science

Gibbs Free Energy: The Universe's Decision Function

Every reaction is a contest between two drives: release energy, or spread out. Gibbs free energy is the referee, and temperature holds the whistle.

Gibbs free energy (G) is the energy in a system that is available to do useful work at constant temperature and pressure. Its change, ΔG = ΔH − TΔS, weighs the heat a reaction releases or absorbs (ΔH) against the entropy it creates (ΔS), with temperature (T) deciding how much the entropy counts. If ΔG is negative, the reaction can proceed on its own; if it is positive, it can't without an input of energy. And ΔG says nothing at all about how fast anything happens.

That is the whole idea. The rest of this essay is about why one subtraction carries so much weight.

The one number that decides what happens

In 1871, Yale appointed Josiah Willard Gibbs professor of mathematical physics, the first such chair in the United States. He had not yet published anything, and he was hired without a salary. He lived on inherited income and stayed unpaid for nine years, until Johns Hopkins tried to lure him to Baltimore in 1880 and Yale finally found the money to keep him (Wikipedia; Encyclopedia.com).

Those unpaid years were not idle. Gibbs wrote the memoir On the Equilibrium of Heterogeneous Substances, published in two parts in 1876 and 1878 in the Transactions of the Connecticut Academy (Scientific Papers of J. Willard Gibbs, Vol. 1). Few readers of that small journal could follow it. One who could was James Clerk Maxwell, who sculpted a clay model of Gibbs's thermodynamic surface with his own hands, made plaster casts of it, and mailed one to New Haven. That cast is on display in the Yale physics department (Wikipedia).

Gibbs defined the function that now bears his name. Its job is to settle an argument that chemistry kept having with itself. Some reactions run because they release heat. Others run while absorbing heat, as long as they spread matter and energy out. Neither rule works alone. Gibbs gave a single quantity that includes both, and it works.

Reading the equation term by term

The definition is G = H − TS. For a process at constant temperature and pressure, the change is:

ΔG = ΔH − TΔS

Read it one piece at a time.

ΔG is the verdict: the change in Gibbs free energy, in kJ/mol. Its sign tells you the direction a process can go. Negative means it can run forward on its own. Positive means the reverse is favoured instead. Zero means the system is at equilibrium. Why the sign carries all that weight is the subject of the next essay in this series, on what ΔG actually tells you.

ΔH is the enthalpy change, the heat absorbed or released at constant pressure. Negative ΔH means the reaction gives off heat (exothermic). Positive means it soaks heat up (endothermic). Bonds broken cost energy; bonds formed release it.

T is absolute temperature, in kelvin. It never goes negative, so it can't flip a sign by itself. What it does is scale the entropy term.

ΔS is the entropy change. The tempting gloss is "the disorder term," but that picture misleads more than it helps. A better reading: ΔS measures how much more widely energy and matter are spread over available microstates after the change than before it. Melting, dissolving, mixing and making gas from a solid all tend to raise it. Freezing and crystallizing tend to lower it.

The minus sign is doing real work. When ΔS is positive, −TΔS is negative and pulls ΔG down. That is the second law's preference for spreading out, written into the bookkeeping.

Units, and the classic trap

Tables give ΔH in kJ/mol but S and ΔS in J/(mol·K). The OpenStax table of standard thermodynamic data, for example, lists liquid water at S° = 70.0 J/(mol·K) and its enthalpy of formation at −285.83 kJ/mol (OpenStax, Appendix G). Multiply ΔS by T and you get J/mol, so convert before you subtract. Forgetting the factor of 1,000 is the most common way to get the sign of ΔG wrong on an exam.

Enthalpy vs entropy, with temperature as referee

There are two drives in every reaction. One pulls toward lower energy: release heat, form strong bonds. The other pulls toward spreading out: more ways for energy and matter to be arranged. Often they agree. When they disagree, the equation settles the matter, and temperature sets the exchange rate between them.

At low temperature, TΔS is small and ΔH dominates. Heat-releasing processes win, even ones that tidy matter up, such as water freezing.

At high temperature, TΔS grows until it can outweigh ΔH. Processes that spread things out win, even ones that absorb heat, such as ice melting.

