Question

(a) Using a graphing calculator or graphing software, such as that on the Web site for this book, calculate the activation energy for the acid hydrolysis of sucrose to give glucose and fructose from an Arrhenius plot of the data shown here. (b) Calculate the rate constant at \(37^{\circ} \mathrm{C}\) (body temperature). (c) From data in Appendix 2A, calculate the enthalpy change for this reaction, assuming that the solvation enthalpies of the sugars are negligible. Draw an energy profile for the overall process. $$ \begin{array}{cc} \text { Temperature }\left({ }^{\circ} \mathrm{C}\right) & k\left(\mathrm{~s}^{-1}\right) \\ \hline 24 & 4.8 \times 10^{-3} \\ 28 & 7.8 \times 10^{-3} \\ 32 & 13 \times 10^{-3} \\ 36 & 20 . \times 10^{-3} \\ 40 . & 32 \times 10^{-3} \\ \hline \end{array} $$

Short answer

The activation energy is calculated from the slope of the Arrhenius plot, the rate constant at body temperature is calculated using the Arrhenius equation with this activation energy, and the enthalpy change can be estimated from the activation energy and temperature.

Step-by-step solution

Understanding the Arrhenius Equation

To calculate the activation energy, we need to understand the Arrhenius Equation which is given by: \(k = A \times e^{-E_a/(RT)}\) where \(k\) is the rate constant, \(A\) is the frequency factor, \(E_a\) is the activation energy, \(R\) is the gas constant, and \(T\) is the temperature in Kelvin.

Preparing the Data for the Arrhenius Plot

Create a table where ln(k) is plotted against 1/T in Kelvin. Convert the Celsius temperatures to Kelvin by adding 273.15 and compute the ln(k) for each rate constant provided.

Plotting the Arrhenius Plot

Using graphing software, plot ln(k) as the y-axis against 1/T (in Kelvin) as the x-axis. The slope of the line will be equal to \(-E_a/R\).

Calculating the Activation Energy

Determine the slope of the Arrhenius plot and calculate the activation energy using the relation \(E_a = -slope \times R\) where \(R = 8.314\, J/(mol\text{K})\).

Calculate the Rate Constant at 37°C

Once we have \(E_a\), we can calculate the rate constant at body temperature (37°C or 310.15 K) using the Arrhenius equation. First rearrange the equation to solve for \(k\) and plug in the values.

Calculate the Enthalpy Change

The enthalpy change \(\Delta H\) can be approximated using the relation \(\Delta H = E_a - RT\), where \(T\) is the average temperature for which the \(E_a\) was calculated.

Drawing the Energy Profile

The energy profile of the reaction can be drawn with the reactants at one level, the transition state at the level of \(E_a\) above that of the reactants, and the products at the level of \(E_a - \Delta H\) relative to the reactants.

Key concepts

Activation Energy

Activation energy, denoted as \(E_a\), refers to the minimum energy required for a chemical reaction to take place. It's a vital concept in the study of kinetics which explains why certain reactions occur only at elevated temperatures.

This energy acts as a barrier that reactants must overcome to transform into products. The higher the \(E_a\), the slower the reaction proceeds at a given temperature, because fewer molecules have the requisite energy to collide effectively and undergo the reaction.

Using the Arrhenius Equation, \(k = A \times e^{-E_a/(RT)}\), the plot of ln(k) against \(1/T\) gives us a straight line whose slope is related to the activation energy by the equation \(E_a = -slope \times R\), with the gas constant \(R\) known. Through this method, students can compute \(E_a\) from empirical data, revealing the intrinsic energy hurdle specific to their reaction of interest.

Rate Constant

The rate constant of a chemical reaction, symbolized by \(k\), is an essential parameter indicating the speed at which a reaction progresses. It is impacted by several factors, including temperature, pressure, and the presence of catalysts.

According to the Arrhenius Equation mentioned earlier, the rate constant and temperature are exponentially related, indicating that even small temperature changes can significantly alter the reaction rate. This relationship underscores the importance of temperature control in chemical processes.

For instance, to calculate the rate constant at body temperature, rearrange the Arrhenius equation to solve for \(k\), substituting in the values of \(E_a\), \(R\), and the temperature in Kelvin. Understanding how to make these calculations is fundamental for students studying reaction kinetics and is critical for applications in fields like pharmaceuticals, where reaction rates at physiological temperatures are crucial.

Enthalpy Change

Enthalpy change, or \(\Delta H\), for a reaction represents the total heat content change occurring as reactants convert to products. It's a thermodynamic quantity that is indicative of whether a reaction is exothermic (releases heat) or endothermic (absorbs heat).

Calculating \(\Delta H\) typically requires calorimetry or analysis of bond energies, but in the context of the Arrhenius equation, it is approximated using \(\Delta H = E_a - RT\), with \(T\) representing the average temperature at which \(E_a\) was determined. This approximation is useful for students exploring the relationship between activation and reaction energies without conducting extensive calorimetric studies.

An energy profile diagram can help visualize this change, highlighting the energy levels of reactants, transition states, and products. Such visual aids not only help students understand the energies involved but also emphasize the difference between activation energy and the overall energy change in a reaction.