Question

The activation energy for the reaction: $$ \mathrm{N}_{2} \mathrm{O}(g) \longrightarrow \mathrm{N}_{2}(g)+\mathrm{O}(g) $$ is \(2.4 \times 10^{2} \mathrm{~kJ} / \mathrm{mol}\) at \(600 \mathrm{~K}\). Calculate the percentage of the increase in rate from \(600 \mathrm{~K}\) to \(606 \mathrm{~K}\). Comment on your results.

Short answer

The rate increases by approximately 2728.18% when temperature rises from 600 K to 606 K.

Step-by-step solution

Understand the Arrhenius Equation

The rate of a chemical reaction often depends on temperature, following the Arrhenius equation: \[ k = A e^{-Ea/(RT)} \]where:- \( k \) is the rate constant,- \( A \) is the frequency factor,- \( Ea \) is the activation energy,- \( R \) is the universal gas constant \( = 8.314 \) J/mol·K,- \( T \) is the temperature in Kelvin. By analyzing this equation, a change in temperature affects the rate constant \( k \).

Calculate Rate Constants at Two Temperatures

To determine how the rate of reaction changes with temperature, first calculate the rate constants at both initial and final temperatures (\( T_1 = 600 \) K and \( T_2 = 606 \) K). Use the Arrhenius equation to calculate \( k_1 \) and \( k_2 \):Start with the expression for the ratio of rate constants:\[ \frac{k_2}{k_1} = e^{(-Ea/R)(1/T_2 - 1/T_1)} \]

Substitute Values and Compute Ratio

Substitute the values into the equation:\[ \frac{k_2}{k_1} = e^{(-2.4 \times 10^5 \, \text{J/mol}) / 8.314 \, \text{J/mol·K} \times (1/606 - 1/600)} \]Calculate the exponent part:\[ 1/606 - 1/600 = -0.0000274725 \]This gives:\[ \frac{k_2}{k_1} = e^{-2.4 \times 10^5 \, (-0.0000274725)/8.314} \].

Evaluate the Exponential Component

The exponent value can be calculated as:\[ (-2.4 \times 10^5) \times (-0.0000274725)/8.314 = 7.91 \]Thus, the ratio becomes:\[ \frac{k_2}{k_1} \approx e^{7.91} \approx 2728.18 \].

Key concepts

Activation Energy

Activation energy is a crucial concept in understanding chemical reactions. It represents the minimum amount of energy required for reactants to undergo a chemical change into products. Picture activation energy as a barrier that reactants must surmount for a reaction to occur.

This energy requirement is often visualized in energy profile diagrams where the reactants need enough energy to reach the top of a figurative hill, called the activation energy barrier. Think of it as a hurdle that molecules need to clear for a successful reaction. The height of this barrier influences how easily a reaction occurs; lower barriers mean easier reactions.

In the Arrhenius equation, activation energy is denoted by the symbol \( Ea \). It plays a vital role in determining the rate at which a reaction proceeds. The higher the \( Ea \), the slower the chemical reaction will be, as fewer molecules will have sufficient energy to surpass the activation barrier.

Rate Constant

The rate constant, symbolized as \( k \) in the Arrhenius equation, is an essential part of chemical kinetics. It quantifies the speed of a chemical reaction at a given temperature.

Understanding the rate constant helps predict how fast a reaction will occur under specific conditions. The rate constant is influenced by the frequency factor \( A \) and how sensitive the reaction is to changes in temperature.

A higher rate constant indicates a faster reaction. It's important to note that the rate constant is not a constant in the usual sense. It varies with temperature, reflecting the fact that reactions tend to speed up as temperature increases. By knowing the rate constant, chemists can estimate how quickly reactant concentrations will change over time, providing insight into the efficiency and feasibility of a reaction.

Temperature Dependence

Temperature dependence in chemical reactions refers to how changes in temperature influence the rate and extent of a reaction. Generally, increasing the temperature speeds up reactions.

Here's why: Higher temperatures provide reactant molecules with more kinetic energy, allowing more molecules to overcome the activation energy barrier needed to transform into products.

The Arrhenius equation highlights this relationship, where an increase in temperature results in an exponential increase in the rate constant \( k \). This explains why slight changes in temperature can lead to significant changes in reaction rates. Understanding temperature dependence is crucial for controlling reactions in various fields, from industrial processes to biological systems.

Chemical Kinetics

Chemical kinetics is the study of reaction rates and the factors that affect them. It gives insights into how quickly reactions proceed, the steps involved, and the overall mechanisms.

In kinetics, factors like concentration, temperature, and the presence of catalysts are examined to understand their impact on the speed of reaction.

The study of chemical kinetics involves using equations like the Arrhenius equation to predict how different parameters affect the rate of reactions. Kinetics not only helps in designing chemical processes efficiently but also in understanding natural processes, such as metabolic pathways in living organisms.