Understanding the high-temperature behavior of nitrogen oxides is essential for controlling pollution generated in automobile engines. The decomposition of nitric oxide (NO) to N2 and O2 is second order with a rate constant of 0.0796 M-1s-1 at 737 _x001E_C and 0.0815 M-1s-1 at 947 _x001E_C. Calculate the activation energy for the reaction.

Verified step by step guidance

Step 1: Use the Arrhenius equation, which relates the rate constant (k) to the activation energy (Ea), temperature (T), and the pre-exponential factor (A): k = A * e^(-Ea/(RT)), where R is the gas constant (8.314 J/mol·K).

Step 2: Take the natural logarithm of both sides of the Arrhenius equation to linearize it: ln(k) = ln(A) - Ea/(RT).

Step 3: Set up two equations using the given rate constants and temperatures. Convert the temperatures from Celsius to Kelvin by adding 273.15.

Step 4: Use the two equations to eliminate ln(A) and solve for Ea. This can be done by subtracting one equation from the other, resulting in: ln(k2/k1) = -Ea/R * (1/T2 - 1/T1).

Step 5: Rearrange the equation to solve for Ea: Ea = -R * ln(k2/k1) / (1/T2 - 1/T1). Substitute the known values for k1, k2, T1, and T2 to find the activation energy.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rate Laws and Reaction Order

Rate laws describe the relationship between the concentration of reactants and the rate of a chemical reaction. The order of a reaction indicates how the rate is affected by the concentration of reactants. In this case, the decomposition of nitric oxide (NO) is second order, meaning that the rate is proportional to the square of the concentration of NO. Understanding reaction order is crucial for applying the appropriate mathematical models to calculate reaction rates and activation energy.

Arrhenius Equation

The Arrhenius equation relates the rate constant of a reaction to temperature and activation energy. It is expressed as k = A * e^(-Ea/RT), where k is the rate constant, A is the pre-exponential factor, Ea is the activation energy, R is the universal gas constant, and T is the temperature in Kelvin. This equation is fundamental for calculating activation energy from rate constants at different temperatures, as it shows how temperature influences reaction rates.

Activation Energy

Activation energy (Ea) is the minimum energy required for a chemical reaction to occur. It represents the energy barrier that reactants must overcome to form products. A higher activation energy indicates that fewer molecules have sufficient energy to react at a given temperature, leading to a slower reaction rate. Calculating Ea is essential for understanding the temperature dependence of reaction rates and for designing strategies to control reactions, such as those involved in pollution control.