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In electrochemistry the Nernst equation says Gibbs=-nFE and the reversible heat=TdS= T*partialE partialT * -nF but the Bernardi heat equation is q= I(Voc-V+TpartialE partialT) when we know that the heat of reaction/ enthalpy = dG+TdS,= IVoc+ ITpartialE partialT. Why is Bernardi heat equation using reversible heat not enthalpy?

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because the enthalpy is the heat of reaction only at constant temperature and pressure in the absence of non-pv work and in a closed system. In the case of a battery we are definitely not in the absence of non-pv work

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