You remember that \(\Delta G^{\circ}\) is related to \(R T \ln (K)\) but cannot remember if it's \(R T \ln (K)\) or \(-R T \ln (K) .\) Realizing what \(\Delta G^{\circ}\) and \(K\) mean, how can you figure out the correct sign?

\(\Delta G^{\circ}\) is the standard change in Gibbs free energy, which is a measure of the maximum reversible work that can be done by a system at constant temperature and pressure. A negative \(\Delta G^{\circ}\) indicates a spontaneous reaction, while a positive \(\Delta G^{\circ}\) implies a non-spontaneous reaction. \(K\) is the equilibrium constant, representing the ratio of the concentrations of products to reactants at equilibrium. If \(K > 1\), the reaction favors the formation of products, while if \(K < 1\), the reaction favors the formation of reactants.

We can reason that if a reaction has a negative \(\Delta G^{\circ}\) (spontaneous), it should favor the formation of products, so \(K > 1\) in this case. Conversely, if a reaction has a positive \(\Delta G^{\circ}\) (non-spontaneous), it should favor the formation of reactants, so \(K < 1\) in this situation. From these observations, we can conclude that a more negative \(\Delta G^{\circ}\) corresponds to a larger value of \(K\), and a more positive \(\Delta G^{\circ}\) corresponds to a smaller value of \(K\)

Now, let's consider the relationship between \(\Delta G^{\circ}\) and \(R T \ln(K)\). Based on Step 2, if the value of \(K\) increases, the value of \(\Delta G^{\circ}\) should become more negative. If the relationship were given as \(\Delta G^{\circ} = R T \ln(K)\), then as \(K > 1\), we would have a positive \(\ln(K)\). In this case, the product of \(R T \ln(K)\) would also be positive, which contradicts our observation that a larger \(K\) should correspond to a more negative \(\Delta G^{\circ}\). However, if the relationship were given as \(\Delta G^{\circ} = -R T \ln(K)\), then as \(K > 1\), we would again have a positive \(\ln(K)\), but this time the negative sign in the relationship would make \(\Delta G^{\circ}\) more negative, which is consistent with our observation in Step 2. Therefore, the correct relationship between \(\Delta G^{\circ}\) and \(R T \ln(K)\) is: \[\Delta G^{\circ} = -R T \ln(K)\]