Somewhere in between, the two contributions cancel and ΔG = 0. Setting ΔH − TΔS = 0 and solving gives the crossover:

T = ΔH / ΔS

This works only when ΔH and ΔS have the same sign. When they have opposite signs, there is no crossover and temperature can't change the verdict.

The four sign cases

Because T is always positive, the signs of ΔH and ΔS fix the possibilities. OpenStax lays out the same four cases (OpenStax Chemistry 2e, 16.4). The examples below use only standard data from the OpenStax tables, so you can check every one.

ΔH	ΔS	ΔG at low T	ΔG at high T	Example
−	+	negative	negative	Diamond → graphite (ΔH = −1.89 kJ/mol, ΔS = +3.36 J/(mol·K)): favoured at every temperature at 1 bar
+	−	positive	positive	Graphite → diamond at 1 bar: favoured at no temperature
−	−	negative	positive	Water freezing: favoured below 0 °C
+	+	positive	negative	Ice melting: favoured above 0 °C

The diamond numbers come from the standard values for carbon: graphite S° = 5.740 J/(mol·K); diamond ΔH°f = 1.89 kJ/mol and S° = 2.38 J/(mol·K) (OpenStax, Appendix G). At 298.15 K that gives ΔG = −1.89 − (298.15 × 0.00336) ≈ −2.89 kJ/mol for diamond turning into graphite.

So every diamond is losing an argument with thermodynamics. It just isn't losing it quickly. A process can be favoured and still effectively frozen, which is the whole story of why diamonds are thermodynamically doomed but kinetically stuck. The first two rows are the easy ones: enthalpy and entropy agree. The last two are where temperature decides the winner.

Worked example: why ice melts at 0 °C

Take H₂O(s) → H₂O(l) at 1 atm. The enthalpy of fusion of ice is ΔH = +6.01 kJ/mol (OpenStax Chemistry 2e, 10.3). The entropy change for melting is ΔS = +22.1 J/(mol·K) (OpenStax Chemistry 2e, 16.3, Example 16.4).

Both signs are positive, so this is the fourth row of the table. Melting absorbs heat, which opposes it. Melting spreads water molecules over far more arrangements, which favours it. The crossover is:

T = ΔH / ΔS = 6,010 J/mol ÷ 22.1 J/(mol·K) ≈ 272 K

That is within about a degree of the real melting point, 273.15 K. The small gap is rounding in the tabulated values: run the arithmetic backwards and the melting point implies ΔS = 6,010 ÷ 273.15 ≈ 22.0 J/(mol·K).

Now check either side of it.

At +10 °C (283.15 K): ΔG = 6,010 − (283.15 × 22.1) ≈ −248 J/mol. Negative. Ice melts.

At −10 °C (263.15 K): ΔG = 6,010 − (263.15 × 22.1) ≈ +194 J/mol. Positive. Ice stays put, and water freezes instead.

The melting point isn't a property someone looked up and memorized. It is the temperature where the heat cost and the entropy gain exactly cancel. The same logic, run on solids, liquids and gases together, explains why phase changes happen at sharp temperatures.

ΔG vs ΔG°: the mix-up that costs marks

ΔG° (read "delta G naught") is the free-energy change under standard conditions: 1 bar for gases, 1 M for solutes, at a stated temperature, usually 298 K. It is a fixed number for a given reaction, and it is what the tables give you.

ΔG is the free-energy change under the conditions you actually have. The two are linked by the reaction quotient Q, the ratio of products to reactants right now:

ΔG = ΔG° + RT ln Q

As a reaction runs, Q changes and ΔG drifts toward zero. When ΔG reaches zero, the reaction has nowhere further to go: that is equilibrium, Q equals the equilibrium constant K, and the equation becomes ΔG° = −RT ln K (OpenStax Chemistry 2e, 16.4). A negative ΔG° means K > 1 and products dominate at equilibrium. A positive ΔG° means K < 1, which is not the same as "never happens." It means the reaction stops early. What equilibrium looks like up close, and why it is not static, is its own essay on chemical equilibrium and the point where ΔG = 0.

