Problem 47 - Chemistry The Molecular Science

Question

Suppose a reaction rate constant has been measured at two different temperatures, \(T_{1}\) and \(T_{2}\), and its values are \(k_{\perp}\) and \(k_{2}\), respectively. (a) Write the Arrhenius equation at each temperature. (b) By combining these two equations, derive an expression for the ratio of the two rate constants, \(k_{1} / k_{2}\). Use this expression to answer the next four questions.

Step-by-Step Solution

Answer

\( \frac{k_1}{k_2} = \exp \left[ \frac{E_a}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right) \right] \)

Step 1: Arrhenius Equation for Temperature T1

The Arrhenius equation for the rate constant at temperature \( T_1 \) is given by: \[ k_1 = A \exp \left( -\frac{E_a}{RT_1} \right) \] where \( k_1 \) is the rate constant at \( T_1 \), \( A \) is the pre-exponential factor, \( E_a \) is the activation energy, and \( R \) is the ideal gas constant.

Step 2: Arrhenius Equation for Temperature T2

Similarly, the Arrhenius equation for the rate constant at temperature \( T_2 \) is: \[ k_2 = A \exp \left( -\frac{E_a}{RT_2} \right) \] where \( k_2 \) is the rate constant at \( T_2 \) and other terms remain the same as in Step 1.

Step 3: Forming the Ratio of Rate Constants

To find the expression for the ratio of the rate constants, \( \frac{k_1}{k_2} \), divide the two Arrhenius equations: \[ \frac{k_1}{k_2} = \frac{A \exp \left( -\frac{E_a}{RT_1} \right)}{A \exp \left( -\frac{E_a}{RT_2} \right)} \] Simplifying, we get: \[ \frac{k_1}{k_2} = \exp \left[ -\frac{E_a}{RT_1} + \frac{E_a}{RT_2} \right] \]

Step 4: Simplify the Expression

The exponent can be rewritten as: \[ \frac{k_1}{k_2} = \exp \left[ \frac{E_a}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right) \right] \] This expression shows the relationship between the ratio of the rate constants and the temperature change.

Key Concepts

Reaction Rate Constant

The reaction rate constant, often denoted by \( k \), is a crucial factor in chemical kinetics. It's a number that helps determine how fast a reaction proceeds. Its value is influenced by several factors, such as the presence of a catalyst, the specific conditions of the reaction, and most notably, the temperature. The rate constant defines the proportionality factor in the rate equation for a reaction and determines the speed at which reactants are transformed into products.

• The higher the rate constant, the faster the reaction.
• It varies for different reactions, indicating that different chemical reactions have inherently different speeds.
• The unit of the rate constant varies depending on the order of the reaction, but it generally relates the concentration of reactants to the rate of reaction.

By utilizing the Arrhenius equation, one can comprehend and predict how the rate constant changes with temperature.

Activation Energy

Activation energy \((E_a)\) is the minimum energy required to initiate a chemical reaction. It acts as an energy barrier that reactants must overcome for a successful collision that leads to products. The value of activation energy helps explain why some reactions happen quickly, while others proceed at a snail’s pace or need specific conditions to occur.

• A lower activation energy indicates that reactants can easily overcome the energy barrier, leading to a faster reaction.
• Conversely, high activation energies result in slower reactions without an external energy source, like heat or light.
• Activation energy can be influenced by catalysts, which lower this energy barrier, making it easier for reactions to proceed.

In the Arrhenius equation, activation energy is directly related to the exponential factor that impacts the rate constant. As a result, a small change in activation energy can significantly impact the reaction rate.

Temperature Dependence of Reaction Rate

The temperature dependence of a reaction rate is a fundamental concept in chemistry, illustrating how temperature changes influence how fast reactions take place. According to the Arrhenius equation, as temperature increases, the reaction rate typically increases; this happens because molecules move faster, leading to more frequent and energetic collisions.

• An increase in temperature often results in an increase in the reaction rate constant \((k)\).
• The Arrhenius equation quantifies this relationship, where an increase in temperature leads to a decrease in the negative exponent \((-E_a/RT)\), enhancing \(k\).
• Even small temperature changes can have a significant effect on the reaction rate due to the exponential nature of the temperature in the Arrhenius equation.

Understanding how temperature affects the reaction rate is vital for controlling reaction conditions in industrial processes, laboratories, and even in understanding biological processes.