Spontaneous Reaction and Gibbs Free Energy I know how to derive the expression for the equilibrium constant starting from $\mathrm dG=0$,
$$\ln(K)=-\frac{\Delta G ^\circ}{RT}$$

However, if we are not at equilibrium, $\mathrm dG$ is not necessarily zero. In this case:
$$\frac{\partial G}{\partial \epsilon} = \sum \mu_i \nu_i = \Delta G ^\circ + RT \ln \left( \prod a_i ^{\nu_i} \right)$$

Where $\nu_i$ are the stoichiometric coefficients, $\mu_i$ are the chemical potentials, $\Delta G^\circ$ is the Gibbs free energy of reaction at standard state and $a_i$ are the activities.

Hence,
$$\frac{\partial G}{\partial \epsilon} = RT \ln \left(\frac{Q}{K} \right)$$

Where $Q=\prod a_i ^{\nu_i}$ is the reaction quotient.

So the question is, if we want to kown if a reaction is spontaneous, do we look at the ratio $Q/K$? If this is correct, if $Q &gt; K$ then $\frac{\partial G}{\partial \epsilon} &gt; 0$ and the reaction would be non-spontaneous, whereas if $Q &lt; K$ then $\frac{\partial G}{\partial \epsilon} &lt; 0$ and the reaction is spontaneous.

If so, if we start with reactants only, $Q=0$ (since the activites of the products are zero because there are none) and the reaction would be spontaneous, and we would always get at least a little bit of products.

However, various pages in internet say that if $\Delta G ^\circ &lt; 0$ then the reaction is spontaneous, and it is not if $\Delta G ^\circ&gt;0$.

So how do you know, then, if you start with only reactants, if a reaction is going to happen?
I think you are confusing two related concepts here. Spontaneity of a reaction is determined by the net change in free energy, $\Delta G$ - notice the capital Delta.

Which direction you will move along a reaction coordinate is determined by $dG$ - note the little $d$.

$\Delta G$ is the overall change in free energy when comparing products to reactants. It tells you whether the reactants are "uphill" or "downhill" of the products.

$dG$ is the slope of the free energy surface at your current position on the reaction coordinate. In the equation you gave, that's given by Q/K. K basically tells you what the equilibrium ratio between reactants and products is, and Q tells you what the current ratio is. The ratio of Q/K is an indication of how far to the left or right you are of equilibrium.

This picture might help explain it a little better:



Image the free energy surface as a gravitational potential energy surface (we can get away with this because free energy is the thermodynamic potential for systems at constant T &amp; P). The ball represents the reacting system, and the x-axis is a reaction coordinate.

$dG$ (the differential) tells us the slope at a given point - it tells us where the system is going to go right now. $\Delta G$ tells us the overall change - it tells us where the system is going to eventually end up.

When you look up standard state $\Delta G$ values for a reaction, you are determining whether an overall reaction is spontaneous or not - you are figuring out which side of the equilibrium has more stuff - reactants or products.

When you look at Q, you are figuring out which direction things need to go to get back to equilibrium, based on where you currently are.