Gibbs Free Energy

Gibbs free energy (G) is the state function that ties together everything else in this chapter — temperature, enthalpy, and entropy — into a single quantity that predicts whether a reaction happens on its own. This article covers the ΔG formula, exergonic and endergonic reactions, how the signs of ΔH and ΔS together determine spontaneity, standard free energy and standard free energy of formation, and how ΔG connects quantitatively to the equilibrium constant and reaction quotient.

Key Takeaways

• Gibbs free energy (G) combines temperature, enthalpy, and entropy: ΔG = ΔH − TΔS, indicating spontaneity at constant temperature and pressure.

• Exergonic reactions release energy (products lower-energy than reactants, e.g. combustion); endergonic reactions absorb energy (products higher-energy than reactants, e.g. photosynthesis).

• Spontaneity doesn't mean fast — a spontaneous reaction can still proceed very slowly without a catalyst.

• The signs of ΔH and ΔS together determine spontaneity: same-direction signs (ΔH−/ΔS+ or ΔH+/ΔS−) are temperature-independent; opposite-direction signs (ΔH−/ΔS− or ΔH+/ΔS+) depend on temperature.

• Standard free energy (ΔG°rxn) is measured at standard-state conditions (1 M solutions); standard free energy of formation (ΔG°f) of an element is zero, and ΔG°rxn = ΣnΔG°f(products) − ΣnΔG°f(reactants).

• ΔG°rxn = −RT ln Keq connects standard free energy to the equilibrium constant — a larger Keq means a more negative ΔG°rxn and a more spontaneous reaction.

• For a reaction not at standard conditions, ΔGrxn = ΔG°rxn + RT ln Q, using the reaction quotient Q in place of Keq.

What Is Gibbs Free Energy?

Gibbs free energy combines three quantities already covered elsewhere in this chapter — temperature, enthalpy, and entropy:

ΔG = ΔH − TΔS

ΔG, the change in Gibbs free energy, measures the change in enthalpy and entropy as a system undergoes a process at constant temperature and pressure, and it indicates whether that process is spontaneous or non-spontaneous. Specifically, ΔG is the maximum amount of energy released by a constant-temperature, constant-pressure process that's available to do useful work. (The formula itself, its sign rule, and the distinction between spontaneity and reaction rate are covered in detail earlier in this course — see Free Energy.)

Exergonic and Endergonic Reactions

Two terms describe reactions based on the direction of energy flow:

• Exergonic reactions release energy: the products end up at lower energy than the reactants. The energy difference is released, usually as heat, light, or sound. Burning wood in a fireplace is exergonic — the wood's chemical energy is released as heat and light, leaving behind ash at lower energy than the original wood.

• Endergonic reactions absorb energy: the products end up at higher energy than the reactants, with the reaction drawing energy in from its surroundings. Photosynthesis is endergonic — plants absorb light energy to convert carbon dioxide and water into glucose and oxygen, and the resulting glucose holds more energy than the CO₂ and water it was built from.

Spontaneity Doesn't Mean Speed

It's worth repeating: a spontaneous reaction is one that can occur without an input of energy — it says nothing about how quickly that reaction happens. Some spontaneous reactions proceed very slowly without a catalyst to speed them along. Spontaneity and rate are two separate questions, answered by two separate branches of chemistry (thermodynamics and kinetics, respectively).

How ΔH and ΔS Determine Spontaneity

Because ΔG = ΔH − TΔS, the signs of ΔH and ΔS — and the temperature — together determine whether ΔG comes out negative (spontaneous) or positive (non-spontaneous):

The two straightforward cases (top two rows) are spontaneous — or not — regardless of temperature, because ΔH and −TΔS push in the same direction. The two temperature-dependent cases (bottom two rows) are a tug-of-war between the ΔH and −TΔS terms, and which one wins depends on how large T is.

Standard Free Energy

The free energy change of a reaction can be measured under standard state conditions — for solutions, this means concentrations of 1 M — to yield the standard free energy of reaction, ΔG°rxn.

The standard free energy of formation, ΔG°f, of a compound is the free energy change when 1 mole of that compound, in its standard state, forms from its elements in their standard states, under standard conditions (298 K and 1 atm). By definition, the ΔG°f of any element in its standard state is zero.

