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Chapter 17: Problem 14

On a cyclohexane ring, an axial carboxyl group has a conformational energy of \(5.9 \mathrm{~kJ}\) (1.4 kcal)/mol relative to an equatorial carboxyl group. Consider the equilibrium for the alternative chair conformations of trans-1,4-cyclohexanedicarboxylic acid. Draw the less stable chair conformation on the left of the equilibrium arrows and the more stable chair on the right. Calculate \(\Delta G^{0}\) for the equilibrium as written, and calculate the ratio of the more stable chair to the less stable chair at \(25^{\circ} \mathrm{C}\).

Short Answer

Expert verified
Answer: The ratio of the more stable chair conformation to the less stable chair conformation for trans-1,4-cyclohexanedicarboxylic acid at 25°C is approximately 357:1.

Step by step solution

01

Drawing chair conformations

To draw the chair conformations of trans-1,4-cyclohexanedicarboxylic acid, imagine a regular cyclohexane ring with two carboxyl groups attached at positions 1 and 4. We'll consider one conformation to have both carboxyl groups in axial positions and the other to have both carboxyl groups in equatorial positions. The less stable chair conformation will have both carboxyl groups in axial positions and the more stable chair conformation will have both carboxyl groups in equatorial positions.
02

Calculating the Gibbs free energy change

The given conformational energy of axial carboxyl group relative to an equatorial carboxyl group is 5.9 kJ/mol. Since there are two carboxyl groups, the overall Gibbs free energy change for this equilibrium is the sum of the energies for the individual carboxyl groups: \(\Delta G^{0} = 2\mathrm{~} \times \mathrm{~} (-5.9 \mathrm{~kJ/mol}) = -11.8 \mathrm{~kJ/mol}\)
03

Calculating the equilibrium constant

To calculate the equilibrium constant, we need to use the Gibbs free energy change and the temperature. The formula for the equilibrium constant (K) is: \(K = e^{-\frac{\Delta G^{0}}{RT}}\) where R is the gas constant (8.314 J/(mol·K)) and T is the temperature in Kelvin (25°C = 298.15 K). Plug in the values: \(K = e^{\frac{11.8 \times 10^3 \mathrm{~J/mol}}{(8.314 \mathrm{~J/(mol·K)})(298.15 \mathrm{~K})}}\) \(K \approx 357\)
04

Calculating the ratio of the more stable chair to the less stable chair

The equilibrium constant is equal to the ratio of the more stable chair to the less stable chair. Therefore, the ratio is approximately 357:1. The more stable chair conformation with both carboxyl groups in equatorial positions is 357 times more favorable than the less stable chair conformation with both carboxyl groups in axial positions at 25°C.

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Most popular questions from this chapter

The reaction of an \(\alpha\)-diketone with concentrated sodium or potassium hydroxide to give the salt of an \(\alpha\)-hydroxyacid is given the general name benzil-benzilic acid rearrangement. It is illustrated by the conversion of benzil to sodium benzilate and then to benzilic acid. Propose a mechanism for this rearrangement. O=C(O)C(=O)C(=O)c1ccccc1 O=C(O)C(O)(c1ccccc1)C(O)(c1ccccc1)c1ccccc1 Benzil Sodium benzilate Benzilic acid

Select the stronger acid in each set. (a) Phenol \(\left(\mathrm{p} K_{\mathrm{a}} 9.95\right)\) and benzoic acid \(\left(\mathrm{p} K_{\mathrm{a}} 4.19\right)\) (b) Lactic acid \(\left(K_{a} 8.4 \times 10^{-4}\right)\) and ascorbic acid \(\left(K_{a} 7.9 \times 10^{-5}\right)\)

