AP Chemistry 9.3-9.4: Gibbs Free Energy and Thermodynamic Control

Gibbs free energy

Gibbs free energy is a thermodynamic state function defined by $G = H - TS$. At constant temperature and pressure, it provides a system-only criterion for spontaneity.

What equation relates the free-energy change of a process to its enthalpy and entropy changes at constant temperature?

$\Delta G = \Delta H - T\Delta S$, where $T$ is the absolute temperature in kelvins.

How is Gibbs free energy related to the entropy change of the universe?

$\Delta G = -T\Delta S_{\text{univ}}$ at constant temperature and pressure. Thus, the signs of $\Delta G$ and $\Delta S_{\text{univ}}$ are opposite.

What does the sign of $\Delta G$ indicate about spontaneity at constant temperature and pressure?

$\Delta G < 0$ indicates a spontaneous forward process, $\Delta G > 0$ indicates a nonspontaneous forward process, and $\Delta G = 0$ indicates equilibrium.

Does a negative $\Delta G$ indicate that a reaction will be fast?

No. A negative $\Delta G$ indicates thermodynamic favorability, not reaction rate. Kinetics and activation energy determine how quickly the process occurs.

Why is Gibbs free energy useful for evaluating spontaneity?

Unlike $\Delta S_{\text{univ}}$, $\Delta G$ can be calculated using properties of the system alone: $\Delta G = \Delta H - T\Delta S$.

How does $\Delta G$ relate to useful work from a spontaneous process?

For a reversible process, the maximum non-expansion work obtainable is $-\Delta G$ when $\Delta G < 0$. Real processes produce less useful work because they are irreversible and practical devices are not perfectly efficient.

What minimum work is required to drive a nonspontaneous process under reversible conditions?

The minimum non-expansion work that must be supplied is $\Delta G$ when $\Delta G > 0$.

State function

A property whose change depends only on the initial and final states, not on the pathway. Gibbs free energy is a state function, so free-energy changes for coupled or multistep reactions can be added.

What is the standard free-energy change, $\Delta G^\circ$?

$\Delta G^\circ$ is the free-energy change when reactants and products are in their standard states. It depends on temperature, so the temperature should be specified when needed.

How can $\Delta G^\circ$ be calculated from standard enthalpy and entropy changes?

Use $\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ$. Convert $\Delta S^\circ$ to energy units consistent with $\Delta H^\circ$, such as converting joules to kilojoules.

What is the standard Gibbs free energy of formation, $\Delta G_f^\circ$?

It is the free-energy change for forming one mole of a substance from its elements in their standard states.

What is the value of $\Delta G_f^\circ$ for an element in its standard state?

$\Delta G_f^\circ = 0$ by definition for an element in its standard state, such as $\mathrm{O_2(g)}$ or a pure solid element in its standard form.

How is the standard free-energy change of a reaction calculated from standard free energies of formation?

$\Delta G^\circ = \sum \nu\Delta G_f^\circ(\text{products}) - \sum \nu\Delta G_f^\circ(\text{reactants})$, where each formation free energy is multiplied by its stoichiometric coefficient.

What determines whether spontaneity depends on temperature?

The signs and magnitudes of $\Delta H$ and $\Delta S$ in $\Delta G=\Delta H-T\Delta S$ determine whether changing $T$ can change the sign of $\Delta G$.

For a process with $\Delta H>0$ and $\Delta S>0$, how does spontaneity depend on temperature?

It is favored at sufficiently high temperatures because the negative $-T\Delta S$ term can outweigh positive $\Delta H$. It is nonspontaneous at sufficiently low temperatures.

For a process with $\Delta H<0$ and $\Delta S<0$, how does spontaneity depend on temperature?

It is favored at sufficiently low temperatures because the negative $\Delta H$ term dominates. At high temperatures, the positive contribution from $-T\Delta S$ can make $\Delta G$ positive.

What is the temperature dependence of spontaneity when $\Delta H>0$ and $\Delta S<0$?

$\Delta G$ is positive at every temperature, so the process is nonspontaneous at all temperatures.

What is the temperature dependence of spontaneity when $\Delta H<0$ and $\Delta S>0$?

$\Delta G$ is negative at every temperature, so the process is spontaneous at all temperatures.

At what temperature is a process at the boundary between spontaneous and nonspontaneous behavior when $\Delta H$ and $\Delta S$ have the same sign?

At the temperature where $\Delta G=0$: $T=\dfrac{\Delta H}{\Delta S}$. Use consistent energy units so the result is in kelvins.

Why can the boiling point of a liquid be estimated using $T=\Delta H^\circ/\Delta S^\circ$?

At the boiling point, liquid and vapor are at equilibrium, so $\Delta G=0$. Substituting into $\Delta G=\Delta H-T\Delta S$ gives $T=\Delta H/\Delta S$.

What is thermodynamic control?

Thermodynamic control occurs when the product distribution is determined by relative free energies and the system has sufficient opportunity to approach equilibrium. The more thermodynamically favorable products predominate.

What is kinetic control?

Kinetic control occurs when product distribution is determined primarily by reaction rates. A product with a lower activation energy can form predominantly even if it is not the most thermodynamically favorable product.

How do thermodynamic favorability and kinetic rate differ?

Thermodynamic favorability depends on $\Delta G$ and indicates the direction favored at equilibrium. Kinetic rate depends on activation energy and determines how rapidly the process proceeds; these properties are independent.

What conclusion follows if the calculated $\Delta G^\circ$ for a reaction is positive under standard conditions?

The forward reaction is nonspontaneous under those conditions, but the reverse reaction is thermodynamically favored. The reaction may still proceed if conditions change or if it is coupled to a favorable process.