Calculating ΔG°rxn from ΔG°f Values

Just like standard enthalpy of reaction, the standard free energy of a reaction can be calculated from the standard free energies of formation of its reactants and products:

ΔG°rxn = ΣnΔG°f(products) − ΣnΔG°f(reactants)

Worked example — standard free energy of the Haber process, N₂(g) + 3H₂(g) → 2NH₃(g), using standard values ΔG°f[NH₃(g)] = −16.5 kJ/mol and ΔG°f[N₂(g)] = ΔG°f[H₂(g)] = 0 kJ/mol (elements in their standard states):

• Products: 2 × (−16.5) = −33.0 kJ/mol

• Reactants: 0 + 3(0) = 0 kJ/mol

• ΔG°rxn = −33.0 − 0 = −33.0 kJ/mol

The negative value confirms the Haber process is spontaneous at standard conditions — consistent with why it can run at all, even though industrially it still needs a catalyst and elevated temperature/pressure to proceed at a useful rate (spontaneity says nothing about speed).

ΔG°rxn and the Equilibrium Constant

Standard free energy connects directly to the equilibrium constant, Keq, through:

ΔG°rxn = −RT ln Keq

where R is the gas constant and T is absolute temperature. This equation supports both quantitative calculations of free energy change and qualitative reasoning about spontaneity: the greater the value of Keq, the more positive its natural logarithm; the more positive that logarithm, the more negative ΔG°rxn; and the more negative ΔG°rxn, the more spontaneous the reaction.

ΔG for Reactions in Progress

ΔG°rxn and the equation above only apply under standard-state conditions — specifically, 1 M solution concentrations. Once a reaction actually gets underway, those conditions no longer hold, so Keq must be replaced with a value reflecting where the reaction currently sits on its path toward equilibrium: the reaction quotient, Q (see Equilibrium for a full treatment of Q vs. Keq). To find the free energy change for a reaction in progress, ΔGrxn (not ΔG°rxn) relates to Q by:

ΔGrxn = ΔG°rxn + RT ln Q

This lets you determine the actual free energy change — and therefore spontaneity — of a reaction at any point along its path, not just at standard-state conditions.

Common MCAT Mistakes

• Forgetting ΔG°rxn = −RT ln Keq needs a negative sign check. A large Keq (favoring products) gives a large positive ln Keq, which the leading minus sign turns into a large negative ΔG°rxn — students sometimes flip the sign and conclude large Keq means large positive ΔG°rxn.

• Assuming ΔG°f of a compound is always negative. Only elements in their standard states have ΔG°f = 0 by definition; a compound's ΔG°f can be positive, negative, or zero depending on its own stability relative to its elements.

• Treating a temperature-dependent case (ΔH and ΔS same sign) as always spontaneous or always non-spontaneous. When ΔH and ΔS have the same sign, spontaneity flips at a specific temperature — it's not a fixed answer the way the two straightforward cases are.

• Confusing ΔG°rxn (standard conditions) with ΔGrxn (actual, in-progress conditions). ΔG°rxn = −RT ln Keq only holds at standard state; a reaction already underway needs ΔGrxn = ΔG°rxn + RT ln Q, using Q rather than Keq.

MCAT-Style Concept Check

Question: A reaction has ΔH = −40 kJ/mol and ΔS = −60 J/(mol·K). Which statement best describes its spontaneity?

• A) The reaction is spontaneous at all temperatures, since ΔH is negative.

• B) The reaction is non-spontaneous at all temperatures, since ΔS is negative.

• C) The reaction is spontaneous only at low temperatures, since ΔH and −TΔS push in opposite directions.

• D) The reaction is spontaneous only at high temperatures, since a larger T makes −TΔS more favorable.

Answer: C

Explanation: With ΔH negative (favoring spontaneity) and ΔS also negative (the −TΔS term unfavorable, since −T times a negative ΔS gives a positive contribution to ΔG), the two terms in ΔG = ΔH − TΔS work against each other. At low T, the small −TΔS term can't outweigh the negative ΔH, so ΔG stays negative and the reaction is spontaneous; at high T, the growing −TΔS term eventually makes ΔG positive, shutting off spontaneity. This is the "spontaneous only at low temperatures" case from the ΔH/ΔS sign table.

FAQ

What is Gibbs free energy?

Gibbs free energy (G) is a state function combining temperature, enthalpy, and entropy that predicts whether a process is spontaneous at constant temperature and pressure. Its change is given by ΔG = ΔH − TΔS; a negative ΔG indicates a spontaneous process.

What's the difference between exergonic and endergonic reactions?

Exergonic reactions release energy, ending with products at lower energy than the reactants (e.g. combustion). Endergonic reactions absorb energy, ending with products at higher energy than the reactants (e.g. photosynthesis).

How does ΔG°rxn relate to the equilibrium constant?

ΔG°rxn = −RT ln Keq. A larger Keq produces a more positive ln Keq, which — after the leading negative sign — produces a more negative ΔG°rxn, meaning a more spontaneous reaction.

What's the standard free energy of formation of an element?

By definition, the standard free energy of formation (ΔG°f) of any element in its standard state is zero, since ΔG°f measures the free energy change of forming a compound from its elements — an element forming from itself involves no change.