Problem 18 - Chemistry and Chemical Reactivity

Question

The conversion of cyclopropane to propene (see Example 15.5 ) occurs with a first-order rate constant of \(2.42 \times 10^{-2} \mathrm{h}^{-1} .\) How long will it take for the concentration of cyclopropane to decrease from an initial concentration of \(0.080 \mathrm{mol} / \mathrm{L}\) to \(0.020 \mathrm{mol} / \mathrm{L} ?\)

Step-by-Step Solution

Answer

It will take approximately 57.27 hours for the conversion.

Step 1: Understand the First-Order Reaction Rate Equation

In a first-order reaction, the rate of reaction is directly proportional to the concentration of one reactant. The integrated rate law for a first-order reaction is given by: \( \ln \left( \frac{[A]_0}{[A]} \right) = kt \), where \([A]_0\) is the initial concentration, \([A]\) is the remaining concentration, \(k\) is the rate constant, and \(t\) is time.

Step 2: Substitute the Known Values

We know \([A]_0 = 0.080 \, \mathrm{mol/L}\), \([A] = 0.020 \, \mathrm{mol/L}\), and \(k = 2.42 \times 10^{-2} \, \mathrm{h}^{-1}\). Substitute these values into the integrated rate law to get: \[ \ln \left( \frac{0.080}{0.020} \right) = 2.42 \times 10^{-2} \times t \].

Step 3: Simplify to Solve for Time \(t\)

First calculate \(\ln \left( \frac{0.080}{0.020} \right)\), which simplifies to \(\ln(4)\). Using a calculator, find \(\ln(4) \approx 1.386\). Substitute back into the equation: \[ 1.386 = 2.42 \times 10^{-2} \times t \].

Step 4: Solve for \(t\)

To solve for \(t\), re-arrange the equation \(1.386 = 2.42 \times 10^{-2} \times t\) to find \(t\) by dividing both sides by \(2.42 \times 10^{-2}\): \[ t = \frac{1.386}{2.42 \times 10^{-2}} \approx 57.27 \, \mathrm{hours} \].

Key Concepts

Integrated Rate Law

The integrated rate law is a crucial tool in understanding how the concentration of a reactant changes over time in a chemical reaction. For first-order reactions, this law allows us to determine how long it will take for a reactant to reach a certain concentration from its initial concentration. The equation takes the form:\[\ln \left( \frac{[A]_0}{[A]} \right) = kt\]where:

• \([A]_0\) is the initial concentration of the reactant.
• \([A]\) is the concentration of the reactant at time \(t\).
• \(k\) is the reaction rate constant.
• \(t\) is the time over which the reaction occurs.

This formula is derived from the basic rate law, and it links the concentration of the reactants to the time and the rate constant. By substituting known values into this equation, you can solve for the unknown variable, typically time \(t\).

Reaction Rate Constant

The reaction rate constant, symbolized as \(k\), is a measure of the speed of a reaction. In the context of first-order reactions, \(k\) tells us how quickly the reaction proceeds. For our specific problem involving cyclopropane, the first-order rate constant is given as \(2.42 \times 10^{-2} \, \mathrm{h}^{-1}\).This constant is unique to each reaction and can change depending on several factors such as temperature and the presence of a catalyst.
For first-order reactions:

• The units of \(k\) are usually \(\mathrm{time}^{-1}\), such as \(\mathrm{h}^{-1}\), indicating the fraction of the reactant that is consumed per time unit.
• A larger \(k\) value means a faster reaction.

Knowing \(k\) allows us to quantitatively describe how quickly the concentration of reactants decreases over time.

Concentration Change

Concentration change in reactions is a key aspect of reaction dynamics. In first-order reactions, this change is exponential, meaning the concentration of the reactant decreases at a rate proportional to its current amount. This is different from zero-order reactions, where concentration decreases linearly.For cyclopropane converting to propene:

• The initial concentration \([A]_0\) is \(0.080 \, \mathrm{mol/L}\).


• The concentration \([A]\) we aim to find the time for is \(0.020 \, \mathrm{mol/L}\).

The change in concentration can be predicted using the integrated rate law for first-order reactions. This transform of concentrations over a timeframe helps predict the point in time by which a specific concentration will be achieved.

Kinetics

Kinetics is the branch of chemistry that studies the rate of chemical reactions and the factors that affect them. For first-order reactions, kinetics involves understanding how the concentration of a single reactant influences the reaction rate. Key concepts include:

• Understanding the relationship between reactant concentration and reaction time.


• Applying the rate constant to quantify reaction speed.


• Using the integrated rate law to connect these factors.

Kinetics not only helps in calculating reaction time but also provides insights into reaction mechanisms, which are crucial for both academic study and industrial applications. By developing a firm grasp of kinetics, students can predict and control the outcomes of chemical reactions effectively.