I'm not sure what problem you are working on but if temperature is constant, Carnot efficiency is zero. See my paper on isothermal efficiency if the temperature is constant: http://alum.mit.edu/www/toms/papers/emmgeo/

I don't know if this can help you.

http://www.chem.umd.edu/files/chem/doc/Jarzynski_AnnuRevCondMattPhys_2_329_2011.pdf

http://arxiv.org/pdf/1204.4895.pdf

http://www.bourbaphy.fr/jarzynskitemps.pdf

I hope it will help you.

Best regards and best of luck for your research.

A little remark, the subject relates irreversibility nor the Clausius-Clapeyron equation related with the change of pressure against temperature (dp/dT)in the phase coexistence. In solid-solid phase transitins the pressure was changed by the stress. In your case, the potential for diffusion is, probalby, the chemical potential.

As far as I known, The Clausius-Clapeyron relation should express the slope of the curve in the P-T plane that characterizes discountinuous transitions.

If you want an analogous of the Clausius inequality for open systems

(in which particles can leave and arrive into the systems)

maybe

dQ -(mu_int - mu_ext) dN= TdS < 0

where mu is the chemical potential of the specie(internal and external)

and dN the number of particles exchanged...

(see also "modern Thermodynamics" by Prigogine and Kondepudi)

thank you Daniele, all

I'll start from "modern Thermodynamics"

if I understand your question, I think you are looking for another inequality analogous to dQ/T (away from equilibrium), which Daniele Martino answered.

I am not sure how the Clausius-Clapeyron equation is brought into this question, but if you are looking at a phase change, dN can also relate molecules exchanging between phases. Most generally, dN relates to exchange between thermodynamic systems or subsystems with different chemical potentials. This exchange can happen by diffusion (change of location), phase change or chemical reaction. The phase change provides a link with Clausius Clapeyron, but macroscopic diffusion is usually not considered as the mechanism of exchange between phases.

I am to remark an older book: ter Haar and Wergeland, "Elements of Thermdynamics" Addison Wesley (1968 ?). In less that 200 pages furnish an excellent overview of classical thermodhynamics.

Robert, I never wrote the title of my question.

Clapeyron has nothing to do with my question.

As an engineer I know well the second law when we deal with heat.

So I know the II law as ds=dQ/T +dss, where the last term(>0) is the entropy production (irreversibility). I know irreversibility is not linked only to spontaneous transfer of heat but to all spontaneous processes , for instance to any diffusion processes, eg species diffusion.I know that where we see a spontaneous flux of something if we put some kind of motor in that gradient then we can exploit that and the universe become old more slowly(in some sense). Usually we studied that given two reservoirs, one hot and one cold, we can put a cyclic motor between them and convert a fraction of heat power (energy diffusion) into work. on that system the clausius inequality holds: Integral(cycle)[dQ/T]

In isothermal diffusion process the maximum work relates the change of Gibbs function between the initial composition of the mixture and the equilibrium composition o the mixture. According the second law this is the MAXIMUM work, and the practical work is, obviously lower. This is a classical problem related to a N pure substances to joint in a mixture of N components. What is the maximal work: the change of the GIbbs fucntion. The same is valid from the one initial mixture to one final mixture. If the process is spontaneous work is losed. At the contrary, the mixture is separated you need work.