Problem 39 - Chemistry and Chemical Reactivity

Question

When heated to a high temperature, cyclobutane, \(\mathrm{C}_{4} \mathrm{H}_{8}\) decomposes to ethylene: $$\mathrm{C}_{4} \mathrm{H}_{8}(\mathrm{g}) \rightarrow 2 \mathrm{C}_{2} \mathrm{H}_{4}(\mathrm{g})$$ The activation energy, \(E_{a},\) for this reaction is \(260 \mathrm{kJ} / \mathrm{mol} .\) At \(800 \mathrm{K},\) the rate constant \(k=0.0315 \mathrm{s}^{-1} .\) Determine the value of \(k\) at \(850 \mathrm{K}.\)

Step-by-Step Solution

Answer

At 850 K, the rate constant \( k = 0.313 \) s\(^{-1}\).

Step 1: Understand the Arrhenius Equation

The Arrhenius equation relates the rate constant \( k \) to the temperature \( T \), the activation energy \( E_a \), and the pre-exponential factor \( A \): \( k = A e^{-E_a/(RT)} \), where \( R \) is the gas constant \( 8.314 \, \text{J/mol K} \). We do not need \( A \) explicitly since we will use the ratio form of the equation.

Step 2: Use the Two-Point Form of the Arrhenius Equation

To find the rate constant \( k_2 \) at a new temperature \( T_2 \) using a known rate constant \( k_1 \) at temperature \( T_1 \), we use: \[ \ln \left( \frac{k_2}{k_1} \right) = -\frac{E_a}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right) \] where \( T_1 = 800 \, \text{K} \), \( T_2 = 850 \, \text{K} \), \( E_a = 260,000 \, \text{J/mol} \), and \( k_1 = 0.0315 \, \text{s}^{-1} \).

Step 3: Calculate the Term \( \frac{1}{T_2} - \frac{1}{T_1} \)

First, calculate \( \frac{1}{850} - \frac{1}{800} \):\( \frac{1}{850} = 0.0011765 \) \( \text{K}^{-1} \)\( \frac{1}{800} = 0.00125 \) \( \text{K}^{-1} \)Then, \( \frac{1}{850} - \frac{1}{800} = -0.0000735 \) \( \text{K}^{-1} \).

Step 4: Substitute Values into the Equation

Substitute the values into the equation:\[ \ln \left( \frac{k_2}{0.0315} \right) = -\frac{260,000}{8.314} \times (-0.0000735) \]

Step 5: Calculate the Exponential Part

Calculate the expression \(-\frac{260,000}{8.314} \times (-0.0000735)\):This equals approximately \(2.2976\).

Step 6: Solve for \( k_2 \)

Substitute the result back to solve for \( k_2 \):\[ \ln \left( \frac{k_2}{0.0315} \right) = 2.2976 \]\[ \frac{k_2}{0.0315} = e^{2.2976} \]\[ k_2 = 0.0315 imes e^{2.2976} \]Calculate \( e^{2.2976} \) which is approximately \(9.95\), giving\( k_2 = 0.313 \) \( \text{s}^{-1} \).

Key Concepts

Activation Energy

The concept of activation energy is fundamental in understanding chemical reactions. Activation energy, often denoted as \( E_a \), is the minimum energy that reactant molecules need to have in order to transform into products. For a chemical reaction to occur, the molecules must collide with enough energy to overcome this barrier and react. Activation energy can be thought of as the threshold that must be crossed for a reaction to proceed.
In the decomposition of cyclobutane to ethylene, the given activation energy is \( 260 \, \text{kJ/mol} \). This high energy requirement implies that the reaction requires substantial energy input, usually in the form of heat, for it to proceed.

• It acts as a barrier to the reaction, preventing it from happening spontaneously.
• The higher the activation energy, the slower the reaction, since fewer molecules have the required energy.
• Conversely, a lower activation energy can lead to a faster reaction.

Understanding activation energy allows chemists to modify conditions to speed up or control reactions.

Rate Constant

The rate constant, symbolized as \( k \), is a crucial part of the Arrhenius equation. It represents the proportionality factor in the rate law, which connects the concentration of reactants to the rate of a chemical reaction. The specific value of \( k \) can vary greatly depending on factors like temperature and the nature of the reactants involved.
The Arrhenius equation provides a mathematical relationship showing how the rate constant changes with temperature:\[ k = A e^{-E_{a}/RT} \]where \( A \) is the pre-exponential factor, \( E_a \) is the activation energy, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin.

• The rate constant helps in determining how quickly a reaction proceeds.
• A higher \( k \) value indicates a faster reaction.
• The rate constant is specific to each reaction and varies with conditions.

For the decomposition of cyclobutane to ethylene, the rate constant at \( 800 \, \text{K} \) is \( 0.0315 \, ext{s}^{-1} \). Using the Arrhenius equation, we can predict how it changes as temperature increases.

Temperature Dependence

Temperature is a critical factor influencing the rate of chemical reactions. Generally, an increase in temperature results in an increase in reaction rate. This is because molecules move faster and collide more frequently at higher temperatures, increasing the likelihood of overcoming the activation energy barrier.
The Arrhenius equation captures this relationship by showing how the rate constant \( k \) changes with temperature. In the Arrhenius equation:\[ \ln \left( \frac{k_2}{k_1} \right) = -\frac{E_a}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right) \]we see that if the temperature \( T_2 \) is higher than \( T_1 \), the exponent will be more positive, thus increasing \( k_2 \).

• Higher temperatures provide molecules with more kinetic energy, increasing collision frequency and energy.
• Even small temperature increases can have a noticeable effect on reaction rates.
• It is essential in industrial processes to accurately predict how temperature adjustments will affect reaction speed.

For cyclobutane decomposition, increasing the temperature from \( 800 \, \text{K} \) to \( 850 \, \text{K} \) increases the rate constant, demonstrating temperature dependence.

Cyclobutane Decomposition

Cyclobutane decomposition is a first-order chemical reaction that showcases several important chemical principles. The reaction involves the breakdown of cyclobutane \( \text{C}_4 \text{H}_8 \) into two ethylene \( \text{C}_2 \text{H}_4 \) molecules:
\[ \text{C}_4 \text{H}_8 (g) \rightarrow 2 \text{C}_2 \text{H}_4 (g) \]This process is representative of how cyclic compounds can convert to alkenes through the breaking of carbon-carbon bonds.

• The reaction is first-order, meaning that its rate depends linearly on the concentration of cyclobutane.
• It is sensitive to temperature changes, with higher temperatures accelerating the process.
• Understanding this decomposition is valuable for learning about reaction mechanisms and kinetics in organic chemistry.

In practical applications, understanding the decomposition of cyclobutane is crucial for controlling and optimizing reactions in synthetic and industrial chemistry.