Why does the equation for chemical potential apply to liquids and aqueous solutes?

The equation for chemical potential of a gas can be derived as such:

At constant temperature,

$$\mathrm dG = V\,\mathrm dP \label{eqn:1}\tag{1}$$

Substituting with the ideal gas law $$PV = nRT$$,

$$\int_{p^\circ}^p\mathrm dG = \int_{p^\circ}^p \frac{nRT}{p}\mathrm dP \label{eqn:2}\tag{2}$$

Then

$$\mu = \mu^\circ + RT\ln\frac{p\phantom{^\circ}}{p^\circ} \label{eqn:3}\tag{3}$$

What I don't get is how this equation, which was derived using the ideal gas law, then becomes applicable to liquids as well as aqueous solutes, where:

$$\mu = \mu^\circ + RT\ln\frac{[\ce{A}]\phantom{^\circ}}{[\ce{A}]^\circ} \label{eqn:4}\tag{4}$$

where $$[\ce{A}]$$ is either $$\ce{A(l)}$$ or $$\ce{A(aq)}$$.

Note: I am aware that for liquids, at equilibrium the chemical potential of the liquid phase is equal to the vapor phase, and by considering Raoult's law:

$$\mu = \mu^\circ + RT\ln\chi_A \label{eqn:5}\tag{5}$$

But how would I go from \eqref{eqn:5} to \eqref{eqn:4}?

I know these are probably quite trivial but I would appreciate any help.

• What is A ? Is it the generic name of any chemical ? Is it the absorbance of a colored stuff ? Why don't you call $p$ as $p$(A) or ${p_A}$ ? What is $\pu{\chi}_A$ ? Is it the molar fraction ? Anyway you can go from $(3)$ to $(4)$ by admitting that the concentration [$\pu{A}$] of a gaseous stuff $\pu{A}$, once dissolved in a liquid is proportional to its pressure $p$ in the gas phase (Henry's law) May 18, 2022 at 19:33
• In general you would use activities or fugacities instead of concentrations or mole fractions. May 30, 2022 at 18:13