Problem 78 - Introductory Chemistry Atoms First

Question

Suppose you increase the temperature of a reaction from \(100^{\circ} \mathrm{C}\) to \(200^{\circ} \mathrm{C}\) and the reaction gets three times as fast. (a) Would the rate constant for the reaction at \(100^{\circ} \mathrm{C}\) be equal to the rate constant for the reaction at \(200^{\circ} \mathrm{C}\) ? (b) Suppose you took a ratio \(k_{200^{\circ} \mathrm{C}} / k_{100^{\circ} \mathrm{C}}\). According to the information given in part (a), what would you expect the value of this ratio to be?

Step-by-Step Solution

Answer

The rate constant at \(100^{\circ}\mathrm{C}\) is not equal to the rate constant at \(200^{\circ}\mathrm{C}\) because the reaction rate increases with temperature. The ratio between the rate constants at these two temperatures, \(\frac{k_{200^{\circ}\mathrm{C}}}{k_{100^{\circ}\mathrm{C}}}\), is 3, as the reaction at \(200^{\circ}\mathrm{C}\) is three times faster than the reaction at \(100^{\circ}\mathrm{C}\).

Step 1: Given that the reaction gets three times as fast when increasing the temperature from \(100^{\circ}\mathrm{C}\) to \(200^{\circ}\mathrm{C}\), it indicates that the rate constants are not equal at these two temperatures. The rate constant is directly related to the reaction rate, so if the rate increases, so does the rate constant. Hence, the rate constant for the reaction at \(100^{\circ}\mathrm{C}\) is not equal to the rate constant for the reaction at \(200^{\circ}\mathrm{C}\). #b) Calculating the ratio#

From the information given in part (a), we know that the rate constant at \(200^{\circ}\mathrm{C}\) is three times the rate constant at \(100^{\circ}\mathrm{C}\). So, we can write this as: \[k_{200^{\circ}\mathrm{C}}= 3 \times k_{100^{\circ}\mathrm{C}} \] Now, we can find the ratio between these two rate constants: \[\frac{k_{200^{\circ}\mathrm{C}}}{k_{100^{\circ}\mathrm{C}}} = \frac{3 \times k_{100^{\circ}\mathrm{C}}}{k_{100^{\circ}\mathrm{C}}}\] By simplifying, we get the value of the ratio: \[\frac{k_{200^{\circ}\mathrm{C}}}{k_{100^{\circ}\mathrm{C}}} = 3\] Therefore, the value of the ratio between the rate constant at \(200^{\circ}\mathrm{C}\) and the rate constant at \(100^{\circ}\mathrm{C}\) is 3.

Key Concepts

Temperature Effect on Reaction Rate

Understanding how temperature affects reaction rates is a fundamental aspect of chemical kinetics. Generally, increasing the temperature of a reaction system leads to an increase in the reaction rate. This is due to the fact that particles at higher temperatures have more kinetic energy, making them move faster and collide more frequently. Not only does the frequency of collisions increase, but also the energy with which the particles collide is greater, increasing the likelihood of overcoming the activation energy barrier, which is the minimum energy required for a reaction to occur.

For instance, suppose you have a reaction that gets three times faster when the temperature is increased from 100 degrees Celsius to 200 degrees Celsius. This signifies a direct relationship between temperature and the speed of the reaction. According to the Arrhenius equation, the rate constant \( k \) of a reaction is affected by temperature, showing an exponential increase with temperature. Most reactions adhere to the rule of thumb that a 10-degree Celsius increase in temperature approximately doubles the reaction rate. However, the increase in this example is much larger, indicating a highly temperature-sensitive reaction.

In summary, higher temperatures generally mean higher reaction rates due to increased kinetic energy, resulting in more effective collisions among reactant molecules, and therefore, an increased reaction rate constant.

Chemical Kinetics

Chemical kinetics is the study of the rates of chemical processes and the factors that influence them. It involves the analysis of how different conditions, such as concentration, temperature, and the presence of catalysts, affect the speed at which reactants are converted into products. One of the primary objectives of studying kinetics is to understand the sequence of steps, or the reaction mechanism, through which a reaction proceeds. This can entail looking at the intermediate species formed during the reaction and their stability.

In the context of temperature's effect on rate, the focus is on how reactant molecules must collide with sufficient energy and proper orientation to form products. Kinetics also involves the calculation of the rate constant \( k \), which provides a quantitative measure of how quickly a reaction proceeds under specific conditions. The rate constant is unique for each reaction at a given temperature and can be influenced by changes in conditions, as seen in the initial exercise where the rate constant changes with temperature.

Rate Constants Comparison

Comparing rate constants at different temperatures gives insight into how sensitive a reaction is to thermal changes. The rate constant, \( k \) is not only a measure of the speed of a reaction but also an indicator of its dependence on temperature. For example, an increase in temperature from 100 degrees Celsius to 200 degrees Celsius resulting in a tripling of the reaction rate signifies a significant sensitivity. To quantify this, a ratio of the rate constants at these two temperatures, \( k_{200^{\text{\degree}C}} / k_{100^{\text{\degree}C}} \) can be used.

In the exercise provided, this ratio is calculated to be 3, indicating that the rate at 200 degrees Celsius is three times that at 100 degrees Celsius. Such a comparison is useful when predicting how a reaction will behave under different thermal conditions, making it critical for the design of chemical processes and safety assessments. This ratio is a practical representation of temperature's influence on reaction rates and highlights the logarithmic relationship between temperature and rate constants described by the Arrhenius equation.