In Chapter 14 we established that the maximum work obtainable from a chemical reaction at constant temperature and pressure is given by Gibbs’s equation:
Maximum Work = −ΔGrxn = −(ΔHrxn − TΔSrxn)
This tells us that ΔHrxn, measured directly in a reaction calorimeter, is not itself the maximum work available. A portion — TΔSrxn — is unavailable for useful purposes. What, physically, is this portion?
This question took the better part of a century to resolve, and resolving it required an unwelcome new property of matter that much of the chemistry community initially wanted nothing to do with.
An Unwelcome Newcomer
When entropy entered thermodynamics through Clausius’s work in the 1850s and 60s, it did not arrive to applause. As Gilbert Lewis and Merle Randall observed in 1923, the Second Law “seemed in no recognizable way to accord with existing thought and prejudice” — unlike the conservation laws, which had been anticipated long before their formal acceptance. [1] Entropy was something genuinely new, and to many chemists it was, in the words of one historian, a “ghostly quantity” — confusing, unmeasurable directly, and seemingly unnecessary. [2]
A real divide formed. Mathematical physicists — Gibbs, Helmholtz, Duhem, Planck — built their work around entropy. Many practicing chemists — van’t Hoff, Nernst, even Lewis himself in his early career — saw no need for it. Van’t Hoff’s influential 1898–1900 physical chemistry textbook contained, by design, not a single mention of entropy. [3] If useful equations could be built from heat and work alone, why introduce a quantity nobody could fully explain?
The answer came from a problem entropy alone could solve: the spontaneous endothermic reaction.
The Thermal Theory of Affinity and Its Limits
By the mid-1800s, Julius Thomsen and later Marcellin Berthelot had proposed what became known as the thermal theory of affinity: a reaction’s spontaneity is governed entirely by ΔHrxn. [4] An exothermic reaction releases energy and proceeds on its own; an endothermic reaction requires energy and therefore cannot be spontaneous. This theory fit a great deal of data. It also fit a clean physical story — a strong attractive force holds the products together, and the calorimeter directly measures the strength of that force as heat released.
But the theory failed completely whenever it encountered a spontaneous endothermic reaction — a reaction that absorbs energy and yet proceeds on its own. According to the theory of affinity, this should not happen. It happened anyway, and by the early 1880s, the data could no longer be ignored.
Two Phenomena Hidden Inside ΔHrxn
The resolution requires recognizing that a calorimeter, while extremely useful, was never designed to separate two distinct physical phenomena that both contribute to the heat it measures.
The first phenomenon is the rearrangement of orbital electrons — bonds breaking in the reactants, bonds forming in the products — as the system moves toward its most probable electronic configuration. This rearrangement changes the chemical potential energy of the system.
The second phenomenon is the change in intermolecular forces between the reactant system and the product system — the attraction and repulsion between atoms and molecules, distinct from the chemical bonding itself.
Both phenomena change potential energy. Both, in a closed system, convert that potential energy change into a corresponding change in kinetic energy — and thus temperature. The calorimeter, maintaining constant temperature by exchanging thermal energy with its surroundings, measures the combined effect of both:
ΔHrxn = ΔGrxn + TΔSrxn
where ΔGrxn corresponds to the first phenomenon — the electron rearrangement — and TΔSrxn corresponds to the second — the change in intermolecular forces and atomic configuration. [5]
The Physical Meaning of TΔS
Recall from Chapter 12 that absolute entropy is built from integrating heat capacity from absolute zero: S = ∫δQ/T. This δQ accounts for two things — the energy needed to set atoms into motion, and the energy needed for them to overcome attractive forces and separate from one another, such as during a phase change. TΔSrxn, then, quantifies the difference in energy required to establish the configuration of the products relative to that of the reactants — the spatial and momentum arrangement of the atoms themselves, separate from the chemical bonding captured by ΔG.
The sign carries direct physical meaning, and it traces to intermolecular attraction.
When TΔS is negative, the attractive forces between atoms in the products are stronger than in the reactants. Stronger attraction pulls atoms together, accelerating them toward each other and increasing their kinetic energy. This released energy must be removed — absorbed by the surrounding medium — to hold temperature constant.
