Gibbs Free Energy

Table of Contents

Definition and Meaning of Gibbs Free Energy

In chemical thermodynamics, the Gibbs free energy $G$ is a state function that combines a system’s enthalpy and entropy in such a way that it predicts whether a process at constant temperature and pressure will occur spontaneously.

The Gibbs free energy is defined as
$$
G = H - TS
$$
where

$H$ = enthalpy of the system
$T$ = absolute temperature (in kelvin)
$S$ = entropy of the system

Because $G$ is a state function, its value depends only on the current state of the system (composition, temperature, pressure), not on the path taken to reach that state.

For a process or reaction, we are usually interested in the change in Gibbs free energy:
$$
\Delta G = \Delta H - T \Delta S
$$

Here, $\Delta H$ is the enthalpy change and $\Delta S$ is the entropy change for the process at the temperature $T$.

Criterion for Spontaneity at Constant $T$ and $p$

At constant temperature and pressure (the typical conditions for many chemical reactions in the laboratory and in living organisms), the sign of $\Delta G$ gives a direct criterion for spontaneity:

$\Delta G < 0$: the process can occur spontaneously (thermodynamically favorable).
$\Delta G = 0$: the system is at equilibrium; no net change occurs.
$\Delta G > 0$: the process is non-spontaneous in the forward direction; the reverse process is spontaneous.

This criterion is specific to constant temperature and pressure. Other conditions require different potentials, but these belong to other parts of thermodynamics.

Standard Gibbs Free Energy Change

For chemical reactions it is practical to use standard reference conditions (e.g. $p^\circ = 1\ \text{bar}$, standard concentrations $c^\circ = 1\ \text{mol·L}^{-1}$). Under these conditions, one defines the standard Gibbs free energy of formation and the standard Gibbs free energy change of a reaction.

Standard Gibbs Free Energy of Formation

The standard Gibbs free energy of formation $\Delta_\mathrm{f} G^\circ$ of a substance is the Gibbs free energy change when 1 mol of the substance is formed from its elements in their reference states, at standard conditions.

By convention, the elements in their reference states (e.g. $\mathrm{H_2(g)}$, $\mathrm{O_2(g)}$, $\mathrm{C(graphite)}$ at $p^\circ$) have
$$
\Delta_\mathrm{f} G^\circ = 0
$$

Values of $\Delta_\mathrm{f} G^\circ$ for compounds are tabulated and are used to calculate reaction Gibbs energies.

Standard Reaction Gibbs Free Energy

For a chemical reaction written in terms of stoichiometric coefficients $\nu_i$ (positive for products, negative for reactants), the standard Gibbs free energy change of reaction $\Delta_\mathrm{r} G^\circ$ is given by

$$
\Delta_\mathrm{r} G^\circ = \sum_{\text{products}} \nu_i \Delta_\mathrm{f} G_i^\circ - \sum_{\text{reactants}} \nu_i \Delta_\mathrm{f} G_i^\circ
$$

Because elements in their reference states have $\Delta_\mathrm{f} G^\circ = 0$, they do not contribute directly to the sum.

A negative $\Delta_\mathrm{r} G^\circ$ under standard conditions means that, starting from standard state reactants, the reaction mixture tends to evolve in the forward direction toward products.

Gibbs Free Energy and Maximum Useful Work

At constant temperature and pressure, the Gibbs free energy change is directly connected to the maximum amount of non-expansion work that can be extracted from a process.

For a reversible process at constant $T$ and $p$:
$$
\Delta G = w_{\text{non-expansion,max}}
$$
where $w_{\text{non-expansion,max}}$ is the maximum work other than $pV$ (pressure–volume) work (for example, electrical work in an electrochemical cell).

For a spontaneous process in reality (which is always irreversible), the actual work obtained is less in magnitude than this maximum:
$$
w_{\text{non-expansion,actual}} > \Delta G
$$
(keeping sign conventions in mind; work done by the system is usually taken as negative).

This connection is particularly important for electrochemical cells, where the electrical work is related to cell voltage; that connection is treated in detail elsewhere.

