The correct relationship between free energy change in a reaction and the corresponding equilibrium constant K_C is

The Gibb's free enrgy change and E_cell are related as 

$$\begin{gathered} ∆G=∆G^o+\hspace{0.17em}RT\hspace{0.17em}\ln Qc \\ Qc = reaction quotient \end{gathered}$$

At state of equilibrium 

$Qc= Kc and ∆G= 0$

Then the equation is written as 

$$\begin{gathered} 0=∆G^o+\hspace{0.17em}RT\hspace{0.17em}\ln Kc \\ -∆G^o= RT \ln Kc \\ ∆G^o =-RT \ln Kc \end{gathered}$$