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Internal energy, enthalpy and Gibbs free energy are all units of energy. So, for any particular process where energy is lost from the system the same amount is given off to the surroundings. Am I right in thinking that a change in internal energy/free energy/enthalpy is all a measure of how the energy of the system changes in a process and as a result should be equal to each other for the same process, albeit under different conditions?

Let me expand slightly: $$dU<TdS-pdV$$ So for a spontaneous change to occur, the internal energy must decrease provided that the entropy and the volume are kept constant. The internal energy will continue to fall until equilibrium is reached. The exact amount of energy lost by the system is gained by the surroundings. As a side question: how exactly is the energy lost to the surroundings? $$dH<TdS+VdP$$ The enthalpy of the system decreases for a spontaneous change, provided that the entropy and pressure are kept constant. Energy lost by system = energy gained by surroundings. $$dG<VdP-SdT$$ The Gibbs free energy of the system decreases for a spontaneous change, provided that the temperature and pressure are kept constant. Energy lost by system = energy gained by surroundings.

For a given process, surely the change in internal energy (at constant S,V) be equal to the change in enthalpy (at constant P,S) and Gibbs free energy (at costant P,V), right? The above argument seems to make sense to me but enthalpy can be defined as $H=U+PV$ which would suggest that the two values would be different wouldn't it?

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While enthalpy, internal energy, and Gibbs's energy are all types of energy, they are not all measuring the same quantities. Internal energy is the energy of the molecules in your system. Enthalpy takes into account the internal energy plus whatever work is being done by or on your system. And Gibbs's energy is the maximum amount of non-PV work that you can get from your system. While these three quantities are all in units of energy, they are not just different terms for the total energy. Since enthalpy and Gibbs's energy involve work, and hence exchange of energy with the surroundings, these quantities will be different than just the internal energy of the system alone.

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