Classical thermodynamics doesn’t prescribe how to design a process but does set boundaries on what the process can achieve. These boundaries quantify either the maximum work generated by or the minimum work required for the process. The calculations behind these boundaries are invaluable for assessing the economic feasibility and practical potential of ideas. Gibbs helped introduce the fundamentals behind this approach.

Gibbs to the rescue

Classical thermodynamics generally deals with equilibrated systems.  But for many industrial systems such as power generation, the initial system isn’t equilibrated, for if it were, no power generation would be possible.  For example, when we burn coal to generate steam to do work, the initial mixture of air and coal is clearly not equilibrated.  So how do we approach such problems?

The ability to produce useful work starts with an energy gradient of some kind.  A waterfall – mechanical.  A hot gas (like steam) – thermal.  An internal combustion engine – chemical.  The nature of the gradient doesn’t matter; instead what does matter is the fact that the initial system is not equilibrated and so will naturally move towards its most probable (equilibrated) state. This natural tendency can be taken advantage of.  As Callen stated, “The propensity of physical systems to increase their entropy can be channeled to deliver useful work.” [1]

The problem to solve in any of these situations is how to do the actual calculations. It sounds rather complicated, doesn’t it?  Fortunately, Gibbs provided us a way out, an approach that doesn’t depend on the process chosen but instead depends only on the initial and final states of the system.

Gibbs’s Maximum Work Equation

Gibbs’s maximum work equation for a chemical reaction, the derivation of which I shared in Chapter 37 of my book (here), is based solely on the state properties of the beginning and end systems, e.g., reactants and products. In other words, the maximum amount of work possible is fixed by the two states and thus independent of the path taken.

            Maximum Work = – ∆G_rxn = -(∆H_rxn – T∆S_rxn) (constant T,P)

G = Gibbs energy. ∆G is the change in Gibbs energy. This is also referred to as Gibbs free energy, as I discussed previously here, or “availability.”

H = enthalpy. ∆H is the change in enthalpy as quantified, for example, in a reaction calorimeter.

S = entropy. ∆S is the change in entropy, specifically in the above equation, for a chemical reaction.

T,P = temperature, pressure

This equation became very useful and practical as most reactions in the industrial world operate at constant T,P. One example of this equation’s usefulness is that it became the means by which to determine reaction spontaneity. When Maximum Work (W_max) > O, meaning ∆G_rxn < O, the reaction can generate positive work and so be considered spontaneous.

The value of the electrochemical cell to thermodynamics

When Gibbs created this equation in the 1870’s, calculation of ∆G_rxn was difficult, for while ∆H_rxn values were readily available from reaction calorimetry experiments, such was not the case for ∆S_rxn values. There was no means at that point in time to quantify the change in entropies of the beginning and end states and thus no means to calculate ∆G_rxn using the above equation.

Breakthrough occurred when Gibbs turned his eyes towards the electrochemical cell and realized that 1) ∆G_rxn could be directly quantified by the voltage across the cell since this voltage represented the maximum work possible for the reaction, and 2) T∆S_rxn could also be directly quantified as the heating/cooling requirement to maintain the cell at constant temperature. (Naturally, this approach only worked for reactions that could be carried out in the cell.) Even if one weren’t able to directly measure the heating/cooling requirements in (2), which admittedly is a difficult measurement to obtain, one could still quantify ∆S_rxn based on knowledge of both ∆H_rxn and ∆G_rxn using the above equation. So, unintentionally, one way or the other, the electrochemical cell became a means by which to quantify the change in entropy of the reaction.

Here’s what fascinates me about all of this

Now here’s the fascinating part. Calculation of absolute entropy became possible in the early 1900s when Walther Nernst’s heat theorem led to the proposition that the entropies of crystalline substances at absolute zero equal zero, thus enabling calculation of absolute entropy by integrating the following equation from absolute zero to a given reference temperature .

wherein, δQ_rev​=C(T) dT, and thus

∆S_rxn values experimentally determined from electrochemical cells could then be compared against those determined by the above integration process involving heat capacity data. [2] Suprise, suprise! They matched reasonably well as I discuss here. The fact that ∆S_rxn values calculated from two totally different approaches–one based on the heating/cooling requirements of an electrochemical cell and the other based on integrating heat capacity data from absolute zero to the reference temperature—match each other (reasonably well) is what fascinates to me.

Explore more!

Check out this practical application of Gibbs’s work in more detail in Chapter 37 of my book, Block by Block – The Historical and Theoretical Foundations of Thermodynamics.

Post Script

The electrochemical cell also affords the opportunity to interpret the two natural effects contributing to ∆H_rxn, as revealed in this re-arrangement of Gibbs’s maximum work equation:

∆H_rxn = ∆G_rxn + T∆S_rxn (constant T,P)

I hypothesize that the enthalpy change of a reaction is the sum of two different effects, that associated with ∆G_rxn, which determines reaction spontaneity and can be directly quantified by cell voltage, and that associated with T∆S_rxn, which can be directly measured by the heating/cooling requirements of the cell to maintain constant temperature. I discussed this in detail, especially regarding the physical meaning of T∆S_rxn, in these posts here and here.

References

[1] Callen, Herbert B., and Herbert B. Callen. 1985. Thermodynamics and an Introduction to Thermostatistics. 2nd ed. New York: Wiley. p. 103.

[2] The comparison could also be done by using entropy values calculated from statistical mechanics, as demonstrated by the work of Hugo Martin Tetrode and Otto Sackur in the early 1900s.

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