Problem 110 - Chemistry The Central Science

Question

(a) Two reactions have identical values for \(E_{a} .\) Does this ensure that they will have the same rate constant if run at the same temperature? Explain. (b) Two similar reactions have the same rate constant at \(25^{\circ} \mathrm{C}\), but at \(35^{\circ} \mathrm{C}\) one of the reactions has a larger rate constant than the other. Account for these observations.

Step-by-Step Solution

Answer

In summary, having identical activation energies does not ensure the same rate constant for two reactions run at the same temperature due to possible differences in their frequency factors. It is also conceivable for two similar reactions with the same rate constant at one temperature to have different rate constants at another temperature, which can be attributed to differences in their activation energies or frequency factors at different temperatures.

Step 1: Understanding Activation Energy and Rate Constants

Activation energy (Ea) is the minimum energy required for a chemical reaction to proceed. The rate of the reaction depends on the activation energy and the temperature at which the reaction is run. The Arrhenius equation relates the rate constant (k) with activation energy and temperature as shown below: \[k = Ae^{-\frac{Ea}{RT}}\] Here: - k = rate constant - A = pre-exponential factor or frequency factor (constant for any reaction) - Ea = activation energy - R = gas constant (8.314 J/mol.K) - T = temperature in Kelvin Now, let's answer the questions mentioned in the exercise.

Step 2: Answering part (a)

The question asks if two reactions with identical values of Ea will have the same rate constant if run at the same temperature. To answer this, let's apply our knowledge of the Arrhenius equation. For two reactions with the same Ea and temperature (T), the rate constants (k1 and k2) will be given by the following expressions: \[k_1 = A_1e^{-\frac{Ea}{RT}}\] \[k_2 = A_2e^{-\frac{Ea}{RT}}\] If A1 and A2 (the frequency factors) are also the same for both reactions, then k1 and k2 will be equal. However, if A1 and A2 are different, k1 and k2 will also be different. Therefore, having identical values for Ea alone does not ensure that the rate constants will be the same for the two reactions.

Step 3: Answering part (b)

The question states that two similar reactions have the same rate constant at \(25^{\circ}\mathrm{C}\), but at \(35^{\circ}\mathrm{C}\), one of the reactions has a larger rate constant than the other. We'll need to explain these observations. At \(25^{\circ}\mathrm{C}\), the rate constants (k1 and k2) for the reactions are the same, which means: \[k_1 = A_1e^{-\frac{Ea_1}{R(298)}}\] \[k_2 = A_2e^{-\frac{Ea_2}{R(298)}}\] Here, 298 K is the temperature in Kelvin corresponding to \(25^{\circ}\mathrm{C}\). Since k1 = k2, \[A_1e^{-\frac{Ea_1}{R(298)}} = A_2e^{-\frac{Ea_2}{R(298)}}\] At \(35^{\circ}\mathrm{C}\), one reaction has a larger rate constant than the other. This means either the activation energy is different, or the frequency factors are different for the two reactions at this higher temperature. To verify this, we can look at the Arrhenius equation again: \[k_1 = A_1e^{-\frac{Ea_1}{R(308)}}\] \[k_2 = A_2e^{-\frac{Ea_2}{R(308)}}\] Here, 308 K is the temperature in Kelvin corresponding to \(35^{\circ}\mathrm{C}\). At this temperature, we know k1 ≠ k2, so \[A_1e^{-\frac{Ea_1}{R(308)}} ≠ A_2e^{-\frac{Ea_2}{R(308)}}\] The difference in rate constants could arise from differences in activation energy or frequency factors at different temperatures. The temperature dependence of frequency factors may also contribute to the observed difference in rate constants. In summary, having identical activation energies does not ensure the same rate constant for two reactions run at the same temperature, and it is possible for two similar reactions with the same rate constant at one temperature to have different rate constants at another temperature. This can be attributed to differences in their activation energies or frequency factors at different temperatures.

Key Concepts

Arrhenius Equation

The Arrhenius Equation is a crucial formula in understanding how the speed of a chemical reaction, or its rate constant (\(k\)), changes with temperature and activation energy (\(E_a\)). This equation is expressed as:\[k = Ae^{-\frac{E_a}{RT}}\]

• k: Rate constant.
• A: Pre-exponential or frequency factor, a constant that indicates the frequency of collisions with the correct orientation for the reaction to proceed.
• E_a: Activation energy, the minimum energy required for the reaction to occur.
• R: Ideal gas constant, 8.314 J/mol.K.
• T: Temperature in Kelvin.

This equation demonstrates that the rate constant is exponentially dependent on the temperature and the activation energy. A higher temperature or a lower activation energy results in a higher rate constant, meaning the reaction proceeds faster. The Arrhenius Equation offers insights into why some reactions are sensitive to temperature changes and others are not.

Rate Constant

The Rate Constant (\(k\)) provides a measure of the speed at which a reaction occurs. It is a variable specific to each chemical reaction and is influenced heavily by the conditions under which the reaction takes place.Several factors affect the rate constant:

• **Activation energy:** The higher the activation energy, the slower the reaction, because fewer molecules have sufficient energy to overcome this barrier.
• **Temperature:** Generally, increasing the temperature increases the rate constant, since more molecules have the necessary energy for the reaction.
• **Frequency factor (A):** Represents the number of times reactants approach the activation barrier per unit time. A higher frequency factor can lead to a higher rate constant.

This highlights that even when two reactions have the same activation energy, their rate constants can differ if their frequency factors differ. Different experimental conditions, like the presence of catalysts or changes in pressure, can also affect the rate constant, further emphasizing its variability.

Temperature Dependence

Temperature Dependence of reaction rates is a fundamental concept which explains how reaction speeds typically increase with an increase in temperature. This phenomenon is rooted in the way temperature influences molecular energy.

Why Temperature Matters

Temperature affects the kinetic energy of molecules. When the temperature increases, molecules move faster, and the number of collisions with enough energy to surpass the activation energy barrier increases. This is why, as demonstrated in the Arrhenius Equation, the rate constant (\(k\)) generally increases with temperature.For the example in the exercise, two reactions can behave very differently with temperature changes:

• At one temperature, they might have the same rate constant.
• A slight increase in temperature might cause one of them to have a larger increase in rate constant compared to the other.

This occurs because reactions can have different activation energies and frequency factors, both of which contribute to how the rate constant changes with temperature. It is this intricate relationship between energy, molecular movement, and reaction conditions that makes studying temperature dependence essential in chemistry.