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.
(a)

Verified step by step guidance

1. Step 1: Identify the substituent involved in the chair–chair interconversion. In this case, the substituent is a methyl group (CH₃). The methyl group is transitioning between the equatorial and axial positions on the cyclohexane ring.
2. Step 2: Refer to the table of A values provided in your textbook or study materials. The A value for a methyl group is typically around 1.74 kcal/mol. This represents the ∆G° for the transition from equatorial to axial position.
3. Step 3: Use the formula ∆G° = -RT ln(Keq) to calculate the equilibrium constant (Keq). Here, R is the gas constant (1.987 cal/mol·K), and T is the temperature in Kelvin (usually assumed to be 298 K unless otherwise specified). Rearrange the formula to solve for Keq: Keq = e^(-∆G°/RT).
4. Step 4: Substitute the A value (∆G° = 1.74 kcal/mol) into the formula. Convert ∆G° to cal/mol (1.74 kcal/mol = 1740 cal/mol) and plug in the values for R and T. Perform the calculation to determine Keq.
5. Step 5: Interpret the result. Since the A value is positive, the equilibrium will favor the equatorial position of the methyl group, meaning Keq will be greater than 1. This indicates that the equatorial position is more stable than the axial position.

Key Concepts

Chair Conformation of Cyclohexane

Cyclohexane can adopt a chair conformation, which minimizes steric strain and torsional strain. In this conformation, substituents can occupy either equatorial or axial positions. The equatorial position is generally more stable for larger substituents due to reduced steric hindrance with adjacent hydrogen atoms, making it crucial for understanding the stability of different substituents in cyclohexane.

A Values

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. These values are essential for calculating the equilibrium constant (K_eq) for chair-chair interconversions, as they quantify the preference of substituents for their respective positions.

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. Understanding K_eq is vital for predicting the stability of different conformations and the distribution of substituents in cyclohexane.