# Chemical potential and it's implications in an open system

Take the fundamental equation for the Gibbs free energy of a system:

$$dG=VdP-SdT+\sum_i\mu_idn_i$$

Thus, the chemical potential can be defined in terms of the Gibbs free energy as follows:

$$\mu_i=(\frac{\partial G}{\partial n_i})_{P,T}$$

Now, assuming that the chemical potential is positive (i'm not sure if that is always true) if I add $dn$ of component $i$, $dG>0$ which means (given that the Gibbs free energy is the capacity of a system to do non PV work) I have enabled the system to do more work. Consequently, I have done work on the system by adding more substance to it because i think I have increased it's potential - like raising a mass to a greater height.

It makes sense to me that adding a substance to a system would increase it's capacity to do work however, it doesn't make sense to me that work must be done to add moles of a substance. It's more clear when I think about the converse...

If I remove a small amount of the substance I reduce it's capacity for work and the system must therefore give out energy. This is confusing to me because surely i'd have to do work to remove a portion of a substance from a system - breaking the intramolecular forces and so on. However, for an ideal gas, there aren't any intramolecular interactions so unless the chemical potential for an ideal gas is zero (which I assume it's not) I must be missing something?

Too long for a comment, so here goes.

It's quite subtle and I'm really not sure how to explain in words, but the gist is that the chemical potential does not reflect a real, physical process of adding a substance that can be carried out. If that were the case, then the chemical potential would be the difference in the Gibbs free energies of the two systems:

• System 1: $n$ moles of substance and $\mathrm{d}n$ moles of substance, separated by some imaginary barrier
• System 2: $n + \mathrm{d}n$ moles of substance

and measuring the chemical potential would simply mean removing the imaginary barrier, allowing intermolecular attractions to form, etc. But that's not the case.

The chemical potential is the difference between these two systems:

• System 1: $n$ moles of substance
• System 2: $n + \mathrm{d}n$ moles of substance

That's all there is to it. Where do the extra $\mathrm{d}n$ moles come from? It doesn't come from anywhere. There's no real physical process going from System 1 to 2 here, so don't think of it as one.

Since Gibbs free energy is an extensive property, the $\mathrm{d}n$ moles of substance carry their own Gibbs free energy, just like they carry their own internal energy. That is why the chemical potential is positive - yes, the formation of intermolecular forces, etc. will affect the exact value of the chemical potential, and in a sense that is why the chemical potential is a partial derivative - but remember just simply having more of the substance means that there is more Gibbs free energy.

• Surely there must be a physical process? I'm not sure what you mean by there isn't one Jan 11, 2016 at 20:16
• No, there isn't. Physical processes must obey conservation laws - in going from system 1 to system 2 you are "adding extra substance that comes from nowhere". Don't think of it as having to represent some process, because by virtue of its mathematical definition, it just doesn't. It's weird! Jan 12, 2016 at 9:10