After doing some research and reading I found some problems which I will try to state as clearly as possible.
The definition of Gibbs Free Energy says "the greatest amount of mechanical work which can be obtained from a given quantity of a certain substance in a given initial state, without increasing its total volume or allowing heat to pass to or from external bodies, except such as at the close of the processes are left in their initial condition. (I have taken the definition from the wikipedia). The equation of the Gibbs energy $$ G= H -TS$$ is also clear.
But the way I understand it is this: the maximum non-expansion work , like transporting an electron, breaking a chemical bond, moving real life things etc. which can be obtained from a system . These things made sense during the study of Second Law of thermodynamics but as I moved to Chemical Equilibrium these concepts began to tremble (for me).
In elementary classes it is said that Equilibrium is a state where the composition of reactants and products do not change over time, but in higher chemistry classes it is said that equilibrium is a condition corresponds to $$ \Delta G = 0$$ So, my first question is how do these two concepts mean same thing? The next thing which causes problem is chemical potentials. If we adhere to the formal meaning of potential i.e. something which is stored and can be used when proper conditions are met, so chemcial potential would mean the potential of substance to react and again this is related to Gibbs energy whose definition I gave above. So, how do chemical potential and Gibbs energy can have any relation?
A question which is off topic over here but I want to mention it, why do we bother so much about standard things like $$ \mu_\mathrm A = \mu_\mathrm A^\circ + RT \ln(p_\mathrm A)$$ why we wanted to express it in that standard (that little circle) form ?
I want to make myself clear that the conception of Gibbs energy was quite clear to me in the context of thermodynamics, we simply meant it to be the work which can be extracted from a substance, but it all-pervading use has made me to doubt myself, just like in mathematics the number $\mathrm e$ appears in odd places. Even if Gibbs energy (according to the understanding that gave above in bold) is appearing mathematically then also it would be having some physical meaning because Thermodynamics and Equilibrium are natural sciences and not the mathematics.
Thank you, any help will be much appreciated.