Question

The rate constants of some reactions double with every \(10^{\circ}\) rise in temperature. Assume that a reaction takes place at \(295 \mathrm{~K}\) and \(305 \mathrm{~K}\). What must the activation energy be for the rate constant to double as described?

Short answer

The activation energy \( E_a \) is approximately 52,880 J/mol.

Step-by-step solution

Understand the Arrhenius Equation

The rate constant of a reaction is given by the Arrhenius equation: \( k = A e^{-\frac{E_a}{RT}} \), where \( k \) is the rate constant, \( A \) is the pre-exponential factor, \( E_a \) is the activation energy, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin. The problem states that the rate constant doubles with a temperature increase of \(10^{\circ}C\), which is equivalent to an increase of \(10 K\) in this context, from \(295 K\) to \(305 K\).

Apply the Arrhenius Equation for Two Temperatures

Write down the Arrhenius equation for the two given temperatures. At \(295 \mathrm{~K}\), we have \( k_1 = A e^{-\frac{E_a}{R \times 295}} \). At \(305 \mathrm{~K}\), we have \( k_2 = A e^{-\frac{E_a}{R \times 305}} \). Since \( k_2 = 2k_1 \), insert this relationship into the equation.

Derive the Relationship for Activation Energy

Divide the second equation by the first: \( \frac{k_2}{k_1} = \frac{A e^{-\frac{E_a}{R \times 305}}}{A e^{-\frac{E_a}{R \times 295}}} = 2 \). This simplifies to: \( e^{-\frac{E_a}{R \times 305}} = 2 e^{-\frac{E_a}{R \times 295}} \). Taking the natural logarithm of both sides, we get: \( -\frac{E_a}{R \times 305} + \frac{E_a}{R \times 295} = \ln(2) \).

Isolate the Activation Energy \(E_a\)

Rearrange the equation to solve for \(E_a\): \( E_a \left( \frac{1}{295} - \frac{1}{305} \right) = R \ln(2) \). Solve: \( E_a = \frac{R \ln(2)}{\frac{1}{295} - \frac{1}{305}} \).

Calculate the Activation Energy

Substitute the value of the universal gas constant \( R = 8.314 \) J/mol·K and calculate \( E_a \). Find \( \frac{1}{295} - \frac{1}{305} = \frac{305 - 295}{295 \times 305} \approx \frac{10}{90025} \) and simplify: \( E_a = \frac{8.314 \times \ln(2)}{\frac{10}{90025}} \). Calculate \( E_a \approx 52880 \) J/mol.

Key concepts

Arrhenius Equation

The Arrhenius Equation is a key formula in chemistry that describes how the rate of a chemical reaction changes with temperature. It can be expressed as \( k = A e^{-\frac{E_a}{RT}} \), where:

• \( k \) is the rate constant, which tells us how fast a reaction proceeds.
• \( A \) is the pre-exponential factor, representing the frequency of collisions with the correct orientation.
• \( E_a \) is the activation energy, the minimum energy needed for a reaction to occur.
• \( R \) is the universal gas constant, approximately \( 8.314 \text{ J/mol·K} \).
• \( T \) is the temperature in Kelvin.

Understanding the Arrhenius Equation helps in predicting how temperature impacts reaction rates. When temperature increases, the exponential term \( e^{-\frac{E_a}{RT}} \) falls, leading to a higher rate constant \( k \), which in turn speeds up the reaction.
The equation's exponential nature shows that even small temperature changes can significantly affect the reaction rate. This is particularly useful in understanding how increasing the temperature can double the reaction rate, as demonstrated in the exercise where the rate constant doubles with a 10 K increase.

Temperature Effect on Reaction Rate

Temperature plays a crucial role in determining the rate of chemical reactions. As the temperature of a system increases, the kinetic energy of the molecules also increases. This results in more frequent and energetic collisions among reactant molecules.
The rule of thumb is that for many reactions, the rate constant approximately doubles with every 10-degree Celsius (or 10 K) increase in temperature. This is a consequence of the Arrhenius Equation, specifically how the exponential term changes with temperature.
When a reaction occurs at higher temperatures, more molecules have the energy required to overcome the activation energy barrier. Thus, higher temperatures mean:

• Increased molecular motion.
• Higher collision frequency.
• Greater energy per collision, leading to a greater proportion of collisions that result in a reaction.

In the example provided, the increase from 295 K to 305 K causes the rate constant to double, emphasizing the powerful effect temperature holds over reaction rates and the importance of activation energy in this process.

Rate Constant

The rate constant, denoted as \( k \), is a fundamental component in chemistry that defines the speed of a chemical reaction. It's part of the rate law, which relates the rate of reaction to the concentrations of reactants, but the rate constant itself is independent of concentration.
Within the Arrhenius Equation, the rate constant is shown to depend on temperature and activation energy. Its value increases with higher temperatures or lower activation energies, meaning the reaction will occur faster. This dependence is why the concept of the rate constant is so crucial when studying chemical kinetics.
Rate constants have distinct units depending on the order of the reaction, a reflection on how different reactions respond uniquely to concentration changes. For instance, a first-order reaction has units of \( s^{-1} \), whereas a second-order reaction might have units of \( M^{-1}s^{-1} \). Understanding the rate constant through the lens of temperature and the Arrhenius Equation provides insight into designing and controlling reactions in labs and industries.

Thermodynamics

Thermodynamics involves the study of energy changes, particularly the transformation of heat into work and vice versa, in chemical processes. When discussing reaction rates and activation energy, thermodynamics helps in understanding the energy landscape of a reaction.
The activation energy \( E_a \) is a thermodynamic property that signifies the energy barrier which reactants must overcome for a reaction to proceed. It plays a major role in defining the rate of a reaction. Lower activation energies mean that fewer molecules need to overcome the energy barrier, resulting in a faster reaction at a given temperature.
Thermodynamics also helps predict the feasibility of reactions based on changes in Gibbs free energy and equilibrium constants, but when focusing on rates, it's the activation energy derived from the Arrhenius Equation that's of primary concern.
By harnessing principles of thermodynamics, chemists can manipulate reaction conditions to optimize speed and efficiency, crucial in fields like materials science and pharmacology.