When TΔS is positive, the attractive forces in the products are weaker than in the reactants. The atoms experience less acceleration, lower kinetic energy results, and the system cools. Energy must be added from the medium to hold temperature constant.
This is the sense in which TΔS is, as Helmholtz described it in 1882, “bound” energy — present in the structural arrangement of the system, exchanged with the surroundings as heat, and unavailable for useful work. [6] Helmholtz’s terminology, free versus bound energy, was likely shaped by Clausius’s earlier concept of disgregation — an attempt to quantify the degree to which the molecules of a body are separated from one another, captured in Clausius’s relation dQ = d(vis viva) + TdZ, where TdZ represented the internal work associated with that separation. [7] TΔS is the conceptual descendant of disgregation, now given a precise statistical foundation through Boltzmann’s later work.
Why the Electrochemical Cell Was Needed
The calorimeter cannot separate these two phenomena — both arrive mixed together as a single heat measurement. The electrochemical cell can, and this is precisely why it became so valuable to thermodynamics.
In an electrochemical cell, electrons flow through an external wire from high potential to low, driving a motor and generating useful work, rather than dissipating directly as heat inside a single vessel. The voltage difference between the electrodes is a direct measure of the electron rearrangement alone. Separately, the cell — held in a constant-temperature bath — exchanges heat with that bath to maintain isothermal conditions, and this exchange is the direct measure of TΔS. The two phenomena, hopelessly tangled together in the calorimeter, are cleanly separated by the cell.
This is exactly what nineteenth-century thermochemists discovered when their energy balances around electrochemical cells refused to close — until they placed the entire cell inside a constant-temperature bath calorimeter and accounted for the additional heat exchanged there. [8] Gibbs, reviewing this body of work, devoted nineteen pages of his third paper to the thermodynamic theory of the electrochemical cell, noting that the heating and cooling requirements needed to maintain isothermal conditions were “frequently neglected” in existing analyses. [9] He showed that a “perfect electrochemical apparatus” — one in which an external voltage exactly cancels the cell’s own voltage, holding current at zero — validates the equality in Maximum Work = −ΔGrxn = −(ΔHrxn − TΔSrxn) exactly.
Testing This Against Real Data
Bertram Dodge’s 1944 textbook reports measurements for the reaction Hg(l) + ½Cl₂(g) → Hg₂Cl₂(s) at 25°C: a calorimetric ΔHrxn of −31,300 cal/mol, and an electrochemically measured external work of 25,140 cal/mol. [10] Using Gibbs’s equation:
25,140 = 31,300 + TΔSrxn TΔSrxn = −6,160 cal/mol ΔSrxn = −20.7 cal/(mol·K)
This value comes entirely from electrochemical and calorimetric measurement — no absolute entropy data involved. It can be checked independently using NIST’s tabulated absolute entropies for mercury, chlorine, and Hg₂Cl₂, which give:
ΔSrxn = 192.5 − 75.9 − ½(223) = −91.2 J/(mol·K) = −21.8 cal/(mol·K)
The two values agree to within roughly 5 percent. [11] Lewis and Gibson performed a comparable check in 1917 using the entropy of chlorine gas and found similarly close agreement. [12] This is genuine, quantitative confirmation that the heat exchanged with a constant-temperature bath to hold an electrochemical cell isothermal is the same physical quantity as the entropy change calculated from the statistical, heat-capacity-based definition developed in Chapter 12.
Temperature Dependence — The Gibbs-Helmholtz Equation
Both the electrochemical cell and chemical equilibrium are sensitive to temperature, and understanding how ΔG shifts with temperature required, once again, entropy. Starting from G = H − TS and differentiating at constant pressure, the middle terms cancel, leaving:
(dG/dT)P = −S
Applying this to both reactants and products and subtracting:
(dΔG/dT)P = −ΔS
This is the Gibbs-Helmholtz equation — first derived not by Gibbs but by Helmholtz. [13] Applied to an electrochemical cell, where ΔG corresponds to the difference in electrode potential, measuring that potential across a range of temperatures allows ΔHrxn to be calculated indirectly, often more accurately than direct calorimetry permits. [14] The same relation, applied to a chemical equilibrium constant, shows how equilibrium itself shifts with temperature — a topic developed further in later chapters.
