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murillo137

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- TL;DR Summary
- Why exactly is the Gibbs energy minimized at equilibrium, even when the number of particles/molecules varies?

I have a rather basic question regarding the chemical potential (##\mu##) in thermodynamics and its relation to the Gibbs free energy (##G##). All thermodynamics textbooks I've looked at (Landau & Lifshitz, Kittel...) derive the fact that, at constant temperature ##T## and pressure ##p##, the Gibbs free energy is minimized. The proof is also in Wikipedia (https://en.wikipedia.org/wiki/Gibbs_free_energy). It uses the definition ##G = U +pV - TS ##, such that at constant ##T## and ##p##:

$$dG = dU + pdV -TdS$$.

The First Law of thermodynamics reads ##dU = \delta Q - pdV + \sum_{i} \mu_{i} dN_{i}##. The Second Law reads ##TdS \geq \delta Q##, giving:

##TdS \geq dU + pdV - \sum_{i} \mu_{i} dN_{i} ##

##dU + pdV - TdS \leq \sum_{i} \mu_{i} dN_{i} ##

##dG \leq \sum_{i} \mu_{i} dN_{i} ##

If the number of particles stays constant, then clearly ##dG\leq 0##, such that ##G## is minimized. My issue is, usually textbooks seem to then just assume that ##G## is minimized, even when the number of particles can change (e.g. in a chemical reaction), such that at equilibrium ##dG=0=\sum_{i} \mu_{i} dN_{i}##. For a reaction like ##A + B \to C + D##, one can thus derive a relation between chemical potentials:

##\mu_{A} + \mu_{B} - \mu_{C} - \mu_{D} = 0##

My issue is that in the proof the ##G## is minimized, it seems necessary to assume a constant number of particles, yet textbooks derive the relationship between the chemical potentials assuming that this is true regardless of the fact that naturally the number of particles of different types changes in a chemical reaction. How does one reconcile this? Can anyone give a rigorous proof of the minimization of ##G##, even in the case where chemical reactions can occur?

$$dG = dU + pdV -TdS$$.

The First Law of thermodynamics reads ##dU = \delta Q - pdV + \sum_{i} \mu_{i} dN_{i}##. The Second Law reads ##TdS \geq \delta Q##, giving:

##TdS \geq dU + pdV - \sum_{i} \mu_{i} dN_{i} ##

##dU + pdV - TdS \leq \sum_{i} \mu_{i} dN_{i} ##

##dG \leq \sum_{i} \mu_{i} dN_{i} ##

If the number of particles stays constant, then clearly ##dG\leq 0##, such that ##G## is minimized. My issue is, usually textbooks seem to then just assume that ##G## is minimized, even when the number of particles can change (e.g. in a chemical reaction), such that at equilibrium ##dG=0=\sum_{i} \mu_{i} dN_{i}##. For a reaction like ##A + B \to C + D##, one can thus derive a relation between chemical potentials:

##\mu_{A} + \mu_{B} - \mu_{C} - \mu_{D} = 0##

My issue is that in the proof the ##G## is minimized, it seems necessary to assume a constant number of particles, yet textbooks derive the relationship between the chemical potentials assuming that this is true regardless of the fact that naturally the number of particles of different types changes in a chemical reaction. How does one reconcile this? Can anyone give a rigorous proof of the minimization of ##G##, even in the case where chemical reactions can occur?