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.




The kinetic theory of matter says that matter consists of individual particles be they atoms, molecules, ions, nanoparticles, even elephants; the state of aggregation can have an effect. Let's restrict to molecules and ions and worry about the elephants when they recover in number. The second factor is that these molecules, ions etc. have thermal energy or heat content that is caused by the random velocity of the molecules [tempered by changes in quantum mechanical states, rotational, vibrational electronic]. The total energy of something is a constant under constant conditions and is determined by the temperature[or vice versa]. It has been noticed that reactions take place only under certain conditions. You must use your sense of observation, imagination, education etc to see what these conditions might be. One factor is actual physical state, another is distance between molecules, another is the identity of adjacent molecules [ie concentration or dispersion]. The way of describing all these factors is the nebulous chemical potential or activity of a chemical moiety in a mix. For ideal behavior these are the pressures, concentrations[molarity, molality], mole fractions. In real live they are chemical activities or chemical potentials. It takes a while to accept this. For now use it and try to think about what is going on and discard preconceived notions and what you learned in high school or college or even grad school It was all incomplete and possibly wrong, but still a fair start.


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