice problems and tables for standard Gibbs free energies of formation, which are essential for mastering thermodynamic calculations in General Chemistry.