Problem 121 - Chemistry

Question

The "boat" form and "chair" form of cyclohexane \(\left(\mathrm{C}_{6} \mathrm{H}_{12}\right)\) interconverts as shown here: 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 \(\longrightarrow\) 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 \(K_{\mathrm{c}}\) for the reaction is \(9.83 \times 10^{3}\) at \(298 \mathrm{~K}\).

Step-by-Step Solution

Answer

The rate constant (\(k_1\)) at \(298 \mathrm{~K}\) is approximately \(7.9 \times 10^{-3} ~\text{s}^{-1}\).

Step 1: Write down the Arrhenius equation

The Arrhenius equation is written as \[k = A e^{( -E_a / RT)} \] where \(k\) is the rate constant, \(A\) is the pre-exponential factor or frequency factor, \(E_a\) is the activation energy, \(R\) is the gas constant, and \(T\) is the temperature in Kelvin.

Step 2: Substitute the given values into the formula

Next, substitutes the given values into the Arrhenius equation. Here, \(A = 1.0 \times 10^{12} ~\text{s}^{-1}\), \(E_a = 41 ~\text{kJ/mol} = 41 \times 10^3 ~\text{J/mol}\), \(R = 8.314 ~\text{J/mol}\cdot\text{K}\), and \(T = 298 ~\text{K}\). The conversion from \(\text{kJ}\) to \(\text{J}\) is important since \(R\) is in \(\text{J}\). So the equation becomes \[k_1 = (1.0 \times 10^{12} ~\text{s}^{-1}) \times e^{(- 41 \times 10^3 ~\text{J/mol}) / (8.314 ~\text{J/mol}\cdot\text{K} \times 298 ~\text{K})} \]

Step 3: Solve for \(k_1\)

Finally, plug the numbers into a calculator to solve for \(k_1\). You'll get \(k_1 \approx 7.9 \times 10^{-3} ~\text{s}^{-1}\).

Key Concepts

Arrhenius Equation

The Arrhenius equation plays a fundamental role in the study of chemical kinetics, which is the branch of chemistry that deals with how fast chemical reactions occur. At its core, the equation relates the rate of a chemical reaction to temperature, providing valuable insight into the reaction's dynamics.

The equation is given by: \[k = Ae^{(-E_a/RT)}\]
where:

• \(k\) is the rate constant, indicating the speed of the reaction,
• \(A\), known as the pre-exponential factor or frequency factor, relates to the number of times reactant particles collide in the correct orientation per unit time,
• \(E_a\) represents the activation energy required to initiate the reaction,
• \(R\) is the universal gas constant, and
• \(T\) is the temperature in Kelvin (K).

This equation allows us to understand how changes in temperature (and to some extent, the presence of catalysts) affect the speed at which a reaction will proceed.

Activation Energy

Activation energy (\(E_a\)) is a crucial concept when studying chemical reactions. It is the minimum amount of energy that reactant molecules must possess for a reaction to occur. Essentially, it's the energy barrier that must be overcome for reactants to transform into products.

The magnitude of the activation energy impacts how fast a reaction will happen. A lower activation energy means that more molecules in a population have sufficient energy to react when they collide, thus increasing the reaction rate. Conversely, a high activation energy implies that fewer molecules have the needed energy, resulting in a slower reaction.

In the exercise given, the activation energy for cyclohexane conversion from chair to boat form is \(41 kJ/mol\), which influences the rate constant (\(k_1\)) at a given temperature.

Rate Constants

Rate constants are critical factors in chemical kinetics, symbolized by \(k\), which provide the rate at which a reaction progresses per unit concentration of reactant. The value of the rate constant is determined by the specifics of the reaction, including the activation energy and temperature.

For a first-order reaction, the rate constant is the proportionality factor connecting the rate of reaction to the reactant concentration. It is essential to note that the rate constant is not constant in all conditions—it changes with temperature, as illustrated by the Arrhenius equation.

The calculation of the rate constant for the chair to boat conversion of cyclohexane involves the Arrhenius equation and produces a value that governs the speed of this interconversion at a given temperature.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that is concerned with the study of the rates of chemical reactions and the mechanisms by which they occur. It not only looks at how fast reactions happen but also sheds light on the different steps that take place from reactants to products—these are referred to as the reaction mechanism.

The speed of a reaction can be influenced by various factors, including concentration of reactants, temperature, surface area of solid reactants or catalysts, and the presence of a catalyst. Chemical kinetics employs mathematical equations, such as the Arrhenius equation, to model these rate changes and to predict how a reaction will progress under different conditions.