16.4 Free Energy

Gibbs Free Energy and Spontaneity

Gibbs free energy and spontaneity

Gibbs free energy ($G$) is a thermodynamic quantity that predicts whether a process will be spontaneous at constant temperature and pressure. It combines enthalpy and entropy into a single value that tells you the direction a reaction naturally wants to go.

• Spontaneous processes occur without external intervention and release free energy. Rusting of iron is a classic example: it just happens on its own given enough time.
• Non-spontaneous processes require energy input to proceed. Electrolysis of water, where you run electric current through water to split it into hydrogen and oxygen, won't happen without that external energy.

The change in Gibbs free energy ($\Delta G$) is what you actually calculate to determine spontaneity:

• $\Delta G < 0$: Reaction is spontaneous in the forward direction (e.g., combustion of fuel)
• $\Delta G > 0$: Reaction is non-spontaneous in the forward direction (e.g., photosynthesis requires energy from sunlight)
• $\Delta G = 0$: Reaction is at equilibrium, with no net change occurring (e.g., a saturated solution)

Free energy calculations from formation data

The standard Gibbs free energy of formation ($\Delta G_f^\circ$) is the change in free energy when one mole of a compound forms from its elements in their standard states (25°C and 1 atm). By definition, $\Delta G_f^\circ$ for any element in its standard state is zero.

You calculate the standard change in Gibbs free energy for a reaction using this equation:

$\Delta G^\circ = \sum \Delta G_f^\circ(\text{products}) - \sum \Delta G_f^\circ(\text{reactants})$

This works just like Hess's Law calculations you've done with enthalpy. Multiply each compound's $\Delta G_f^\circ$ by its stoichiometric coefficient, then subtract the reactant total from the product total.

For example, in the combustion of methane ($CH_4 + 2O_2 \rightarrow CO_2 + 2H_2O$), you'd use 1 mol of $CO_2$ and 2 mol of $H_2O$ on the product side, and 1 mol of $CH_4$ on the reactant side. The $O_2$ term drops out because $\Delta G_f^\circ$ for elements in their standard state equals zero.

Free energy from enthalpy and entropy

Gibbs free energy change connects directly to enthalpy and entropy through this key equation:

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

• $\Delta H$: Change in enthalpy (heat released or absorbed, in J or kJ)
• $T$: Absolute temperature in Kelvin (convert from °C by adding 273.15)
• $\Delta S$: Change in entropy (measure of disorder, in J/mol·K)

To calculate $\Delta G$ using this equation:

1. Determine $\Delta H$ and $\Delta S$ for the reaction (from tables or calculation).
2. Make sure your units match. If $\Delta H$ is in kJ and $\Delta S$ is in J/K, convert one so they're consistent. This is a very common mistake on exams.
3. Multiply the absolute temperature ($T$) by $\Delta S$.
4. Subtract $T\Delta S$ from $\Delta H$ to get $\Delta G$.

Temperature effects on spontaneity

Temperature affects spontaneity by changing how much the entropy term ($T\Delta S$) contributes relative to the enthalpy term ($\Delta H$). At low temperatures, $T\Delta S$ is small, so $\Delta H$ dominates. At high temperatures, $T\Delta S$ grows and can overpower $\Delta H$.

This creates four possible sign combinations:

A good way to think about this: ice forming (exothermic, decreasing entropy) is spontaneous only at low temperatures. Water evaporating (endothermic, increasing entropy) is spontaneous only at high temperatures. These everyday examples map directly onto the table above.

Free energy changes vs equilibrium constants

Standard Gibbs free energy change is related to the equilibrium constant ($K$) by this equation:

$\Delta G^\circ = -RT \ln K$

• $R$: Gas constant (8.314 J/mol·K)
• $T$: Absolute temperature in Kelvin
• $\ln K$: Natural logarithm of the equilibrium constant

You can rearrange to solve for $K$:

$K = e^{-\Delta G^\circ / RT}$

The relationship between $\Delta G^\circ$ and $K$ makes intuitive sense:

• $\Delta G^\circ < 0$ corresponds to $K > 1$: products are favored at equilibrium
• $\Delta G^\circ > 0$ corresponds to $K < 1$: reactants are favored at equilibrium
• $\Delta G^\circ = 0$ corresponds to $K = 1$: neither side is favored

One thing to keep straight: $\Delta G^\circ$ tells you about the reaction under standard conditions, while $\Delta G$ (without the degree symbol) tells you about the reaction at whatever conditions you're actually at. A reaction can have a positive $\Delta G^\circ$ but still proceed forward if the concentrations are far from equilibrium.

Thermodynamics and Free Energy

Thermodynamics is the study of energy transformations in physical and chemical processes. Gibbs free energy, developed by Josiah Willard Gibbs, is one of its most useful tools because it wraps up both the energy and entropy aspects of a reaction into a single, calculable number.

Two related concepts worth knowing:

• Chemical potential is the change in Gibbs free energy when the amount of a substance in a system changes. It helps predict which direction matter will flow in a system.
• Reversibility refers to idealized processes that can be reversed without any net change in the system or surroundings. Real processes are always at least slightly irreversible, but reversible processes set the theoretical limit for efficiency.