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The spontaneity condition for a system at constant temperature and pressure in which the only type of work accomplished is of the $pV$ type can be expressed as:

$$\Delta G\le0\tag1$$

In case of other types of work (such as electrical, friction, etc.) there is an additional term and the previous equation becomes:

$$\Delta G\le W_\text{other}\tag2$$

In case of an electrochemical process the maximum work is the electrical work and can be expressed as:

$$W_\text{other}=-nFE\tag3$$

  • Question: Why it is still said that the electrochemical reaction is spontaneous if the change in Gibbs free energy is lower than zero, thus using Equation $(1)$, even though the proper equation to be used is number $(2)$?
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Most likely the source is wrong, or it has purposfully omitted the correct equation to pseudosimplify a problem. It might be that an electrochemical reaction was under discussion, not a cell. A further possibility is pursued in the appendix. I recommend adding a source where you came upon the quoted exchange to avoid a straw man argument; even though in this case I have observed (and possessed) the misconception first hand.

One definition of spontaneity

I define the true spontaneity condition as such when there is a tendency toward net 'something different' (chemical reaction, expansion etc) to establish state 2 instead of some original state 1. An example could be a chemical reaction where there is a higher reaction extent than at equilibrium, so a net reverse reaction takes place.

You are correct that the true spontaneity equation (from Clausius's inequality) in the thermodynamic limit is, for net-constant pressure and temperature in the system,

$$\operatorname{d} G-\delta w_\pu{other}\leq 0.\tag1\label1$$

When electrochemical work is the only component besides expansion work, this implies (due to $|\operatorname{d} n_\pu{e}|=|\nu_\pu{e}|\operatorname{d}\xi$)

$$\Delta_\pu{r} G_{T,P} +|\nu_\pu{e}|FE\leq0\tag2\label2$$

where $\Delta_\pu{r} = \partial/\partial \xi$; here $\xi$ is extent of the reaction. The Greek $|\nu_\pu{e}|$ signifies an absolute value of the stoichiometric coefficient of an electron in some half-reaction. Equation $\eqref2$ assumes that only one net reaction occurs. The term $|\nu_\pu{e}|FE$ should be a good indicator of electrical work.

It might also be that they are discussing an electrochemical reaction, not the cell itself. We can have the process

$$\ce{Zn(sln) + Cu^2+(sln) -> Cu(sln) + Zn^2+(sln)}\tag3\label3$$

without harnessing its electrical work via an external circuit. So $\operatorname{d} G\leq 0$ would hold for spontaneity.

Appendix: An alternative

A different definition of spontaneity might be in play. Namely, the term spontaneity is also used to mean a large enough standard equilibrium constant (especially in biochemistry).

$$RT\ln \frac1K = \Delta_\pu{r} G^\circ_{T,P} \tag4\label4$$

Equation $\eqref4$ is technically a definition of the standard equilibrium constant. When non-negative absolute temperatures are quaranteed, the LHS of equation $\eqref4$ will become non-positive for all $K$ big enough, i.e., $K\ge1$ . That implies

$$\Delta_\pu{r} G^\circ_{T,P} \leq 0. \tag5\label5$$

Note, however, that $K\ge1$ doesn't necessarily imply much about the reaction balance itself because the equilibrium constant comprises of activities (not concentrations), and because the stoichiometric coefficients in the denominator may drown out the coefficients in the numerator. (Equation $\eqref6$ assumes that surrounding fugacity is equal to an agreed fugacity in the standard state, denoted here and elsewhere by '$^\circ$'.)

$$K(\pu{in solution, solvent \ce{A}}) = \left[a(\ce{A})_\pu{eq}\right]^{\nu_\ce{A}}\prod_i \left[a(\ce{B_$i$})_\pu{eq}\right]^{\nu_i}. \tag6\label6$$

Reactants–products (other than solvent) are designated by $\ce{B_$i$}$. But still, for $K$ big enough (2nd definition of spontaneity), equation $\eqref5$ will hold by definition. Also keep in mind $\Delta_\pu{r} G^\circ_{T,P} \neq \Delta G$ (even their dimensions are different!). So the source is presumably still at fault for poor notation.

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  • $\begingroup$ Some details (such as derivations and motivations for definitions) have been omitted for brevity. If something is unclear, please point it out in the comments, and I will do my best to clarify (edit) as soon as possible. $\endgroup$ – Linear Christmas Feb 2 '18 at 14:38
  • $\begingroup$ Seems a matter of definitions. "The" fundamental equation that I'm familiar with is $\Delta G = -S dT + (work terms)$, and if you have electric fields, magnetic fields, surface tension, gravity, etc. etc., then that is part of the definition of $dG$. So with that definition, $dG \le 0$ is still a valid spontaneity criterion. If your definition of $dG$ only includes $PV$ work, then YMMV. $\endgroup$ – Curt F. Feb 2 '18 at 18:27
  • $\begingroup$ @CurtF. Thank you for your comment. I took some time before this reply to think, and think some more. The conclusion: I stand my ground (for now). As long as the definition of Gibbs energy remains $G = U + PV - TS$ (is this challenged?), then $\operatorname{d} G \leq 0$ is a necessary and sufficient condition for spontaneity iff (1) a thermodynamic limit is applied, (2) net-constant pressure, (3) net-constant temperature, (4) only non-expansion work is done. The effect of external electromagnetic fields, surface tension, gravity, friction and so on have to be included in $w_\pu{other}$. $\endgroup$ – Linear Christmas Feb 2 '18 at 21:09
  • $\begingroup$ see also: the appendix (b) of my answer here $\endgroup$ – Linear Christmas Feb 2 '18 at 21:12
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    $\begingroup$ And to emphasize, I'm not saying your answer is wrong at all, in fact +1 from me. Just depends on your definition of $G$. $\endgroup$ – Curt F. Feb 2 '18 at 22:27
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By spontaneous we normally mean that the reaction is thermodynamically allowed WITHOUT forcing it (that is, without doing non-PV work from the outside). This is described by your eq. (1).

A non-spontaneous reaction we can make thermodynamically allowed by adding enough work, this is your eq. (2). A typical chemistry lab example of the last is electrochemical splitting of water into oxygen and hydrogen gases by means of two electrodes and an external power supply.

Eq. (2) also tells the thermodynamic limit (i.e. maximum) electrical work one can possibly extract from the given electrochemical reaction, if it is spontaneous and fulfills eq. (1).

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