Textbook Question
For the following acid–base reactions studied in Assessment 5.25, draw a likely transition state. Be sure to indicate in your drawing the degree to which bonds are broken or formed.
(a)
6. Thermodynamics and Kinetics
Gibbs Free Energy
7:59 minutesProblem 20a
Textbook Question
Calculate the percentage of isopropylcyclohexane molecules that have the isopropyl substituent in an equatorial position at equilibrium. (Its ∆G° value at 25 °C is -2.1 kcal/mol.)
Verified step by step guidance1
Step 1: Understand the problem. The question asks for the percentage of isopropylcyclohexane molecules with the isopropyl group in the equatorial position at equilibrium. This involves using the Gibbs free energy change (∆G°) to calculate the equilibrium constant (K) and then determining the percentage distribution between equatorial and axial positions.
Step 2: Recall the relationship between ∆G° and the equilibrium constant (K). The formula is: , where R is the gas constant (1.987 cal/(mol·K)) and T is the temperature in Kelvin (25 °C = 298 K). Rearrange the formula to solve for K: .
Step 3: Substitute the given values into the formula. Use ∆G° = -2.1 kcal/mol (convert to cal: -2100 cal/mol), R = 1.987 cal/(mol·K), and T = 298 K. Calculate K using the formula: . This will give the equilibrium constant, which represents the ratio of molecules in the equatorial position to those in the axial position.
Step 4: Use the equilibrium constant (K) to determine the percentage of molecules in the equatorial position. The ratio of equatorial to axial molecules is given by K, and the total number of molecules is the sum of equatorial and axial molecules. The percentage of equatorial molecules can be calculated using the formula: .
Step 5: Interpret the result. Once the percentage is calculated, it represents the proportion of isopropylcyclohexane molecules with the isopropyl group in the equatorial position at equilibrium. This is the more stable conformation due to steric hindrance being minimized in the equatorial position.
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Video duration:
7mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Equatorial vs. Axial Position
In cyclohexane derivatives, substituents can occupy equatorial or axial positions. Equatorial positions are generally more stable due to reduced steric strain and torsional strain, allowing larger groups to be farther from other atoms. Understanding the preference for equatorial positioning is crucial for predicting the stability of isopropylcyclohexane.
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04:02
Equatorial Preference
Gibbs Free Energy (∆G°)
Gibbs Free Energy (∆G°) indicates the spontaneity of a reaction or process at standard conditions. A negative ∆G° value, such as -2.1 kcal/mol, suggests that the formation of the equatorial isopropylcyclohexane is favored at equilibrium. This concept is essential for calculating the distribution of conformers in the equilibrium state.
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05:02Breaking down the different terms of the Gibbs Free Energy equation.
Equilibrium Constant (K)
The equilibrium constant (K) relates the concentrations of reactants and products at equilibrium. It can be derived from the Gibbs Free Energy change using the equation ∆G° = -RT ln(K). By calculating K from the given ∆G° value, one can determine the ratio of equatorial to axial isopropylcyclohexane molecules, which is key to answering the question.
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02:19The relationship between equilibrium constant and pKa.
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