# Gibbs free energy ΔG=ΔG°+RTln(Kp) different expressions for ΔG

ΔG=ΔG°+RTln(Kp)

If I understand it correctly, the ΔG in this equation gives the "distance" in Gibbs free energy of a chemical system at a particular moment in a reaction (encoded by the Kp) to the gibbs free energy of the chemical system at equilibrium. So when we write this in symbols for a general equilibrium reaction A + B <=> C + D:

ΣG(products at eq) + ΣG(reactants at eq) - (ΣG(products at k) + ΣG(reactants at k)) with k denoting a random point with certain concentrations in the reaction and eq denoting the equilibrium point.

However, ΔG is also ΣG(products at k) - ΣG(reactants at k).

So how are these two expressions for ΔG related? Basically how would you prove that

ΣG(products at eq) + ΣG(reactants at eq) - (ΣG(products at k) + ΣG(reactants at k)) = ΣG(products at k) - ΣG(reactants at k)

• I can comprehend distance measured in seconds, but distance in Gibbs free energy is something new to me. Is it something related to the reaction coordinate? Also I'm not sure what exactly $ξ$ is here; I'm used to $ξ$ as extent of reaction when used alongside with Gibbs energy, but here it seems more like an arbitrary reaction coordinate. – andselisk Jan 20 '19 at 11:59
• You're right, that's confusing so I edited it. I meant "distance" in the sense that this site uses it chem.libretexts.org/Bookshelves/… – delivosa Jan 20 '19 at 12:02
• Oh. Then maybe you should've also quoted the word "distance" as your source did; it's more like a metaphor rather than a physical quantity. – andselisk Jan 20 '19 at 12:06

$$K_p$$ is normally reserved for equilibrium thus only apples when $$\Delta G=0$$ and conventionally $$Q$$ is used otherwise. In a plot of free energy $$G$$ vs extent of reaction $$\zeta$$, the slope is $$(\partial G/\partial \zeta)_{T,p} =\Delta G$$, and equilibrium occurs when the slope is zero then $$\Delta G^\mathrm{o} =-RT\ln(K_p)$$. (The extent of reaction is zero with only reactants present, and 1 when one mole of reactants has become product.)
Thus to speak of 'distance' does not mean the energy difference from equilibrium as $$\Delta G$$ is the slope not the value of the energy $$G$$ vs $$\zeta$$.
In the sums you ask about, always calculate products - reactants. The only experimental quantities you have are the concentrations or pressures so it would be possible to calculate $$\Delta G$$, using your first equation, if the values are known at equilibrium and at some other point. This seems to be what you ask about.