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?

First, let's recall that ∆G° is the standard Gibbs free energy change which indicates whether a reaction is thermodynamically favorable or unfavorable. A negative ∆G° implies a spontaneous reaction, and a positive ∆G° means a non-spontaneous reaction. K is the equilibrium constant, which represents the ratio of the concentrations (or activities) of the products to the reactants at equilibrium. A larger K value indicates that the reaction favors the formation of products, while a smaller K value means that the reactants are favored. When K is greater than 1, the reaction favors products, and when K is less than 1, the reaction favors reactants.

Now let's consider the possible cases to understand the relationship between ∆G° and K: 1. If ∆G° is negative (favorable reaction), the reaction will proceed spontaneously, meaning the products are favored over the reactants. In this case, we expect K to be greater than 1. 2. If ∆G° is positive (unfavorable reaction), the reaction is not spontaneous, indicating that the reactants are favored over the products. In this case, we expect K to be less than 1.

We have two possible expressions, either ∆G° = RTln(K) or ∆G° = -RTln(K). We will consider both cases to see which one matches the relationship between ∆G° and K deduced in Step 2. Case 1: ∆G° = RTln(K) If ∆G° is negative, then ln(K) must also be negative, which implies K < 1. However, this goes against our understanding of the relationship between ∆G° and K, as a negative ∆G° should favor products (K > 1). Case 2: ∆G° = -RTln(K) If ∆G° is negative, then ln(K) must be positive, which implies K > 1. This matches our understanding of the relationship between ∆G° and K as deduced in Step 2. Based on this analysis, we conclude that the correct sign in the expression is negative, and the correct relationship is: ∆G° = -RTln(K)