Gibbs energy and Gibbs free energy (G and dG) – J. Willard Gibbs

The arrival of enthalpy as a composite property (H = U + PV) was soon followed by the arrival of Gibbs energy (G = H − TS). Here is how J. Willard Gibbs arrived at this composite property.

Gibbs recognized two things:

• (1) When two systems come together and enable any gradient that exists between them–temperature, pressure, chemical–to dissipate, the entropy of the combined equilibrated system is greater than the sum of the entropies of the two initial systems. In short, S_combined $\ge$ $\sum$ S_parts.
• (2) Many commercial chemical processes occur at constant temperature and pressure.

Putting these together, he proposed a new property of matter that would be useful when analyzing constant T,P processes. Here is how he did it.

Assume that a large constant-temperature bath — called a medium — and a body are placed inside an isolated, constant-volume vessel. The medium and the body are separated by a membrane that conducts heat and responds to mechanical pressure, maintaining the body at the temperature and pressure of the medium at all times. The body itself may not be internally equilibrated — it may, for example, contain reactants undergoing a chemical reaction. Thus:

dU_body + dU_medium = 0 total energy inside the isolated vessel is constant

dV_body + dV_medium = 0 total volume inside the isolated vessel is constant

dS_body + dS_medium ≥ 0 total entropy inside the isolated vessel increases to a maximum

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

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

This inequality becomes an equality once the body reaches equilibrium.

A detailed physical understanding of the process leading to this inequality is more complicated than can be dealt with here and is saved for a later chapter.

A word on terminology. 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.

Maximum Work

One of the most important consequences of the Gibbs energy framework is its connection to maximum work. To see how this connection arises, return to the inequality derived above:

dG_T,P ≤ 0 for the body

This inequality was derived by assuming the total system to be comprised of the body plus a large medium maintaining constant T and P. As the body equilibrates, G decreases to a minimum and dGT,P equals zero. Equilibration is one way to turn the inequality into an equality.

There is a second way. Rather than allowing the body to equilibrate freely, consider instead creating a precise balance between the non-equilibrated body and the lifting of an external weight — analogous to balancing a lever by placing a weight 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.

To quantify it, return to the First Law written in its most general form. 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 an electrical current through a motor. A more general statement is:

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

Rearranging:

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

In finite terms, since the total amount of external work possible is fixed by the difference between the initial and final states of the body:

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

The physical reasoning is direct. The total energy released 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 and is therefore unavailable for conversion to useful work. What remains, ΔH − TΔS = ΔG, is the maximum convertible to useful work.

A critical point about this reversible process: the entropy of the total system — body plus medium — remains constant throughout. 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.

If ΔG is negative the process can in principle deliver work spontaneously. If ΔG is positive work must be supplied to drive it. If ΔG is zero the system is at equilibrium and no external work is possible.

The maintenance of constant T and P does something else valuable beyond simplifying the mathematics — it converts the analysis from path-dependent to state-dependent. The maximum work is fixed entirely by the difference in Gibbs energy between the initial and final states of the body, regardless of the path taken between them.

A concrete illustration of maximum work in practice is the perfect electrochemical cell. When a reverse voltage exactly equal to the cell’s own voltage is applied, the cell chemistry is held at the precise edge of equilibrium — neither proceeding forward nor backward. In this condition the cell is performing reversible work. One could equivalently connect a motor to the cell and raise a weight, achieving the same result. The maximum work extracted equals −ΔG for the cell reaction. Real electrochemical cells fall short of this ideal due to irreversibilities, but −ΔG remains the theoretical upper bound. This will be examined in detail in Chapter 10.

This framework — born from Gibbs’s analysis of a body equilibrating inside an isolated vessel — will reappear when we address chemical reactions, electrochemical cells, and phase equilibria in later chapters.

Gibbs illustrated these concepts as I discussed previously here.

A deeper probe into terminology: Gibbs energy versus Gibbs free energy

The change in Gibbs energy resulting from a process was originally called “available energy” — Gibbs’s own preferred term in his 1875 publication. It quantifies the energy stored in the internal gradients of a non-equilibrated system and thus represents the maximum work that can possibly be generated by reversibly dissipating those gradients.