The common error is to read the sign of ΔG° as the final verdict. It isn't. The sign of ΔG, under your concentrations, is.

Why "free"? And why constant temperature and pressure?

"Free" here means available, not costless. Of the energy a reaction releases as heat (ΔH), some has to go into the surroundings as TΔS to satisfy the second law. What is left, ΔG, is the energy free to do useful work. Run the process reversibly and ΔG is the maximum work you can extract (OpenStax Chemistry 2e, 16.4).

Constant temperature and pressure are the conditions of most chemistry on Earth: open flasks, living cells, the atmosphere pressing down at about one bar. Under those conditions, a negative ΔG for the system is equivalent to an increase in the entropy of the universe. That is why Gibbs's function is so useful. It lets you track only your system and still obey the second law. At constant volume instead of constant pressure, chemists use the closely related Helmholtz free energy.

Coupled reactions

A reaction with positive ΔG can still happen if it is chained to one with a larger negative ΔG, because free-energy changes add. Cells do this constantly. Hydrolysing ATP to ADP and phosphate has ΔG° of −30.5 kJ/mol, and making the amino acid glutamine from glutamic acid and ammonia needs +14.2 kJ/mol, so the enzyme glutamine synthetase couples the two and the combined reaction runs downhill (Alberts et al., Essential Cell Biology, 6th ed., ch. 3). How much ATP really delivers inside a cell, and why that number differs from the textbook one, is the subject of ATP hydrolysis and the −30.5 kJ/mol that runs your body.

A negative ΔG is permission, not a deadline

One more warning, because it trips up nearly everyone. "Spontaneous" in thermodynamics means "can proceed without outside energy." It does not mean fast, and it does not mean soon. ΔG compares where a reaction starts and where it ends; it says nothing about the hill in between. That hill, the activation energy, is what makes a diamond last and a match need striking. If you only remember one thing from this series, make it that a negative ΔG doesn't mean a reaction is fast.

FAQ

What does a negative ΔG mean?

It means the reaction is thermodynamically favoured under the stated conditions: it can proceed without an outside energy input, and the products have lower free energy than the reactants. It does not tell you how quickly that will happen.

Is ΔG the same as ΔG°?

No. ΔG° is the change under standard conditions (1 bar, 1 M). ΔG is the change under actual conditions, given by ΔG = ΔG° + RT ln Q. A reaction with positive ΔG° can still run if the actual concentrations make ΔG negative.

Why is it called "free" energy?

"Free" means available to do work. ΔG is the part of the energy change left over after the entropy cost to the surroundings is paid, and it equals the maximum useful work a reversible process could deliver at constant temperature and pressure.

Does a negative ΔG mean a reaction is fast?

No. Speed is set by kinetics, chiefly the activation energy. Diamond converting to graphite has a negative ΔG at room temperature and still doesn't happen on any timescale you will notice.

What units is Gibbs free energy measured in?

Usually kJ/mol. Watch the entropy term: ΔS is normally tabulated in J/(mol·K), so convert it to kJ before combining it with ΔH.

Who was Josiah Willard Gibbs?

An American scientist (1839-1903) who spent his career at Yale. His 1876-1878 memoir on heterogeneous equilibrium laid the foundations of chemical thermodynamics. He received the Royal Society's Copley Medal in 1901 (Wikipedia).

Where to go next: the Gibbs Free Energy series

Gibbs Free Energy Explained (you are here)
What Is Delta G? The sign that decides direction.
Spontaneous Reactions Why "spontaneous" isn't about speed.
The Gibbs-Helmholtz Equation How heating changes which reactions are favoured.
Chemical Equilibrium Where ΔG reaches zero.
ATP Hydrolysis The energy currency, by the numbers.
Protein Folding Finding the free-energy minimum.
Phase Transitions Why ice melts at 0 °C.
Diamonds and Graphite Favoured is not the same as fast.
Membrane Potential Free energy stored across a barrier.
Synthesis: Free Energy as the Logic of Becoming

The pebble: a reaction doesn't choose between lower energy and more freedom. It takes whichever deal ΔG says is better at the temperature on offer.

Gibbs Free Energy: The Universe's Decision Function

The one number that decides what happens