(a) The standard free-energy difference between 2,2...

(a) The standard free-energy difference between 2,2 -dimethylpropane and pentane is $6.86 \mathrm{~kJ} \mathrm{~mol}^{-1}$ (1.64 kcal mol $^{-1}$ ); 2,2 -dimethylpropane is the more stable compound. If the two were present in an equilibrium mixture, what would be the percentage of each in the mixture at $25^{\circ} \mathrm{C} ?$ (b) The energy difference between anti-butane and either one of the gauche-butane conformations is $2.8 \mathrm{~kJ} \mathrm{~mol}^{-1}$ $\left(0.67 \mathrm{kcal} \mathrm{mol}^{-1}\right)$ (Fig. $2.5, \mathrm{p} .53$ ). Treating this difference as a standard free energy, calculate the amounts of gauche-and anti-butane present in equilibrium in one mole of butane at $25^{\circ} \mathrm{C}$. (Remember that there are two gauche conformations.)

(a) The standard free-energy difference between 2,2 -dimethylpropane and pentane is $6.86 \mathrm{~kJ} \mathrm{~mol}^{-1}$ (1.64 kcal mol $^{-1}$ ); 2,2 -dimethylpropane is the more stable compound. If the two were present in an equilibrium mixture, what would be the percentage of each in the mixture at $25^{\circ} \mathrm{C} ?$
(b) The energy difference between anti-butane and either one of the gauche-butane conformations is $2.8 \mathrm{~kJ} \mathrm{~mol}^{-1}$ $\left(0.67 \mathrm{kcal} \mathrm{mol}^{-1}\right)$ (Fig. $2.5, \mathrm{p} .53$ ). Treating this difference as a standard free energy, calculate the amounts of gauche-and anti-butane present in equilibrium in one mole of butane at $25^{\circ} \mathrm{C}$. (Remember that there are two gauche conformations.)

Key Concepts

Standard Free Energy Difference

This concept refers to the difference in free energy between two species under standard conditions. It is a measure of the thermodynamic favorability of one species over another and is fundamental in predicting the position of equilibrium in a reaction or conformational change.

Equilibrium Constant and Its Relation to ?G°

The equilibrium constant (K) quantifies the ratio of concentrations of products to reactants at equilibrium and is directly related to the standard free energy difference (?G°) by the equation ?G° = ?RT ln K. This relationship allows one to compute the relative amounts of species present at equilibrium, given their free energy differences.

Boltzmann Distribution

The Boltzmann distribution describes how molecules are distributed among various states as a function of their energy and the temperature. It explains that at thermal equilibrium, lower energy states are more populated than higher energy ones, a principle that is central to understanding the distribution of conformers or isomers.

Conformational Isomerism

This concept involves different spatial arrangements of the same molecule that result from rotations around single bonds. The energy differences between these conformers determine their relative stabilities and populations at equilibrium, making conformational analysis a critical aspect of molecular thermodynamics.

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