Question

The "boat" form and the "chair" form of cyclohexane \(\left(\mathrm{C}_{6} \mathrm{H}_{12}\right)\) interconvert as shown here: $$\underset{k_{-1}}{\stackrel{k_{1}}{\rightleftarrows}}$$ In this representation, the \(\mathrm{H}\) atoms are omitted and a \(\mathrm{C}\) atom is assumed to be at each intersection of two lines (bonds). The conversion is first order in each direction. The activation energy for the chair boat conversion is \(41 \mathrm{~kJ} / \mathrm{mol}\). If the frequency factor is \(1.0 \times 10^{12} \mathrm{~s}^{-1}\), what is \(k_{1}\) at \(298 \mathrm{~K} ?\) The equilibrium constant \(K_{c}\) for the reaction is \(9.83 \times 10^{3}\) at \(298 \mathrm{~K}\).

Short answer

The rate constant \(k_1\) at 298 K is \(6.71 \times 10^4 \mathrm{~s}^{-1}\).

Step-by-step solution

Understand the Relationship

The conversion between the chair and boat forms of cyclohexane can be represented as a reversible first order reaction. This means for each forward (\(k_1\)) and reverse (\(k_{-1}\)) directions, a separate rate constant is defined.

Arrhenius Equation

Use the Arrhenius equation to calculate the rate constant \(k_1\): \[ k = A e^{-\frac{E_a}{R T}} \]where \(E_a = 41 \mathrm{~kJ} / \mathrm{mol} = 41000 \mathrm{~J} / \mathrm{mol} \), \(R = 8.314 \mathrm{~J} / \mathrm{mol \cdot K} \), \(T = 298 \mathrm{K} \), and the frequency factor \(A = 1.0 \times 10^{12} \mathrm{~s}^{-1} \).

Substitute Values in Arrhenius Equation

Substitute the known values into the Arrhenius equation:\[ k_1 = 1.0 \times 10^{12} e^{-\frac{41000}{8.314 \times 298}} \]

Calculate the Exponential Term

Calculate the exponent part first:\[ -\frac{41000}{8.314 \times 298} \approx -16.52 \]This is the power to which \(e\) needs to be raised.

Compute the Exponential Function

Calculate the exponential term: \[ e^{-16.52} \approx 6.71 \times 10^{-8} \]

Determine Rate Constant \(k_1\)

Multiply the frequency factor by the exponential term to find \(k_1\):\[ k_1 = 1.0 \times 10^{12} \times 6.71 \times 10^{-8} = 6.71 \times 10^4 \mathrm{~s}^{-1} \]

Review Equilibrium Constant \(K_c\) (Optional)

The given equilibrium constant \(K_c = 9.83 \times 10^3\) is for checking if needed, but doesn't directly affect the calculation of \(k_1\).

Key concepts

Activation Energy

Activation energy is the minimum amount of energy needed for a chemical reaction to occur. For the conversion between the chair and boat forms of cyclohexane, it's the energy needed for the molecules to change shape. This energy helps the molecules overcome the energy barrier to transform from one conformation to another. Think of it like pushing a boulder up over a hill - you need enough initial force to get it to the top before it can roll down the other side.

In this exercise, the activation energy for the chair-to-boat conversion is given as 41 kJ/mol. This value is crucial for determining the reaction rate because it tells us how much energy is necessary to reach the transition state.

If you were to lower the activation energy, the reaction could proceed more quickly because less energy is needed for the transformation. Conversely, if you increase the activation energy, the reaction slows down because it becomes harder for the molecules to gain enough energy to react.

Arrhenius Equation

The Arrhenius equation is a formula that relates the rate constant of a reaction to the temperature and activation energy. It is expressed as:

• \[ k = A e^{-\frac{E_a}{RT}} \]

In this equation:

• \( k \) is the rate constant.
• \( A \) is the frequency factor, representing the number of times that reactants approach the activation barrier per unit time.
• \( E_a \) is the activation energy.
• \( R \) is the universal gas constant.
• \( T \) is the temperature in Kelvin.

For the cyclohexane conformation change, we use the Arrhenius equation to calculate the rate constant \( k_1 \) at a temperature of 298 K. By substituting the given values, we determine \( k_1 \) using the steps outlined in the solution. The rate constant helps us understand how quickly the chair form converts to the boat form.

Equilibrium Constant

The equilibrium constant, denoted as \( K_c \), provides insight into the balance of a reversible reaction at equilibrium. It is a measure of the ratio of concentrations of products to reactants when the reaction has reached a steady state.

In the case of cyclohexane conformations, the equilibrium constant \( K_c \) is given as \( 9.83 \times 10^3 \) at 298 K. Though it is not directly used in calculating \( k_1 \), it is crucial for understanding the position of equilibrium. It shows that at equilibrium, the chair form is much more prevalent than the boat form. **A higher \( K_c \)** signifies that products (chair form here) are favored when equilibrium is reached.

Knowing \( K_c \) becomes useful if you want to calculate the reverse rate constant \( k_{-1} \) or understand how changes in conditions can affect the balance between forms.

First Order Reaction

A first order reaction means the rate at which the reaction proceeds is directly proportional to the concentration of one reactant. In our example, both the conversion from chair to boat and from boat back to chair are first order reactions. This implies:

• The reaction rate depends on the concentration of cyclohexane molecules undergoing the transformation.
• Each direction of conversion is treated separately with its own rate constant (\( k_1 \) for chair to boat and \( k_{-1} \) for boat to chair).

Understanding first order reactions simplifies analysis because we only need to keep track of the concentration of a single reactant to determine the rate. The mathematical representation of a first order reaction is expressed as:

• \[ \text{Rate} = k [A] \]

where \([A]\) represents the concentration of the reactant. Because cyclohexane converts in both directions at its respective rate constant, these reactions maintain the dynamic equilibrium defined by \( K_c \). This concept allows chemists to predict how quickly equilibriums are reached in such reversible systems.