Problem 50 - Chemistry The Central Science In Si Units

Question

Sucrose \(\left(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right),\) commonly known as table sugar, reacts in dilute acid solutions to form two simpler sugars, glucose and fructose, both of which have the formula \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\). At \(23{ }^{\circ} \mathrm{C}\) and in \(0.5 \mathrm{MHCl}\), the following data were obtained for the disappearance of sucrose: \begin{tabular}{cc} \hline Time (min) & {\(\left[\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{0}_{11}\right](\mathrm{M})\)} \\ \hline 0 & 0.316 \\ 39 & 0.274 \\ 80 & 0.238 \\ 140 & 0.190 \\ 210 & 0.146 \\ \hline \end{tabular} (a) Is the reaction first order or second order with respect to \(\left[\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right] ?(\mathbf{b})\) What is the rate constant? (c) Using this rate constant, calculate the concentration of sucrose at 39 , \(80,140,\) and \(210 \mathrm{~min}\) if the initial sucrose concentration was \(0.316 \mathrm{M}\) and the reaction were zero order in sucrose.

Step-by-Step Solution

Answer

The reaction is first-order. The rate constant is 0.001 min⁻¹. For zero-order, the calculated concentrations deviate from measured concentrations.

Step 1: Determine Order of Reaction

To determine if the reaction is first or second order, we analyze the data assuming a first-order reaction. For a first-order reaction, \\[ \ln\left( \frac{[A]_0}{[A]} \right) = kt \] \where \([A]_0\) is the initial concentration and \([A]\) is the concentration at time \(t\). Calculate \(\ln\left( \frac{[A]_0}{[A]} \right)\) at each time point and observe if they fit a linear trend, indicating a first-order reaction.

Step 2: Calculate Data for First-Order Reaction

Using the formula from Step 1, calculate \( \ln\left( \frac{[0.316]}{[0.274]} \right) \) for 39 minutes, \( \ln\left( \frac{[0.316]}{[0.238]} \right) \) for 80 minutes, and so forth. Check if plotting these results against time gives a straight line. If it does, the reaction is first order.

Step 3: Regression Check for Second-Order

If the data do not fit a linear model in Step 2, attempt a second-order analysis. For second-order reactions, use \\[ \frac{1}{[A]} = \frac{1}{[A]_0} + kt \] \Calculate and plot \( \frac{1}{[A]} \) for each time to see if this fits a linear line. This will confirm second-order behavior.

Step 4: Determine Rate Constant

Given that the reaction is determined to be first-order (if the fit in Step 2 was linear), use the slope of the line from your plot, which represents the rate constant \(k\). Use \\[ k = \frac{\ln([A]_0) - \ln([A])}{t} \] \to calculate using one of the time points, confirming with multiple points for precision.

Step 5: Calculate Sucrose Concentration for Zero-Order

For zero-order reactions, use \\[ [A] = [A]_0 - kt \] \Given \(k\) and initial concentration \([A]_0 = 0.316\) M, calculate \([A]\) at each time: \- For 39 min, \- For 80 min, \- For 140 min, and \- For 210 min.

Key Concepts

First-order Reactions

First-order reactions are characterized by their dependence on the concentration of a single reactant. Essentially, the rate of reaction is directly proportional to the concentration of one reactant. This is a common scenario in many chemical processes.
In mathematical terms, the rate of a first-order reaction can be given by the differential rate equation:

• \( \ln\left( \frac{[A]_0}{[A]} \right) = kt \)

Here, \([A]_0\) denotes the initial concentration, \([A]\) is the concentration after a time \(t\), and \(k\) is the rate constant. By plotting \( \ln([A]_0/[A]) \) against time, a straight line should be observed if the reaction follows first-order kinetics. This linear pattern indicates a constant proportion of the reactant disappearing over equal time periods, which is the hallmark of first-order reactions. This type of graph helps chemists to visually confirm the reaction order before moving on to more detailed calculations.

Second-order Reactions

In second-order reactions, the reaction rate is proportional to either the square of the concentration of a single reactant or the product of the concentrations of two reactants.
For second-order reactions, the related equation is:

• \( \frac{1}{[A]} = \frac{1}{[A]_0} + kt \)

This implies that the plot of \(\frac{1}{[A]}\) versus time should yield a straight line for a second-order reaction, which contrasts with the results from the first-order plot.
The slope of this line will be the rate constant \(k\). Understanding whether a reaction is second-order is crucial for predicting how the concentrations will change over time and for determining the time it takes to reach a certain concentration of reactant or product.

Rate Constant Calculation

The rate constant \(k\) is vital in the field of reaction kinetics as it links the reaction rate with the concentrations present in the system. It serves as a proportionality factor in rate equations, offering insight into the speed of a reaction.
Regardless of the reaction order, calculating \(k\) involves determining the slope from plotted data:

• For first-order reactions, it's found from the slope of \( \ln([A]_0/[A]) \) versus time plots.
• For second-order reactions, it's derived from the slope of \( \frac{1}{[A]} \) versus time plots.

The rate constant's unit changes with the order of reaction, ensuring the equation results in a consistent unit of rate (e.g., mol/L/s). Understanding \(k\) helps predict how quickly the substances involved will be converted and is essential for scaling up reactions for larger applications.

Zero-order Reactions

Zero-order reactions are unique because their rate is independent of the concentration of the reactant. This type of reaction often occurs under conditions where a substance's surface can be completely saturated, such as in catalytic processes or when light supplies the energy.
The zero-order rate equation is:

• \([A] = [A]_0 - kt\)

This equation indicates that the concentration of reactant decreases linearly over time, resulting in a constant rate of reaction. Here, plotting \([A]\) versus time yields a straight line where the slope is \(-k\).
Understanding zero-order reactions is vital for accurately predicting how a reaction progresses over time, especially when external factors like catalysts are involved. It is fascinating as it showcases a scenario where traditional concentration dependencies do not apply, giving a different perspective on reaction kinetics.