Given here are \({ }^{1} \mathrm{H}-\mathrm{NMR}\) and \({ }^{19} \mathrm{C}-\mathrm{NMR}\) spectral data for nine compounds. Each compound shows strong absorption between 1720 and \(1700 \mathrm{~cm}^{-1}\), and strong, broad absorption over the region \(2500-3300 \mathrm{~cm}^{-1}\). Propose a structural formula for each compound. Refer to Appendices 4,5, and 6 for spectral correlation tables. $$ \begin{aligned} &\text { (a) } \mathrm{C}_{5} \mathrm{H}_{10} \mathrm{O}_{2}\\\ &\begin{array}{cc} \hline{ }^{1} \text { H-NMR } & { }^{13} \text { C-NMR } \\ \hline 0.94(\mathrm{t}, 3 \mathrm{H}) & 180.71 \\ 1.39(\mathrm{~m}, 2 \mathrm{H}) & 33.89 \\ 1.62(\mathrm{~m}, 2 \mathrm{H}) & 26.76 \\ 2.35(\mathrm{t}, 2 \mathrm{H}) & 22.21 \\ 12.0(\mathrm{~s}, 1 \mathrm{H}) & 13.69 \\ \hline \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { (b) } \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{2}\\\ &\begin{array}{cc} \hline{ }^{1} \text { H-NMR } & { }^{19} \text { C-NMR } \\ \hline 1.08(\mathrm{~s}, 9 \mathrm{H}) & 179.29 \\ 2.23(\mathrm{~s}, 2 \mathrm{H}) & 47.82 \\ 12.1(\mathrm{~s}, 1 \mathrm{H}) & 30.62 \\ & 29.57 \\ \hline \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { (c) } \mathrm{C}_{5} \mathrm{H}_{8} \mathrm{O}_{4}\\\ &\begin{array}{cc} \hline{ }^{1} \text { H-NMR } & { }^{13} \text { C-NMR } \\ \hline 0.93(\mathrm{t}, 3 \mathrm{H}) & 170.94 \\ 1.80(\mathrm{~m}, 2 \mathrm{H}) & 53.28 \\ 3.10(\mathrm{t}, 1 \mathrm{H}) & 21.90 \\ 12.7(\mathrm{~s}, 2 \mathrm{H}) & 11.81 \\ \hline \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { (d) } \mathrm{C}_{5} \mathrm{H}_{8} \mathrm{O}_{4}\\\ &\begin{array}{cr} \hline{ }^{1} \text { H-NMR } & { }^{19} \text { C-NMR } \\ \hline 1.29(\mathrm{~s}, 6 \mathrm{H}) & 174.01 \\ 12.8(\mathrm{~s}, 2 \mathrm{H}) & 48.77 \\ & 22.56 \\ \hline \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { (e) } \mathrm{C}_{4} \mathrm{H}_{6} \mathrm{O}_{2}\\\ &\begin{array}{cc} \hline{ }^{1} \text { H-NMR } & { }^{13} \text { C-NMR } \\ \hline 1.91(\mathrm{~d}, 3 \mathrm{H}) & 172.26 \\ 5.86(\mathrm{~d}, 1 \mathrm{H}) & 147.53 \\ 7.10(\mathrm{~m}, 1 \mathrm{H}) & 122.24 \\ 12.4(\mathrm{~s}, 1 \mathrm{H}) & 18.11 \\ \hline \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { (f) } \mathrm{C}_{3} \mathrm{H}_{4} \mathrm{Cl}_{2} \mathrm{O}_{2}\\\ &\begin{array}{cc} \hline{ }^{1} \text { H-NMR } & { }^{19} \text { C-NMR } \\ \hline 2.34(\mathrm{~s}, 3 \mathrm{H}) & 171.82 \\ 11.3(\mathrm{~s}, 1 \mathrm{H}) & 79.36 \\ & 34.02 \\ \hline \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { (g) } \mathrm{C}_{5} \mathrm{H}_{8} \mathrm{Cl}_{2} \mathrm{O}_{2}\\\ &\begin{array}{cc} \hline{ }^{1} \text { H-NMR } & { }^{13} \text { C-NMR } \\ \hline 1.42(\mathrm{~s}, 6 \mathrm{H}) & 180.15 \\ 6.10(\mathrm{~s}, 1 \mathrm{H}) & 77.78 \\ 12.4(\mathrm{~s}, 1 \mathrm{H}) & 51.88 \\ & 20.71 \\ \hline \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { (h) } \mathrm{C}_{5} \mathrm{H}_{9} \mathrm{BrO}_{2}\\\ &\begin{array}{cc} \hline{ }^{1} \text { H-NMR } & { }^{13} \text { C-NMR } \\ \hline 0.97(\mathrm{t}, 3 \mathrm{H}) & 176.36 \\ 1.50(\mathrm{~m}, 2 \mathrm{H}) & 45.08 \\ 2.05(\mathrm{~m}, 2 \mathrm{H}) & 36.49 \\ 4.25(\mathrm{t}, 1 \mathrm{H}) & 20.48 \\ 12.1(\mathrm{~s}, 1 \mathrm{H}) & 13.24 \\ \hline \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { (i) } \mathrm{C}_{4} \mathrm{H}_{8} \mathrm{O}_{3}\\\ &\begin{array}{cc} \hline{ }^{1} \text { H-NMR } & { }^{13} \text { C-NMR } \\ \hline 2.62(\mathrm{t}, 2 \mathrm{H}) & 177.33 \\ 3.38(\mathrm{~s}, 3 \mathrm{H}) & 67.55 \\ 3.68(\mathrm{~s}, 2 \mathrm{H}) & 58.72 \\ 11.5(\mathrm{~s}, 1 \mathrm{H}) & 34.75 \\ \hline \end{array} \end{aligned} $$

The \(K_{a 1}\) of ascorbic acid is \(7.94 \times 10^{-5}\). Would you expect ascorbic acid dissolved in blood plasma (pH 7.35-7.45) to exist primarily as ascorbic acid or as ascorbate anion? Explain.

Name the carboxylic acid and alcohol from which each ester is derived. (a) COC(=O)C1CCCCC1 (b) CC(=O)OC1CCC(OC(C)=O)CC1 (c) CCC=CC(=O)OC(C)C (d) CCOC(=O)CCC(=O)OCC
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