Note: If not specified, assume a temperature of 298 K for all problems. (••) The following table of strain energies is associated with a variety of 1,2-gauche interactions. Use this table to answer the questions (i)–(iv). For each of the bond rotations shown, (i) identify which you believe to be more stable, (ii) calculate ∆G°, (iii) calculate the equilibrium constant, and (iv) draw the transition state. (c)

Verified step by step guidance

1. **Step 1: Identify the More Stable Conformation** - Examine the given bond rotations and identify the steric interactions present in each conformation. Use the table of strain energies to determine which conformation has lower energy and is therefore more stable.
2. **Step 2: Calculate $\mathrm{\Delta }{G}^{\circ }$ ** - Use the formula $\mathrm{\Delta }{G}^{\circ }=-RT\ln K_{eq}$ to calculate the free energy difference between the two conformations. Here, $R$ is the gas constant (8.314 J/mol·K) and $T$ is the temperature (298 K).
3. **Step 3: Calculate the Equilibrium Constant $K_{eq}$ ** - Rearrange the formula from Step 2 to solve for $K_{eq}$: $K_{eq}=e^{-\mathrm{\Delta }{G}^{\circ }/RT}$. Substitute the $\mathrm{\Delta }{G}^{\circ }$ value obtained in Step 2 to find $K_{eq}$.
4. **Step 4: Draw the Transition State** - Identify the transition state for the bond rotation. This involves drawing the highest energy conformation that occurs during the rotation, typically a staggered or eclipsed conformation depending on the specific interactions.
5. **Step 5: Analyze the Results** - Review the calculated $\mathrm{\Delta }{G}^{\circ }$ and $K_{eq}$ values to ensure they are consistent with the identified more stable conformation. Confirm that the transition state drawing accurately represents the energy barrier for the rotation.

Key Concepts

Strain Energy

Strain energy refers to the energy stored in a molecule due to the distortion of its normal geometry. In organic chemistry, this often arises from interactions such as steric hindrance or torsional strain, particularly in conformational isomers. Understanding strain energy is crucial for predicting the stability of different conformations, as lower strain energy typically correlates with greater stability.

Gibbs Free Energy (∆G°)

Gibbs Free Energy (∆G°) is a thermodynamic potential that indicates the spontaneity of a reaction at constant temperature and pressure. A negative ∆G° suggests that a reaction is spontaneous, while a positive value indicates non-spontaneity. In the context of bond rotations and conformational analysis, calculating ∆G° helps determine the relative stability of different conformers.

Equilibrium Constant (K)

The equilibrium constant (K) quantifies the ratio of the concentrations of products to reactants at equilibrium for a reversible reaction. It is directly related to the Gibbs Free Energy change (∆G°) through the equation ∆G° = -RT ln(K), where R is the gas constant and T is the temperature in Kelvin. Understanding K is essential for evaluating the stability of different conformations and predicting the favored conformer in a given reaction.