Today this quantity is known by two names: Gibbs energy and Gibbs free energy. As I discussed here, the terminology and its exact definitions have been a source of confusion throughout history. The word “free” has a specific historical meaning — it refers to the portion of the total energy that is free to do useful work, as opposed to the portion bound up in entropy and unavailable. Wilhelm Ostwald and Hermann von Helmholtz both used this language in the late 19th century. The problem is that “free” sounds as though the energy comes at no cost, when in fact it simply means available for work under specified conditions. It is a Wheeler trap — a term that sounds intuitive but misleads. Gibbs energy is the more precise name and the one used throughout this book.

The distinction between Gibbs energy and Helmholtz energy is worth clarifying here, as both are sometimes called “free energy” which compounds the confusion. Gibbs energy G = H − TS applies at constant temperature and pressure — the natural conditions for most chemical and biological processes. Helmholtz energy A = U − TS applies at constant temperature and volume — the natural conditions for a sealed system. Each represents the maximum work extractable under its respective conditions. The two quantities are related but they are not interchangeable, and conflating them under the single label “free energy” obscures a physically important distinction.

It is the difference between two states — not the absolute value at any single state — that quantifies the maximum work potential. This is consistent with the broader theme of this chapter: thermodynamics works with changes, not absolutes.

One further consequence of this framework deserves attention. A reaction with a negative entropy change can still proceed spontaneously at constant T and P and still generate maximum work. This is possible because the medium, in this case, extracts the thermal energy needed to maintain constant temperature. The body’s entropy decrease is exactly compensated by the medium’s entropy increase, keeping total system entropy constant. Left alone in an isolated vessel, a negative-entropy reaction might not proceed at all. In the presence of a constant-temperature medium, it can. The medium is not merely a passive bystander — it is an active participant in enabling the process. This point will be examined further when we address chemical reactions in Chapter xx.

Interpreting ΔG: Spontaneity, Equilibrium, and Direction

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

• When ΔGT,P is negative, the process is spontaneous — it will proceed on its own without external work being supplied. Internal gradients dissipate naturally. As discussed above, 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 does not need to obey that constraint alone.

• When ΔGT,P is positive, the process will not proceed spontaneously in the proposed direction. No external work can be extracted — in fact, external work must be supplied to make it happen. In this case −ΔG becomes negative, which means 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.

• When ΔGT,P equals zero, the situation is 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. In reality, a more careful accounting — one that tracks the Gibbs energy of the entire system as a function of the extent of reaction, including the entropy of mixing — will typically show that G is minimized at some partial conversion. For example, if ΔG = 0 for complete conversion of A to B, then G itself 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, and it will be developed fully in the chapter on chemical reactions.

The electrochemical cell illustrates all three cases in a single physical system. Consider a spontaneous reaction — ΔG < 0 — driving a current through an external circuit. Apply a reverse voltage equal in magnitude to the cell voltage and the reaction stops. The system is balanced at the edge of equilibrium, extracting maximum work. Increase the reverse voltage slightly and the reaction runs backward — a ΔG-positive process is now being driven by external work. Decrease the reverse voltage slightly and the forward reaction resumes. The entire range of spontaneous, equilibrium, and driven behavior is accessible simply by adjusting the external voltage. The electrochemical cell is the clearest physical embodiment of the Gibbs energy framework ever devised, and it will be examined in detail in Chapter 12.

Helmholtz Energy (A) — Hermann von Helmholtz

Not all thermodynamic 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 change. For such systems, enthalpy H and Gibbs energy G are the wrong tools. A different composite property is needed, one suited to constant temperature and constant volume conditions.

Hermann von Helmholtz constructed exactly this property in 1882. Starting from the same logical foundation as Gibbs — the First Law combined with the Second Law inequality — but applying it to a constant T and V system (conducted in a sealed container) rather than a constant T and P system, Helmholtz arrived at:

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 temperature and volume equilibrates. At equilibrium, dAT,V equals zero.

The maximum work extractable from a constant T and V process is:

Maximum Work = −ΔA_T,V = −(ΔU − TΔS)

This is the total maximum work — including both PdV work and any other external work — available from the system. This differs from Gibbs energy in an important way. At constant T and P, the PdV work against the atmosphere is unavailable for useful purposes and is already accounted for in the H = U + PV term. At constant T and V there is no PdV work against a changing pressure, so −ΔA represents all work without that subtraction. For this reason Helmholtz energy is sometimes the more natural property in theoretical physics and statistical mechanics, where systems of fixed volume are commonly analyzed.

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 calculations. Both are sometimes called free energy, which is precisely the source of the confusion addressed in the terminology section above. The conditions assumed — constant P or constant V — determine which property applies, and conflating them under a single name obscures a physically important distinction.

END