# Gibbs free energy: G or ∆G?

One of my objectives in creating a more effective approach to teaching thermodynamics is to bring clarity to some of the confusing terms and concepts embedded in this field. Initially I focused on the concept of heat by pointing out that there is no such thing. I now turn toward free energy.

As a very (very) brief historical summary, Rudolf Clausius created what became known as the 1st Law of Thermodynamics, which I wrote about here, based on energy and its conservation when he wrote the equation: dU = TdS – PdV . J. Willard Gibbs then built upon this by creating a new property of matter, later to be given the symbol G after Gibbs, for which G = H – TS . This energy term became very useful in thermodynamic analyses of physical phenomena and industrial processes that occur at constant temperature and pressure. Of relevance to this post, Gibbs showed that the change in G at constant T,P quantifies the maximum amount of work that can be generated by a given process such as a chemical reaction.

Prominent thermodynamics textbooks, such as Lewis and Randall [3, p. 158] and Smith and van Ness [4, p. 170], named G free energy. Today we often refer to G as Gibbs free energy and associate it with the amount of energy that is free to do useful work.

HOWEVER

Naming G free energy always confused me. While certain thermodynamic properties such as temperature, pressure, and mass are absolute, meaning that they can be referenced to zero, the property internal energy (U) is not. There is no zero for internal energy, which is why the primary focus in thermodynamics is based on changes and not absolutes. We’re largely concerned with changes in energy; absolute energy doesn’t exist. Thus, Gibbs’s G property, which is based on energy since it includes internal energy U, i.e., G = H – TS = U + PV – TS, is meaningless on its own. This is the reason for my confusion with naming G free energy. Free energy has meaning, while G itself does not.

Consider the intent of the two founders of “free energy” – Gibbs & Helmholtz 

To Gibbs, it’s the change in G that’s meaningful, not G itself. It is the distance between a given body’s non-equilibrated energy—non-equilibrated in the sense that the body is either not internally equilibrated or not equilibrated with the environment or both—and its equilibrium state energy that was important to Gibbs. He called this distance, which quantified change, “available energy” [6, pp. 49-54]. Today we view available energy and free energy as synonyms.

Hermann von Helmholtz created his own energy term A = U – TS, which served a similar purpose as Gibb’s G, but for constant temperature processes as opposed to constant temperature and pressure for Gibbs. It was Helmholtz who coined the term “free energy” as shown here from his publication on the matter :

It has long been known that there are chemical processes which occur spontaneously and proceed without external force, and in which cold is produced. Of these processes the customary theoretical treatment, which deals only with the heat developed as the measure of the work-value of the chemical forces of affinity, can give no satisfactory account.

Here Helmholtz is referring to the Thomsen-Berthelot theory of thermal affinity, for which the “heat developed” is quantified by ∆H, the enthalpy change of reaction. This theory suggested that cold-producing endothermic reactions (∆H > 0) should not happen; and yet they did. Continuing…

If we now take into consideration that chemical forces can produce not merely heat but also other forms of energy… then it appears to me unquestionable that… a distinction must be made between the parts of their forces of affinity capable of free transformation into other forms of work, and the parts producible only as heat. In what follows I shall, for the sake of brevity, distinguish these two parts of the energy as the “free” and and as the “bound” energy.

It is clear to me that Helmholtz sought to replace ∆H with a term that quantified his concept of “free” energy. This term had to be similar to ∆H in that it had to quantify change as opposed to absolute. This is how he arrived at ∆A.

In Sum: Free energy was founded on change

That both Gibbs and Helmholtz based their respective concepts of free energy on change as opposed to absolute supports my contention is that G should be known as Gibbs energy and ∆G should be known as Gibbs free energy. In other words: Gibbs free energy (∆G) is the change in Gibbs energy (G).

I propose that textbooks make clear these definitions, especially since some confusingly refer to G as both Gibbs energy and Gibbs free energy. Is the above argument strong enough to justify this? What do you think?

Thank you for reading my post. I go into much greater detail about these concepts in my book Block by Block – The Historical and Theoretical Foundations of Thermodynamics.

References

 dU = Q – W = TdS – PdV. Q = thermal energy added to system = TdS, W = work done by system = PdV, U = internal energy, T = temperature, S = entropy, P = pressure, V = volume. If no thermal energy (i.e., heat) is added to the system and if no work is done by the system, then the internal energy of the system does not change, i.e., dU = 0.

 G = H – TS. H = enthalpy = U + PV. The change in G is thus: dG = dH – d(TS) = dU + PdV + VdP – TdS – SdT. For a constant temperature and pressure process, dT = dP = 0. If the system of interest is equilibrated then dU + PdV – TdS = 0, and thus dG = 0. The property G is particularly useful when considering phase equilibrium. Consider two phases, A and B, in equilibrium with each other. The values of G for the two phases are equal. If you change temperature and pressure together so as to maintain a two phase system, again dU + PdV – TdS = 0 as it’s an equilibrated system, and so dG = VdP – SdT. One can show that dG for A must equal dG for the B. With some re-arrangement, dP/dT = (SB – SA)/(VB-VA), which is a version of the famed Clapeyron and Clausius-Clapeyron equations.

 Lewis, Gilbert Newton, and Merle Randall. 1923. Thermodynamics and the Free Energy of Chemical Species. New York: McGraw-Hill Book Company, Inc.

 Smith, J. M., and H. C. Van Ness. 1975. Introduction to Chemical Engineering Thermodynamics. 3d ed. McGraw-Hill Chemical Engineering Series. New York: McGraw-Hill.

 Gibbs cited the influence of François Massieu on his work that included the creation of G = H – TS.

 Gibbs, J. Willard. 1993. The Scientific Papers of J. Willard Gibbs. Volume One Thermodynamics, Woodbridge, Conn: Ox Bow Press. p. 51.

 Helmholtz, H. von, On the thermodynamics of chemical processes, Physical memoirs selected and translated from foreign sources 1 (1882): 43-97.

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