Question

Consider the hypothetical decomposition \(Z \longrightarrow\) products The rate of the reaction as a function of temperature in \(M / \min\) is $$ \text { rate }=2.7 t-19 $$ where \(t\) is the temperature in \({ }^{\circ} \mathrm{C}\). (a) Calculate the rate of decomposition at \(17^{\circ} \mathrm{C}\) and at \(27^{\circ} \mathrm{C}\) (b) Estimate the activation energy of the reaction. (c) What is the percent increase in rate for a \(10^{\circ} \mathrm{C}\) increase in temperature?

Short answer

Answer: The rate of decomposition at \(17^{\circ} \mathrm{C}\) is 26.9 M/min, and at \(27^{\circ} \mathrm{C}\) is 53.9 M/min. The percent increase in rate for a \(10^{\circ} \mathrm{C}\) increase in temperature is 100%.

Step-by-step solution

(a) Calculate the rate of decomposition at \(17^{\circ} \mathrm{C}\) and at \(27^{\circ} \mathrm{C}\)

To find the rate of decomposition at the respective temperatures, we plug in the temperature values into the given rate equation: $$ \text{rate} = 2.7t - 19 $$ For \(t=17^{\circ} \mathrm{C}\): $$ \text{rate}_{17} = 2.7(17) - 19 = 45.9 - 19 = 26.9\, M/\text{min} $$ For \(t=27^{\circ} \mathrm{C}\): $$ \text{rate}_{27} = 2.7(27) - 19 = 72.9 - 19 = 53.9\, M/\text{min} $$

(b) Estimate the activation energy of the reaction

We are only given rate and temperature, so we cannot directly use the Arrhenius equation to determine the activation energy. Therefore, we can only estimate the activation energy based on the given information. By observing that the rate increases as the temperature increases, we can conclude that the reaction has a positive activation energy. However, we cannot provide an exact numerical value for the activation energy without more information.

(c) Calculate the percent increase in rate for a \(10^{\circ} \mathrm{C}\) increase in temperature

To find the percent increase in rate for a \(10^{\circ} \mathrm{C}\) increase in temperature, we use the rates calculated in part (a): $$ \text{percent increase} = \frac{\text{Rate at final temperature} - \text{Rate at initial temperature}}{\text{Rate at initial temperature}} \times 100\% $$ Using the rates calculated at \(17^{\circ} \mathrm{C}\) and \(27^{\circ} \mathrm{C}\), we have: $$ \text{percent increase} = \frac{53.9 - 26.9}{26.9} \times 100\% = 1.0 \times 100\% = 100\% $$ The percent increase in the rate for this reaction for a \(10^{\circ} \mathrm{C}\) increase in temperature is 100%.

Key concepts

Chemical Kinetics

Chemical kinetics is the branch of chemistry that deals with the rates of chemical reactions—the speed at which they occur and the various factors that influence this speed. Understanding kinetics allows chemists to control reaction conditions to optimize the production of desired products. One important factor that affects the rate of a reaction is temperature, as shown in the given exercise.

In the hypothetical decomposition example considered, the rate equation is linear in terms of temperature, with the rate increasing as the temperature increases. This indicates that the reaction proceeds more rapidly at higher temperatures, a common observation in chemical reactions due to increased molecular motion leading to more frequent and effective collisions among reactant molecules. This relationship between reaction rate and temperature is a fundamental principle in kinetic theory.

When studying kinetics, we also look into the concentration of reactants, the presence of catalysts, the physical state of the reactants, and the surface area. All these factors can significantly alter the rate at which a reaction will proceed.

Arrhenius Equation

The Arrhenius equation provides a quantitative basis for the relationship between the rate of a chemical reaction and temperature. It is a fundamental formula in the study of chemical kinetics, represented as:

\[ k = A \times e^{-\frac{E_a}{RT}} \]
where:

• \( k \) is the rate constant of the reaction,
• \( A \) is the pre-exponential factor, which is a constant for each chemical reaction,
• \( E_a \) is the activation energy, the energy barrier that must be overcome for the reactants to transform into products,
• \( R \) is the gas constant, and
• \( T \) is the temperature (in Kelvin).

This equation shows that the rate constant \( k \) increases exponentially as the temperature rises, provided that the activation energy remains constant.

Although the Arrhenius equation was not used directly in the exercise to calculate the activation energy, it is implicit in understanding why the reaction rate increases with temperature. The equation helps to predict how much faster a reaction will proceed if the temperature is increased. Unfortunately, without additional information like the rate constant at different temperatures, it is not possible to calculate the exact value of the activation energy.

Temperature Dependence of Reaction Rates

The temperature dependence of reaction rates is typically expressed in the form of the Arrhenius equation, as previously described. This relationship is vital to the understanding of chemical processes and can be experimentally determined by measuring reaction rates at various temperatures.

In the exercise provided, the rate of reaction increases linearly with temperature, a simplistic representation of the temperature effect. In reality, the correlation is often more complex. As temperature increases, molecules move faster and collide more often with greater energy, which generally leads to an exponential increase in reaction rate. This is because only a fraction of the collisions have sufficient energy to overcome the activation energy barrier.

The stark 100% increase in rate from a temperature change from \(17^{\text{o}}C\) to \(27^{\text{o}}C\) may also demonstrate an aspect of the exponential nature of this dependence. The significant percent increase shows that even a small temperature increase can dramatically affect the reaction rate. This phenomenon underlies the importance of temperature control in industrial processes, where rates must be kept within specific limits to ensure safety and product quality.