The ability to do work

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.

Reply to
Edward Green
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Hardly an expert opinion, but.... IC engines work on the pressure of the working fluid. This depends on temperature. The cold side of the heat engine is the exhaust, the hot side the temp of the burning fuel. Thus the Carnot cycle determines the maximum work available. Doesn't Helmholtz have the constraint of constant volume - not applicable to IC engines?


Reply to

With the IC cycles, the gas is compressed before the reaction is started...thus the initial state of reactants is not T_e.

Actual engine cycles are very complex, the simplification to a closed cycle of ideal gas is a good start, but start taking into account factors like change in mass ( adding fuel in liquid form in a diesel ) Evaporation of fuel ( petrol engine ), change of gas composition, ( 02 + fuel to CO2 + H2O.... ) and it would be a lot more accurate..... IF you could get close to modelling it as a cycle.

Have you thought of the biological way of extracting energy, reacting chemicals to change structure, and extracting the energy from the structure change... Judging by the amount of exercise I have to take to use up fat it must be very efficient :-(

-- Jonathan

Barnes's theorem; for every foolproof device there is a fool greater than the proof.

To reply remove AT

Reply to
Jonathan Barnes

< snip my own introductory boring and ineffective rant on the value of resort to elementary texts before posting >

If you insist on a 'canonical heat engine,' it is one that runs an ideal gas through a Carnot cycle between a high temperature at T_2 and a sink at T_1:

Process Description Heat input Work Output

1-2 isentropic compression 0 -c_v(T_2-T_1) 2-3 isothermal expansion RT_2ln(v_3/v_2) RT_2ln(v_3/v_2) 3-4 isentropic expansion 0 -c_v(T_4-T_3) 4-1 isothermal compression -RT_4ln(v_4/v_1) -RT_4ln(v_4/v_1)

T_3 = T_2, T_4 = T_1 c_v is the specific heat at constant volume; the ideal gas law applies (v_2/v_1)=(T_1/T_2)^(1/(k-1)), etc.

Add up the net work out and divide it by the heat added at T_2 and you get the familiar

Carnot efficiency = (T_2 - T_1)/T_2

The common model for the IC engine is the Otto cycle:

Process Description

1-2 isentropic compression 2-3 heating at iso volume 3-4 isentropic expansion 4-1 heat rejection at isovolume

If you fill in the blanks for the heat input and work output, add them up, and calculate the efficiency for a ideal-gas Otto cycle, you'll find that it's independent of temperatures. In its simplest form, it's dependent only on the compression ratio.

If you insist, you can show that the Otto efficiency is less than the Carnot efficiency for any finite compression ratio.

Analysis at this level is conducted as though the system is closed. It's something of a hand-waving exercise to get the Otto cycle to fit an open system in which the 4-1 process - the heat rejection at isovolume - is accomplished by exhausting the old charge and replacing it with new at sink temperature.

Helmholz Free Energy at this level of discussion is a red herring.

hth, Fred Klingener

Reply to
Fred Klingener

(1) I had thought at first to ignore that complication, and consider an IC engine which burns initially uncompressed gas, and (2) who says we have to use the state just prior to ignition as the initial state, anyway?

The situation is complicated, but I don't think precompression is the major complication.

The change in composition seems to be a chief complication to modeling the thing as a closed cycle: that is something the working fluid in an ideal heat engine does not do. I hadn't thought of evaporation of fuel.

I take it you are replying from sci.engr.mech. What motivated my cross-post was a kind of pop-science quiz posted to sci.physics which glibly assumed an IC engine mapped easily to a heat engine for the purposes of applying the Carnot limits -- not so fast I say. But the relation of the IC approach in general to extracting work from a chemical reaction to the thermodynamic limits from changes in state functions like Gibb's and Helmholtz free energies is something I've long been puzzled by.

The mechanism of the Na/K ion pump in the cell membrane approachs a Maxwell's demon in atomic specificity. Presumably it's not _actually_ a Maxwell demon ... but if you proposed its mechanism as a possible nano-technology to a scientifically trained but micro-biologically naive audience, it would probably be objected to on those grounds!

Reply to
Edward Green

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