Temperature Dependence of $\Delta G$ and Competing Effects

Because
$$
\Delta G = \Delta H - T \Delta S
$$
the sign of $\Delta G$ depends on both enthalpy and entropy changes and on the temperature. Four qualitative cases are often discussed:

$\Delta H < 0$ and $\Delta S > 0$

$\Delta G$ is negative at all temperatures
Reaction is spontaneous under standard conditions regardless of $T$

$\Delta H < 0$ and $\Delta S < 0$

$\Delta G$ is negative at low $T$ (where $T|\Delta S|$ is small)
Reaction tends to be spontaneous only at lower temperatures

$\Delta H > 0$ and $\Delta S > 0$

$\Delta G$ is negative at high $T$ (where $T\Delta S$ dominates)
Reaction tends to be spontaneous only at higher temperatures

$\Delta H > 0$ and $\Delta S < 0$

$\Delta G$ is positive at all temperatures
Reaction is non-spontaneous under standard conditions regardless of $T$

This simple scheme highlights that endothermic reactions ($\Delta H > 0$) can still be spontaneous if they are accompanied by a sufficiently large increase in entropy and the temperature is high enough.

Gibbs Free Energy and Chemical Reactions

For a chemical reaction, the Gibbs free energy change depends on the extent of reaction and the composition of the reaction mixture. As composition changes, so does $G$; the system evolves spontaneously in the direction that lowers $G$ until a minimum is reached.

Reaction Gibbs Free Energy and Reaction Quotient

For a reaction at constant $T$ and $p$, the (instantaneous) Gibbs free energy change $\Delta_\mathrm{r} G$ can be related to the standard reaction Gibbs energy and the reaction quotient $Q$:
$$
\Delta_\mathrm{r} G = \Delta_\mathrm{r} G^\circ + RT \ln Q
$$
where

$R$ = gas constant
$T$ = absolute temperature
$Q$ = reaction quotient, constructed from activities (often approximated by concentrations or partial pressures)

In this form, the sign of $\Delta_\mathrm{r} G$ depends not only on $\Delta_\mathrm{r} G^\circ$ but also on the current composition of the system, via $Q$.

Direction of Spontaneous Change

Given the expression above:

If $\Delta_\mathrm{r} G < 0$: the reaction proceeds spontaneously in the forward direction (toward products).
If $\Delta_\mathrm{r} G > 0$: the reaction proceeds spontaneously in the reverse direction (toward reactants).
If $\Delta_\mathrm{r} G = 0$: the reaction is at equilibrium.

The system “moves” in composition such that $G$ decreases, and this continues until it reaches a composition where $\Delta_\mathrm{r} G = 0$.

This shows:

A large, negative $\Delta_\mathrm{r} G^\circ$ corresponds to a large equilibrium constant $K$; products are strongly favored.
A large, positive $\Delta_\mathrm{r} G^\circ$ corresponds to a very small $K$; reactants are strongly favored.
$\Delta_\mathrm{r} G^\circ \approx 0$ corresponds to $K$ of order 1; neither reactants nor products are strongly favored.

The detailed treatment of equilibrium and the law of mass action is covered separately; here, the key point is that Gibbs free energy provides the thermodynamic basis for the equilibrium constant.

Thermodynamic Interpretation of $G$ as a Potential

Analogous to how mechanical systems minimize potential energy, a system at constant $T$ and $p$ tends to minimize its Gibbs free energy.

$G$ plays the role of a thermodynamic potential appropriate for conditions of fixed temperature and pressure.
Spontaneous processes at constant $T$ and $p$ proceed in the direction of decreasing $G$.
Equilibrium corresponds to a minimum of $G$ with respect to allowed changes (e.g. composition changes at given $T$ and $p$).

Mathematically, for infinitesimal changes,
$$
\mathrm{d}G \le 0
$$
for spontaneous changes at constant $T$ and $p$, and at equilibrium $\mathrm{d}G = 0$ (with appropriate stability conditions).

Practical Aspects: Using Gibbs Free Energy in Chemistry

In practice, Gibbs free energy is used to answer questions such as:

Will a given reaction be thermodynamically favorable under specified conditions ($T$, $p$, concentrations)?
How does changing temperature affect the favorability of a reaction?
How far will a reaction proceed before reaching equilibrium (in qualitative terms)?
How much maximum useful (non-expansion) work could be obtained from a reaction or process?

To make such predictions:

Use tabulated $\Delta_\mathrm{f} G^\circ$ values to find $\Delta_\mathrm{r} G^\circ$ for the reaction.
If non-standard concentrations or pressures are involved, use
$$
\Delta_\mathrm{r} G = \Delta_\mathrm{r} G^\circ + RT \ln Q
$$
with the appropriate $Q$ for the actual composition.
Judge spontaneity by the sign of $\Delta_\mathrm{r} G$ at the given conditions.

The detailed computational techniques and examples fall under specific applications and equilibrium calculations, which are treated elsewhere; this chapter provides the thermodynamic foundation through the concept of Gibbs free energy.

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