Question

A chemist found the value of the specific rate constant for the decomposition of nitrous oxide, \(2 \mathrm{NO}+\mathrm{N}_{2}+\mathrm{O}_{2}\), at two separate temperatures: \(\mathrm{k}_{,}=14\) liter/mole-sec at \(970^{\circ} \mathrm{K}, \mathrm{k}_{2}\) \(=3.7\) liter mole-sec at \(1085^{\circ} \mathrm{K}\). (a) Calculate the activation energy, \(\mathrm{E}_{\mathrm{a}}\), for the reaction, (b) Calculate the A factor in the Arrhenius equation, (c) Calculate the specific rate constant at \(800^{\circ} \mathrm{K}\).

Short answer

(a) The activation energy (Ea) for the reaction is approximately 71,073 J/mol. (b) The A factor (pre-exponential factor) in the Arrhenius equation is approximately 7.69 x 10¹² L/mol-s. (c) The specific rate constant at 800 K is approximately 9.74 L/mol-s.

Step-by-step solution

Identify the given variables

We are given the following data: k₁ = 14 L/mol-s at T₁ = 970 K k₂ = 3.7 L/mol-s at T₂ = 1085 K

Calculate the activation energy (Ea)

We can use the Arrhenius equation at both given temperatures to calculate the activation energy. This gives us the following equations: k₁ = A * \(e^{-\frac{E_a}{R \times T₁}}\) and k₂ = A * \(e^{-\frac{E_a}{R \times T₂}}\) Divide the first equation by the second to eliminate A: \(\frac{k₁}{k₂} = e^{\frac{E_a}{R}(\frac{1}{T₂}-\frac{1}{T₁})}\) Now we can solve for Ea: \(\ln(\frac{k₁}{k₂}) = \frac{E_a}{R}(\frac{1}{T₂}-\frac{1}{T₁})\) \(E_a = R \times \ln(\frac{k₁}{k₂}) \times (\frac{T₁ \times T₂}{T₂ - T₁})\) Plug in the given values: \(E_a = 8.314 \times \ln(\frac{14}{3.7}) \times \frac{(970)(1085)}{1085 - 970}\) Ea ≈ 71,073 J/mol

Calculate the A factor (pre-exponential factor) in the Arrhenius equation

We can use k₁ and T₁ to find the A factor: k₁ = A * \(e^{-\frac{E_a}{R \times T₁}}\) Solve for A: \(A = \frac{k₁}{e^{-\frac{E_a}{R \times T₁}}}\) Plug in the given values: \(A = \frac{14}{e^{-\frac{71073}{8.314 \times 970}}}\) A ≈ 7.69 x 10¹² L/mol-s

Calculate the specific rate constant at 800 K

Now we have Ea and A, we can use the Arrhenius equation to find k at 800 K: k = A * \(e^{-\frac{E_a}{R \times T}}\) Plug in the given values: k = (7.69 × 10¹²) * \(e^{-\frac{71073}{8.314 \times 800}}\) k ≈ 9.74 L/mol-s The specific rate constant at 800 K is approximately 9.74 L/mol-s.

Key concepts

Arrhenius Equation

The Arrhenius equation is a fundamental formula in chemical kinetics, bringing together critical concepts like temperature, activation energy, and the rate constant. It's generally represented as: \[ k = A \times e^{-\frac{E_a}{RT}} \] where

• \(k\) stands for the rate constant.
• \(A\) is the pre-exponential factor, sometimes called the frequency factor.
• \(E_a\) represents the activation energy.
• \(R\) is the universal gas constant, 8.314 J/mol⋅K.
• \(T\) is the absolute temperature in Kelvin.

The equation showcases how the rate of reaction is influenced by temperature: As temperature increases, the exponential term lessens the effect of activation energy, making reactions faster. By understanding each component of the Arrhenius equation, chemists can predict how fast a reaction will occur under various conditions.

Rate Constant

The rate constant \(k\) in the context of the Arrhenius equation plays a crucial role in determining the speed of a chemical reaction. It's worth noting that the rate constant is not really constant unless the temperature is fixed. For a given reaction, the rate constant can change with different temperature settings. Typically, its units vary depending on the order of the reaction. In our original nitrous oxide decomposition reaction, measured in liter/mole-sec, the rate constant indicates the efficiency and speed of the reaction at specified temperatures. The increase of the rate constant with rising temperatures exemplifies the fundamental principle that higher temperatures typically increase reaction rates. This is due to more energetic collisions between molecules, making it easier to surpass the energy barrier represented by the activation energy.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that focuses on understanding the rates of reactions. It utilizes important concepts such as the rate constant, temperature, and activation energy. In chemical kinetics, reactions are explained in terms of molecular collisions and energy barriers. Understanding kinetics allows chemists to predict how long a reaction will take or how changing conditions will alter the speed of a reaction. For instance, by calculating activation energy and using the Arrhenius equation, the reaction rate at different temperatures can be determined. Ultimately, applying chemical kinetics principles helps in designing new reactions and optimizing existing ones, making processes more efficient in research and industrial applications.

Pre-exponential Factor

The pre-exponential factor \(A\), also known as the frequency factor in the Arrhenius equation, provides insights into the collision frequency and the orientation of reacting molecules. It is linked to the likelihood that collisions result in a reaction. In practice, it represents factors other than temperature and activation energy affecting the reaction rate, such as the sterics and the number of collisions per unit time. Calculated or estimated through experiments, \(A\) often combines several factors including:

• The number of reactive collisions that can occur.
• The spatial orientation of the reactant molecules.
• Catalytic effects that might speed up the reaction.

A precise determination of \(A\) aids in refining reaction parameters and better understanding complex kinetic scenarios.