The A value of a substituent on a cyclohexane ring is essentially the ∆G° for a substituent going from the equatorial to the axial position in a chair–chair interconversion. Because most substituents prefer to be in the equatorial position, A values are, by definition, positive numbers. Use the table of A values to calculate ∆G° and K_eq for the chair–chair interconversions shown.
(b)

Verified step by step guidance

1. Step 1: Understand the concept of A values. A values represent the free energy difference (∆G°) between the equatorial and axial positions of a substituent on a cyclohexane ring during a chair–chair interconversion. Positive A values indicate that the equatorial position is favored.
2. Step 2: Recall the relationship between ∆G° and the equilibrium constant (K_eq). The equation is given by: $\mathrm{∆G}°=-RTlnK\_\mathrm{eq}$, where R is the gas constant (8.314 J/mol·K), T is the temperature in Kelvin, and K_eq is the equilibrium constant.
3. Step 3: Use the provided table of A values to find the ∆G° for the substituent in question. The A value directly corresponds to the ∆G° for the equatorial-to-axial transition.
4. Step 4: Rearrange the equation to solve for K_eq: $K\_\mathrm{eq}=e^{-\mathrm{∆G}°/RT}$. Substitute the values for ∆G°, R, and T into the equation to calculate K_eq.
5. Step 5: Interpret the results. If K_eq > 1, the equatorial position is strongly favored. If K_eq < 1, the axial position is favored. Use the calculated values to understand the preference of the substituent in the chair–chair interconversion.

Key Concepts

Chair Conformation of Cyclohexane

The chair conformation is the most stable arrangement of cyclohexane, minimizing steric strain and torsional strain. In this conformation, carbon atoms adopt a staggered arrangement, allowing substituents to occupy either equatorial or axial positions. Equatorial substituents are generally more stable due to reduced steric hindrance with adjacent hydrogen atoms, making understanding this conformation crucial for analyzing substituent effects.

A Values and Free Energy Change (∆G°)

A values represent the free energy change (∆G°) associated with the transition of a substituent from the equatorial to the axial position in cyclohexane. Positive A values indicate that substituents favor the equatorial position due to lower energy and greater stability. Understanding A values is essential for calculating the equilibrium constant (K_eq) for chair-chair interconversions, as they directly influence the stability of different conformations.

Equilibrium Constant (K_eq)

The equilibrium constant (K_eq) quantifies the ratio of the concentrations of products to reactants at equilibrium for a reversible reaction. In the context of chair-chair interconversions, K_eq can be calculated using the relationship K_eq = e^(-∆G°/RT), where R is the gas constant and T is the temperature in Kelvin. This concept is vital for understanding the relative stability of different conformations and predicting the favored position of substituents in cyclohexane.