The entropy of the electron gas used by Llano and Eriksson (LE) in the definition they used for an absolute half-cell potential (AHCP) is indeterminate. This AHCP is therefore not based on a thermodynamically well-defined process and cannot easily be extended to other thermodynamic functions such as entropy. An alternative approach to the AHCP avoids these difficulties. The present comment also corrects statements and concepts presented by LE about the temperature coefficient of the Fermi level and surface charge of the electrolyte solution.

Llano and Eriksson (LE) recently revisited the concept and definition of the absolute half-cell potential (AHCP) and applied it to certain problems in ion chemistry.^{1} The present Comment corrects some statements in that paper and critically examines the definition of the AHCP that LE used.

LE stated that the temperature coefficient of the Fermi level of a metal is zero. However, since the Fermi level of a degenerate free electron Fermi gas depends on temperature,^{2} it is unlikely that the Fermi level of a metal is independent of temperature. It has been argued that the absolute thermoelectric power of a metal gives the temperature coefficient of the Fermi level.^{3}

LE stated that the free surface of the electrolyte solution is uncharged following equilibration of a redox reaction with an electrode. This would be true for a solution with a dielectric constant equal to that of free space, but an electrolyte is a conductor on the macroscopic scale ($\u2aa2$ Debye length), so we must consider the free surface of the electrolyte solution to be charged.

LE based their definition of the AHCP on earlier work^{4} but added constraints on the number density $\rho $ and temperature $T$ of an electron gas. In this definition the AHCP is the difference between the vacuum level near the surface of the solution and the Fermi level of a metal electrode. Let us call this a “type I” AHCP.

A type I definition focuses on the electron. It relates most closely to the nonisothermal process

with gas phase electrons at $0K$.^{1,5} LE stipulate the gas to be “classical,” implying a nondegenerate free electron Fermi gas, requiring that $\rho \u21920$ as $T\u21920$. This is an improvement over previous type I definitions,^{4} which made no attempt to define the thermodynamic state of the electron gas. However as shown below the LE stipulations are not sufficient to fully define the thermodynamic state of the electron gas.

A weakly degenerate or nondegenerate free electron gas satisfies the following:^{2}

where $h$ is Planck’s constant, $m$ is the electron mass, and $k$ is Boltzmann’s constant. The entropy is^{2}

where $L$ is Avogadro’s number. The last term accounts for the onset of degeneracy.

Consider four possible implicit rules relating $\rho $ to $T$ during the limiting procedure.

where $A,B,C$, and $D$ are arbitrary constants. All four rules conform to the LE stipulations on $\rho $ and $T$. Clearly, $S$ can have any value $\u22732.6Lk$, including infinity, and $TS$ can have any non-negative value, including infinity.

From Eqs. (4)–(15) one concludes that for a type I AHCP: (1) the entropy of the electron gas is undefined, (2) the TS contribution to the free energy of the electron gas is undefined, (3) the electron gas is not in a thermodynamically well-defined state, (4) the type I AHCP does not refer to the Gibbs free energy $\Delta G$ of a well-defined thermodynamic process, and (5) the approach is not easily extended to other thermodynamic functions such as entropy.

To avoid some of these issues one might define the electron gas to be a fictitious degenerate free electron Fermi gas at $T=0$ and a specific nonzero $\rho $, in which case the entropy of the electron gas is zero. However, a better way to avoid these problems is to start with an isothermal absolute half-cell process, such as

and define a “type II” AHCP as $\Delta G$ for this process. This approach focuses on the complete half-cell reaction rather than focusing mainly on the electron. For measurement purposes one would break this process into

and apply methods described elsewhere.^{5} All reactants and products are placed in thermodynamically well-defined states, and standard states are defined for all species. A complete system of absolute half-cell thermodynamics can be developed using this approach, including functions such as entropy and enthalpy.^{6}

Thermodynamic descriptions of processes generally include standard states. These states must be thermodynamically well defined but need not be physically realizable states. For a gas the usual choice is the fictitious state defined by extrapolating from zero pressure to unit pressure assuming ideal gas behavior.^{7–9} A gas need not be ideal, but it must approach ideality as pressure $\u21920$. Thermodynamic properties of this state are fictitious, but they are fully defined by the extrapolation. Standard thermodynamic tables use this convention for the electron gas.^{10} A similar convention was suggested for a type II AHCP.^{5}

The degenerate free electron Fermi gas at a specified temperature and fugacity or pressure^{11} could also be used as a standard state in the type II AHCP. LE discussed this state in their discussions of gas phase ion chemistry but did not apply it to their definition of the AHCP.^{1}

Support from the ARUP Institute for Clinical and Experimental Pathology is gratefully acknowledged.