Created in 1875–78 by J. Willard Gibbs and Hermann von Helmholtz

By Chapter 13 we had a calorimeter and a question it couldn’t answer. The calorimeter measures ΔH_rxn — the total heat effect of a reaction. But as Chapter 13 established, ΔH_rxn bundles together contributions from electron rearrangement, intermolecular force changes, and volume work, arriving as a single number with no indication of how much of it is available to do useful work. This is the problem Gibbs solved — and the property he created to solve it, Gibbs energy, is arguably the single most powerful tool in all of chemical thermodynamics.

Historical context: from Carnot to Thomsen-Berthelot

Sadi Carnot’s 1824 theoretical analysis of the steam engine asked a deceptively simple question: how much work can be generated from a bushel of coal? [1] Conservation of energy dictates that the maximum work obtainable from any process is bounded by the energy difference between input and output. But which energy? By 1750 the answer for the water wheel was clear — mechanical energy. Falling water delivered kinetic and potential energy directly. The steam engine broke this simplicity. Coal in, work out — but the two had no common unit of measure.

In 1854 Danish chemist Julius Thomsen, and independently in 1864 French chemist Marcellin Berthelot, proposed that the chemical energy of a reaction should be quantified by ΔH_rxn — the heat released in a calorimeter. [2] They reasoned that ΔH_rxn represents the maximum work obtainable from a reaction, and that an exothermic reaction (ΔH_rxn < 0) is spontaneous while an endothermic one cannot be. Their thermal theory of affinity worked reasonably well — for many reactions, though not all. Sometimes a single data point is enough to break a theory. Here, that data point was the spontaneous endothermic reaction. According to Thomsen and Berthelot, it wasn’t supposed to exist. And yet it did.

How Gibbs approached the problem

Gibbs recognized two things. First, when systems come together and energy gradients between them — temperature, pressure, chemical — dissipate, the entropy of the combined equilibrated system is greater than the sum of the entropies of the initial systems: S_combined ≥ ΣS_parts. Second, most chemical processes of practical interest occur at constant temperature and pressure. Putting these together, he constructed a new property of matter suited specifically to constant T,P conditions.

Here is how he did it. Assume a large constant-temperature bath — called a medium — and a body are placed inside an isolated, constant-volume vessel. A membrane separates them, conducting heat and responding to mechanical pressure, maintaining the body at the temperature and pressure of the medium at all times. The body may not be internally equilibrated — it may, for example, contain reactants undergoing a chemical reaction. The constraints on the total system are:

dU_body + dU_medium = 0 — total energy is constant

dV_body + dV_medium = 0 — total volume is constant

dS_body + dS_medium ≥ 0 — total entropy increases to a maximum at equilibrium

Because the medium remains fully equilibrated throughout and all its interactions with the body are reversible:

dU_medium = T dS_medium − P dV_medium

Making substitutions and flipping signs:

dU_body + P dV_body − T dS_body ≤ 0

Because both T and P are constant — and it is precisely this constancy that makes the next step possible — they can be drawn inside the differential:

d(U + PV − TS)_T,P ≤ 0

Gibbs isolated a new composite property of matter, later named Gibbs energy. This quantity — U + PV − TS, equivalently H − TS — must decrease to a minimum as the body equilibrates at constant T and P:

dG_T,P ≤ 0 for the body (not the medium, not the total system)

This inequality becomes an equality once the body reaches equilibrium.

Clarifying terminology: Gibbs energy (G) versus Gibbs free energy (ΔG)

G is sometimes called “free energy” — a term that survives from an earlier era and suggests energy that is somehow liberated or freely available. This is partially true but physically misleading. Gibbs energy is the more precise name and the one used throughout this book. Gibbs free energy is defined by dG or ΔG — the change in Gibbs energy. G itself carries no meaning; only the change in G does. This is consistent with the broader theme of this book: thermodynamics works with changes, not absolutes.

Gibbs himself called ΔG “available energy” in his 1875 publication — it quantifies the energy available in the internal gradients of a non-equilibrated system to do productive work. More on this topic here.

Maximum work

One of the most important consequences of the Gibbs energy framework is its connection to maximum work. Return to the inequality:

dG_T,P ≤ 0 for the body

As the body equilibrates, G decreases to a minimum — that is one way to turn the inequality into an equality. There is a second way. Rather than allowing the body to equilibrate freely, consider creating a precise balance between the non-equilibrated body and the lifting of an external weight — analogous to balancing a lever by placing equal weights on each side. When this balance is achieved, the process is reversible and the inequality becomes an equality. The external work extracted in this reversible process is by definition the maximum possible.

