# When are thermodynamical potentials conserved?

I have studied that are 4 thermodynamical potentials wich are useful, which are the internal energy $$U$$, the helmholtz function $$F$$, the Gibbs free energy $$G$$ and the enthalpy $$H$$:

$$dU=TdS-PdV$$ $$dH=TdS+VdP$$ $$dG=VdP-SdT$$ $$dF=-SdT-PdV$$

where, $$T$$ is temperature, $$S$$ is entropy, $$V$$ volume and $$P$$ pressure. But when do we know that these processes are conserved? For example I know (and is somehow intuitive) that in an isolated system internal energy has to be conserved. But what about the others?

• Entropy is conserved in reversible processes taking place in an isolated system. Commented Dec 3, 2018 at 23:54

First, let's write the complete form of these differentials:

Energy $$dU = T dS - P dV + \mu dN$$: To get $$dU = 0$$ we must set $$dS=0$$, $$dV=0$$, $$dN=0$$. This means adiabatic+constant volume+closed.

Enthalpy $$dH = T dS + V dP + \mu dN$$: To get $$dH=0$$ set $$dS=0$$, $$dP=0$$, $$dN=0$$. This means adiabatic+isobaric+closed.

Gibbs energy $$dG = -S dT+V dP + \mu dN$$. To get $$dG=0$$ set $$dT=0$$, $$dP=0$$, $$dN=0$$. This means isothermal+isobaric+closed

Free energy $$dF = - SdT - P dV + \mu dN$$: The get $$dF=0$$ set $$dT=0$$, $$dV=0$$, $$dN=0$$. This means isothermal+constant-volume+closed.

Not much memorization needed if you just follow the rules of calculus.

• I don't think I ever saw $\mu dN$. What is it? Other than that good answer Commented Dec 4, 2018 at 10:45
• $\mu$ is the chemical potential. The internal energy is a function of entropy, volume, and number of particles. When we write $dU = TdS - P dV$, we imply a closed system, i.e., $dN=0$. If we allow $N$ to change, then the full form of the differential is the one I gave above. Sam with all other potentials. Commented Dec 4, 2018 at 11:07