Question

For the reaction $$ \mathrm{NO}(g)+\mathrm{O}_{3}(g) \longrightarrow \mathrm{NO}_{2}(g)+\mathrm{O}_{2}(g) $$ the frequency factor \(A\) is \(8.7 \times 10^{12} \mathrm{~s}^{-1}\) and the activation energy is \(63 \mathrm{~kJ} / \mathrm{mol}\). What is the rate constant for the reaction at \(75^{\circ} \mathrm{C} ?\)

Short answer

Using above provided solution steps, you can calculate the rate constant \(k\) at \(75^{\circ}C\) for the given reaction. The actual value will depend on the exact computations done in the third step.

Step-by-step solution

Organize the information

First, identify the given quantities in the problem. You have the activation energy (\(E_a = 63 kJ/mol\)), the frequency factor (\(A = 8.7x10^{12} s^{-1}\)), and the temperature (\(T = 75^{\circ}C\)). The problem asks for the rate constant \(k\). Note that, the activation energy in the Arrhenius equation should efficiently be in joules per mol, and the temperature should be in Kelvin (K), so these conversions need to be done. The value of the gas constant \(R\) is known to be \(8.314 J/mol*K\).

Convert activation energy and temperature

Convert the given activation energy (\(E_a\)) from kJ/mol to J/mol by multiplying by 1000. Thus, \(E_a = 63 kJ/mol * 1000 = 63000 J/mol\). Similarly, convert the temperature (\(T\)) from Celsius to Kelvin by adding 273.15. Thus, \(T = 75^{\circ}C + 273.15 = 348.15 K\).

Substitute values into the Arrhenius equation and solve

Now, substitute the values into the Arrhenius equation ( \(k = Ae^{-\frac{E_a}{RT}}\)) and calculate the rate constant. So, \(k = 8.7x10^{12} s^{-1} * e^{-\frac{63000 J/mol}{8.314 J/mol*K * 348.15 K}}.\) Simply calculate this expression to find the value of \(k\).

Key concepts

Activation Energy

Activation energy, often denoted as (E_a), is a critical concept in chemical kinetics and plays a vital role in the Arrhenius equation. It is the minimum amount of energy that reacting particles must have for a reaction to occur. In other words, activation energy is like a barrier to the start of a chemical reaction.

Consider an everyday example of striking a match; the friction provides the activation energy needed to initiate the combustion reaction. In the given exercise, the activation energy is initially provided in kilojoules per mole (63 kJ/mol), a common unit in chemistry for expressing this threshold energy.

To use this value in the Arrhenius equation, it needs to be converted into joules per mole because the gas constant (R) used in the equation is in these units. This demonstrates the importance of unit consistency in chemical calculations. The relationship between activation energy and the rate constant can be visualized on a graph known as an energy profile diagram, which illustrates the energy of reactants, the activation energy barrier, and the energy of products during a chemical reaction. High activation energy means a slower reaction rate, as fewer particles will have enough energy to overcome this barrier at a given temperature.

Rate Constant

The rate constant, symbolized as (k), is a proportionality factor that links the rate of a chemical reaction to the concentration of the reactants. It is crucial in the Arrhenius equation, which mathematically relates the rate constant to both the activation energy and the temperature of the reaction.

In the given textbook exercise, you're asked to find the rate constant at a certain temperature (75°C). The magnitude of (k) changes with temperature and activation energy; it is not a fixed value for a given reaction. A higher rate constant typically indicates a faster rate of reaction. The frequency factor (A), another component of the Arrhenius equation, pertains to the frequency of collisions with the proper orientation for a reaction to proceed. Even if particles collide with energy greater than or equal to the activation energy, they may not react unless they are oriented correctly.

After converting the provided temperature to Kelvin and the activation energy to joules per mole, you then apply the exponential function in the Arrhenius equation to calculate (k). This value informs us about the speed of the reaction under specific conditions and can determine how variations in temperature impact the reaction rate.

Chemical Kinetics

Chemical kinetics is the study of reaction rates and the pathways taken during the transformation of reactants into products. It is an essential field in chemistry because it helps us understand the factors affecting how fast a reaction will occur. Temperature, concentration of reactants, surface area, catalysts, and the presence of inhibitors are all variables that can influence the reaction rate.

In the context of this exercise, we apply chemical kinetics principles to calculate the rate constant using the Arrhenius equation. This equation is a fundamental model in kinetics used to understand how changes in temperature and activation energy affect the speed of chemical reactions. It emphasizes the exponential impact of temperature changes on reaction rates; even a small increase in temperature can significantly increase the rate constant, hence speeding up the reaction.

Understanding the kinetics of a chemical reaction is also practically essential across various industrial and research applications, as it can inform decisions on operational conditions, safety measures, and optimization of reaction processes.