When Clausius originally wrote dU = TdS − PdV, the only work considered was PdV — volume change work. But other forms of external work are equally valid, such as lifting a weight or driving a current through a motor. The more general form is:

dU = TdS − PdV − Maximum Work for a reversible process

Rearranging:

Maximum Work = −(dU + PdV − TdS) = −dG_T,P

In finite terms:

Maximum Work = −(ΔH − TΔS) = −ΔG_T,P based on body properties only

The physical reasoning is direct. The total energy exchanged by a process at constant T and P is ΔH. But not all of that energy is available for work. A portion equal to TΔS must be exchanged with the surroundings as heat to maintain constant temperature — removed when TΔS is negative, supplied when TΔS is positive — and is therefore unavailable for conversion to useful work in either case. What remains, ΔH − TΔS = ΔG, is the maximum convertible to useful work.

A critical point: the entropy of the total system — body plus medium — remains constant throughout this reversible process. The body’s entropy change is exactly offset by the medium’s entropy change in the opposite direction. It is this total entropy constancy that defines the process as reversible and the work as maximum.

How to interpret ΔG

The Gibbs energy framework gives us a clean way to interpret the direction and limits of any process at constant T and P. Three cases arise.

• ΔG_T,P < 0 — The process is spontaneous. It will proceed on its own without external work being supplied. Internal gradients dissipate naturally, and external work can be extracted during this process, up to a maximum of −ΔG. The process will proceed even if the entropy of the body itself decreases, because the medium supplies the thermal energy needed to compensate. It is the total system — body plus medium — whose entropy is non-decreasing. The body alone does not need to obey that constraint.
• ΔG_T,P > 0 — The process will not proceed spontaneously in the proposed direction. External work must be supplied to make it happen. In this case −ΔG becomes negative, meaning it is now the minimum work required to drive the process rather than the maximum work extractable from it. Maximum and minimum work are two sides of the same coin.
• ΔG_T,P = 0 — More subtle than it first appears. This does not mean nothing happens. Calculated for complete conversion of reactants to products, ΔG = 0 simply means there is no thermodynamic driving force for complete conversion. A more careful accounting — one that tracks G as a function of the extent of reaction, including the entropy of mixing — will typically show that G is minimized at some partial conversion. If ΔG = 0 for complete conversion of A to B, then G is minimized for a mixture of A and B — perhaps a 50:50 mixture — because that mixture represents the most probable state and maximizes entropy. This is the connection between Gibbs energy minimization and Boltzmann’s most probable distribution, developed fully in the chapter on chemical reactions.

ΔG and the electrochemical cell

Constant T and P do more than simplify the mathematics — they convert the analysis from path-dependent to state-dependent. Maximum work is fixed entirely by the difference in Gibbs energy between initial and final states, regardless of path.

The electrochemical cell is the clearest physical embodiment of this framework, illustrating all three cases of ΔG in a single system. Consider a spontaneous reaction — ΔG < 0 — driving current through an external circuit. Apply a reverse voltage exactly equal to the cell’s own voltage and the reaction stops, held at the precise edge of equilibrium, extracting maximum reversible work equal to −ΔG. Increase that reverse voltage slightly and the reaction runs backward — a ΔG-positive process now driven by external work. Decrease it slightly and the forward reaction resumes. The entire range of spontaneous, equilibrium, and driven behavior is accessible simply by adjusting the voltage. Real cells fall short of this ideal due to irreversibilities, but −ΔG remains the theoretical upper bound.

This framework — born from Gibbs’s analysis of a body equilibrating inside an isolated vessel — reappears in the chapters on chemical reactions, electrochemical cells, and phase equilibria that follow.

Helmholtz Energy (A) — for systems at constant T and V

Not all processes occur at constant pressure. A sealed bomb calorimeter, a rigid pressure vessel, a gas confined to a fixed volume — these are systems where volume is constant and pressure is free to vary. For such systems, H and G are the wrong tools because both embed the assumption of constant pressure in their derivation. A different composite property is needed.

Hermann von Helmholtz constructed exactly this property in 1882, applying the same logical foundation as Gibbs — the First Law combined with the Second Law inequality — but to a constant T and V system rather than a constant T and P system. The result is:

A = U − TS

By an argument parallel to Gibbs’s derivation, the change in Helmholtz energy at constant T and V obeys:

dA_T,V ≤ 0 for the body

Helmholtz energy decreases to a minimum as a system at constant T and V equilibrates. At equilibrium, dA_T,V equals zero. The maximum work extractable from a constant T and V process is:

Maximum Work = −ΔA_T,V = −(ΔU − TΔS)

The relationship between the two properties is direct:

G = A + PV

Each is the right tool for its own conditions. Gibbs energy for constant T and P — the natural conditions of most chemical, biological, and industrial processes. Helmholtz energy for constant T and V — the natural conditions of sealed rigid systems and many theoretical treatments in statistical mechanics. Both are sometimes called free energy, which is precisely the source of the confusion the terminology section above addresses. The conditions assumed — constant P or constant V — determine which property applies, and conflating them under a single name obscures a physically important distinction.

References

[1] Carnot, S., Reflections on the Motive Power of Fire and on Machines Fitted to Develop that Power, translated and edited by R.H. Thurston, John Wiley & Sons, New York, 1897.

[2] Kragh, H., “Julius Thomsen and Classical Thermochemistry,” British Journal for the History of Science, 17(3), 255–272, 1984.