It bears repeating what this equation does and does not explain. It tells us, mathematically and exactly, how ΔG varies with temperature given ΔS. It does not, on its own, explain why a quantity governed by electron rearrangement (ΔG, by hypothesis) should be mathematically locked to a quantity governed by atomic configuration and momentum (ΔS) through this relation. That deeper question — whether this is a fundamental physical connection or a consequence of how these properties were mathematically constructed — remains open, and is addressed as a hypothesis, not a settled answer, in the chapter on chemical reaction spontaneity.
What This Means
TΔS is not a mathematical leftover invented to make Gibbs’s equation balance. It is a physically real heat effect — first encountered as an unexplained discrepancy in nineteenth-century electrochemical experiments, given theoretical structure by Gibbs and Helmholtz, and confirmed numerically against absolute entropy values built independently from heat capacity data. It quantifies the energy bound up in the changing intermolecular configuration of a system’s atoms as reactants become products — energy that must be exchanged with the surroundings to hold temperature constant, and that can never, for that reason, be converted to useful work.
References
[1] Lewis, G.N. and Randall, M., Thermodynamics, McGraw-Hill, 1923, as quoted in Hanlon, R.T., Block by Block, Oxford University Press, 2022, Chapter 39.
[2] Kragh, H., “Julius Thomsen and Classical Thermochemistry,” British Journal for the History of Science, 17(3), 255–272, 1984.
[3] Van’t Hoff, J.H., Vorlesungen über theoretische und physikalische Chemie, 1898–1900, as discussed in Hanlon, Block by Block, Chapter 39.
[4] Thomsen, J. (1854) and Berthelot, M. (1864), thermal theory of affinity, as discussed in Kragh (1984).
[5] This two-phenomena framework — chemical bond rearrangement and intermolecular force change — is developed in Hanlon, Block by Block, Chapter 39, and in Hanlon, R.T., “Deciphering the Physical Meaning of Gibbs’s Maximum Work Equation,” Foundations of Chemistry, 26, 179–189, 2024.
[6] von Helmholtz, H., “On the Thermodynamics of Chemical Processes,” Physical Memoirs Selected and Translated from Foreign Sources, 1, 43–97, 1882.
[7] Clausius, R., disgregation concept, as discussed in Hanlon, Block by Block, Chapter 34.
[8] Ostwald, W., Electrochemistry: History and Theory, Amerind Publishing Co., 1980, pp. 741–1016.
[9] Gibbs, J.W., The Scientific Papers of J. Willard Gibbs, Volume One: Thermodynamics, Ox Bow Press, 1993, pp. 331–349.
[10] Dodge, B.F., Chemical Engineering Thermodynamics, McGraw-Hill, 1944, p. 71.
[11] NIST Chemistry WebBook, SRD 69.
[12] Lewis, G.N. and Gibson, G.E., “The Entropy of the Elements and the Third Law of Thermodynamics,” Journal of the American Chemical Society, 39(12), 2554–2581, 1917.
[13] Helmholtz’s priority on this equation is discussed in Hanlon, Block by Block, Chapter 39.
[14] Comparative accuracy of electrochemical versus calorimetric ΔH determination as discussed in Hanlon, Block by Block, Chapter 39.
A few editorial notes on what I did and why. I dropped the Table 39.1 spontaneity table from this chapter — it leans heavily on ΔG and its electron-based interpretation as the sole determinant of spontaneity, which is exactly the material you asked me to keep out of this chapter. It would be an excellent table for the later chapter on reaction spontaneity instead. I also kept the Gibbs-Helmholtz section but added an explicit closing caveat so the chapter doesn’t accidentally imply the deeper electron/entropy connection has been resolved — consistent with your paper’s own honest admission that it hasn’t. Let me know if you’d like that table built out separately for the future chapter.
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