Why can one set the Gibbs free energy change to 0 when calculating the standard free energy change?

I am trying to understand the following equation:


$$\Delta G = \Delta G^{\circ} + RT \ln\left(\frac{[C][D]}{[A][B]}\right)$$


for a reversible reaction with reactants A and B and products C and D.


The way I understand it is that the Gibbs free energy change $\Delta G$ is built up by two components. One is the standard Gibbs free energy change $\Delta G^{\circ}$ which is something we can measure in the lab so to say and use as a reference point. The second term accounts for the given conditions such as actual temperature and reactant and product concentrations which deviate from those used to calculate/measure $\Delta G^{\circ}$.


My confusion stems from the following. In the textbook I am reading (Biochemistry by Berg et al) they say that:


"A simple way to determine $\Delta G^{\circ}$ is to measure the concentrations of reactants and products when the reaction has reached equilibrium. At equilibrium, there is no net change in reactants and products; in essence, the reaction has stopped and $\Delta G$ = 0." They then set $\Delta G = 0 $ and solve for $\Delta G^{\circ}$.


So they appear to be doing the backwards process of what I described above. Also why is this allowed at all? Isn't setting $\Delta G = 0 $ akin to enforcing a statement about the spontaneity of the reaction? The way I understood $\Delta G$ was that if it is positive the reaction is not spontaneous and if $\Delta G$ is negative the reaction is spontaneous. By extension if $\Delta G=0$ the reaction is at equilibrium, so I guess this means that $K=1$? Those should be things intrinsic to a given reaction, so how can we just set $\Delta G$=0? Also does $\Delta G$ change throughout a reaction at all? I mean, it clearly does, because it is a process that evolves in time but I thought that when we say $\Delta G$ we always take initial and the final free energy, so in that sense it is a constant and not an evolving quantity.


As you have probably gathered from the question I have no chemistry background so any help would be greatly appreciated.





> 
> Isn't setting $\Delta G = 0$ akin to enforcing a statement about the spontaneity of the reaction?
> 
> 
> 


Yes, it is. But they stipulate that the reaction is at equilibrium, which tells you that neither the forward nor backward reaction is spontaneous, which is precisely what allows you to assert that $\Delta G = 0$.\* In other words, **when a system is at equilibrium, we have $\Delta G = 0$**.



> 
> By extension if $\Delta G = 0$ the reaction is at equilibrium, so I guess this means that $K = 1$?
> 
> 
> 


No, that's not right, because we have $\Delta G^\circ = -RT \ln K$ (note the standard sign). $\Delta G$ has no direct relation to $K$.


One way of putting it is that $\Delta G^\circ$ and $K$ are properties of the *reaction*. That is, once you write down a chemical equation, these two are fixed, immutable constants.† However, $\Delta G$ (and $Q$) depend on the exact composition of the reacting system, i.e. they depend on the concentrations of reactants and products at any given point in time.



> 
> Also does $\Delta G$ change throughout a reaction at all?
> 
> 
> 


Yes, it does. As the reaction proceeds towards equilibrium, it goes from whatever value it is initially at, to zero.



> 
> I thought that when we say $\Delta G$ we always take initial and the final free energy
> 
> 
> 


... *Kind of.* You need to be careful here because the difference of a *free energy* should have units of free energy, traditionally kJ. However, $\Delta G$ is usually given in units of molar free energy, i.e. kJ/mol.


The reason is because what you're calling $\Delta G$ is actually not the difference between the free energy of products and free energy of reactants. It's actually a lazy shorthand for $\Delta\_\mathrm{r}G$, the *reaction Gibbs free energy*. Actually, $\Delta\_\mathrm{r}G$ is the difference between the [*chemical potentials*](https://en.wikipedia.org/wiki/Chemical_potential) of reactants and products in the reaction, multiplied by their stoichiometric coefficients.


I wrote about this difference before, see [What is the difference between ΔG and ΔrG?](https://chemistry.stackexchange.com/questions/50569), although I must apologise as it's a bit convoluted. (In fairness to myself, the topic in general is pretty convoluted.)




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\* A quick and dirty definition of spontaneity is that if you set something up and the reaction naturally goes in the net forward direction, the forward reaction is spontaneous. A formal definition would involve $\Delta G$, but this is not suitable in the present case, as it would be a circular argument.


† To be precise, they're constant for a given set of reaction conditions. $K$, and $\Delta G^\circ$, *will* change with temperature or pressure, for example. But that's not under consideration here.


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