I'm perplexed by the relation of several bounds on the ability of a chemical reaction to do work.
Change in Helmholtz free energy F = E - TS places a bound on the total work (in all forms) extractable from a reaction at a constant temperature. The Carnot efficiency places a bound on the maximum work extractable from a heat engine operating between two heat baths.
One way we might extract work from a chemical reaction is to carry it out inside a closed vessel and allow the products to do PV work on the environment: we generically label such a scheme an Internal Combusion (IC) engine. If the reactants in an IC engine started at, and the products returned to, the temperature of an environmental bath T_e, then the applicability of Helmholtz is met, and delta F sets a limit on the work done. A theoretical maximum efficiency for the engine might therefore be defined as (delta F)/(delta E).
It is also common to consider an internal combustion engine as a heat engine, and as such claim it must be subject to the Carnot efficiency limit. An IC engine is certainly not a canonical heat engine (one having a closed cycle engine operating between between heat baths at T_hot and T_cold); it's an open cycle engine operating in a single heat bath at T_e. But it might be argued that if the reactants reached a temperature T_max during each cycle that the efficiency should be bounded by that of a Carnot engine operating between T_max and T_e : (T_max - T_e)/T_max .
Well, these limits certainly have very different forms! One makes no mention of the temperatures (except implicitly as T_e affects delta F and delta E), the other, no mention of the reactants (except implicitly as they affect T_max through delta E and the heat capacities). There is nothing inherently implausible about two independent limiting forms bounding a feasible region, possibly tight in different neighborhoods. Is this the case here? Or are these two ansatz cryptic equivalents?
The Helmholtz limit is seems necessarily theoretically correct to me, while the blunt application of the Carnot still seems fishy: we are a little vague on the "Q" from the non-existant hot reservoir (do we have to worry about the dependence of heat capacity on temperature and reactants vs. products?), and we seem to be ignoring the entropy of reaction in our working fluid. Clearly the IC engine is something like a heat engine, but how like and what something?
Comments, expert and otherwise, welcome.