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, \(E_{x}\) 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 \(850 \mathrm{K}\).

Short answer

At 850 K, the rate constant \(k\) is approximately 0.0397 \text{s}^{-1}."

Step-by-step solution

Identify Known Variables

We have the following known variables:- Activation energy, \(E_a = 260 \text{kJ/mol} = 260,000 \text{J/mol}\) (converted for calculation)- Initial temperature, \(T_1 = 800 \text{K}\)- Final temperature, \(T_2 = 850 \text{K}\)- Rate constant at initial temperature, \(k_1 = 0.0315 \text{s}^{-1}\)

Choose the Arrhenius Equation

To find the rate constant at a different temperature, we will use the Arrhenius equation in its two-point form:\[\ln\left(\frac{k_2}{k_1}\right) = \frac{E_a}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right)\]where \(R\) is the universal gas constant, \(8.314 \text{J/mol K}\).

Plug In the Known Values

Substitute the known values into the two-point Arrhenius equation:\[\ln\left(\frac{k_2}{0.0315}\right) = \frac{260,000}{8.314} \left( \frac{1}{800} - \frac{1}{850} \right)\]

Calculate the Right Side

Calculate the right side of the equation:1. \(\frac{260,000}{8.314} = 31272.5\)2. \(\frac{1}{800} - \frac{1}{850} = 1.25 \times 10^{-4} - 1.176 \times 10^{-4} = 7.4 \times 10^{-6}\)3. Multiply these results: \(31272.5 \times 7.4 \times 10^{-6} = 0.2315\)

Solve for k_2

\[\ln\left(\frac{k_2}{0.0315}\right) = 0.2315\]Convert back from natural logs by calculating the exponential:\[\frac{k_2}{0.0315} = e^{0.2315} \approx 1.2604\]Thus, solve for \(k_2\):\[k_2 = 1.2604 \times 0.0315 = 0.0397 \text{s}^{-1}\]

Conclusion

At \(850 \text{K}\), the rate constant \(k\) is approximately \(0.0397 \text{s}^{-1}\).

Key concepts

Activation Energy

Activation energy is a fundamental concept in chemical kinetics. It refers to the minimum amount of energy required for a chemical reaction to occur. Think of it as the energy barrier that reactants must overcome to be transformed into products. The higher the activation energy, the slower the reaction rate, as fewer molecules will have sufficient energy to surmount this barrier.

When a reaction has a high activation energy, it requires more input energy, like heat, to proceed at a significant rate. This is why many reactions occur slowly at room temperature but can speed up when heated. In the context of cyclobutane decomposition, the activation energy is crucial for determining how readily the reaction proceeds at elevated temperatures like 800 K or 850 K.

• Activation energy is usually denoted as \(E_a\).
• For the given reaction, \(E_a = 260 \text{kJ/mol}\).
• It influences both the reaction rate and temperature dependence.

Rate Constant

The rate constant, represented as \(k\), is another key concept in chemical kinetics. It is a proportionality factor in the rate equation of a chemical reaction that relates the rate of reaction to the concentration of reactants. The rate constant is not constant in the usual sense, as it varies with temperature.

In the Arrhenius equation, \(k\) integrates the effects of temperature and activation energy on the speed of a reaction. Increasing the temperature generally increases \(k\), which in turn speeds up the reaction. In this example, at 800 K, \(k\) for the decomposition of cyclobutane is given as \(0.0315 \text{s}^{-1}\), and at 850 K, it increases to approximately \(0.0397 \text{s}^{-1}\).

• Changes in \(k\) with temperature help us understand the temperature dependence of reaction rates.
• \(k\) is unique for every reaction and depends on the specific conditions like temperature and pressure.

Temperature Dependence

Temperature has a profound effect on chemical reaction rates. As temperature increases, the kinetic energy of molecules also increases. This means that molecules move faster and collide more frequently and with greater energy, making successful reactions more likely.

The Arrhenius equation explicitly shows this relationship. It demonstrates how \(k\), the rate constant, varies with temperature. The equation is:\[\ln(k) = \ln(A) - \frac{E_a}{RT}\]where \(A\) is the pre-exponential factor, \(E_a\) is the activation energy, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.

• A small increase in temperature can significantly increase the rate of reaction by increasing \(k\).
• This principle is utilized in many industrial processes to optimize reaction conditions.
• For cyclobutane decomposition, warming the system from 800 K to 850 K resulted in an increase in \(k\).

Chemical Kinetics

Chemical kinetics is the branch of chemistry that studies the speed or rate of reactions and the factors affecting these rates. By understanding kinetics, we can predict how a reaction behaves under different conditions, which is invaluable across scientific and industrial applications.

The rate of a reaction can provide insights into the mechanism of the reaction or the steps involved in transforming reactants into products. For cyclobutane decomposition, the rate is dependent not only on temperature and activation energy but also on the physical and chemical nature of the substances involved.

• Kinetics helps in controlling reactions to produce desired products effectively.
• By manipulating factors like temperature and concentration, chemists can optimize reactions for efficiency.
• Understanding kinetics is critical for developing new materials, pharmaceuticals, and numerous chemical processes.