Problem 20 - General Chemistry: Principles and Modern Applications

Question

In the first-order reaction \(A \longrightarrow\) products, \([\mathrm{A}]=0.816 \mathrm{M}\) initially and \(0.632 \mathrm{M}\) after \(16.0 \mathrm{min}.\) (a) What is the value of the rate constant, \(k ?\) (b) What is the half-life of this reaction? (c) At what time will \([\mathrm{A}]=0.235 \mathrm{M} ?\) (d) What will [A] be after 2.5 h?

Step-by-Step Solution

Answer

The calculated rate constant \(k\) is \( \frac{1}{16} \cdot ln(\frac{0.816}{0.632}) \approx 0.0204 min^{-1}\). The half-life of the reaction is \( \frac{ln(2)}{k} \approx 34.0 min\). The time it takes for the reactant concentration to reach 0.235 M is \( \frac{1}{k} \cdot ln(\frac{0.816}{0.235}) \approx 52.5 min\). The concentration of A after 2.5 hours or 150 minutes is \(A_0 \cdot e^{-kt} = 0.816 M \cdot e^{-0.0204 min^{-1} \cdot 150 min} \approx 0.093 M\).

Step 1: Calculate Rate Constant (k)

The rate constant \(k\) describes how fast a reaction proceeds. It can be calculated using the formula \(k = \frac{1}{t} \cdot ln(\frac{A_0}{A(t)})\), where \(A_0\) is the initial amount, \(A(t)\) is the amount after time \(t\). Insert the given values \(A_0 = 0.816 M\), \(A(t) = 0.632 M\), and \(t = 16.0 min\) into the formula to calculate \(k\).

Step 2: Calculate Half-Life (t_{1/2})

The half-life \(t_{1/2}\) is the time it takes for half of the substance to react. It can be calculated by the formula \(t_{1/2} = \frac{ln(2)}{k}\). Use the calculated value for \(k\) from Step 1 to calculate \(t_{1/2}\).

Step 3: Calculate the Time to Reach 0.235 M

To find the time it takes for the reactant concentration to reach a certain value, the formula \(t = \frac{1}{k} \cdot ln(\frac{A_0}{A(t)})\) can be used. Insert the given values \(A_0 = 0.816 M\), \(A(t) = 0.235 M\), and the previously calculated \(k\) into the formula to calculate the time.

Step 4: Calculate the Concentration After 2.5 Hours

The equation \(A(t) = A_0 \cdot e^{-kt}\) can be used to calculate the amount at a specific time. Convert 2.5 hours to minutes to match the units of the \(k\), and then use this time, along with \(A_0\) and the previously calculated \(k\), to find the concentration of A after 2.5 hours.

Key Concepts

Rate constant calculation

The rate constant (\( k \)) in a first-order reaction gives insight into the speed of the process. In essence, it defines how fast the reaction progresses over time. To determine the rate constant, you can use the formula:
\[k = \frac{1}{t} \cdot \ln\left(\frac{A_0}{A(t)}\right)\]where:

• \( A_0 \) is the initial concentration of the reactant,
• \( A(t) \) is the concentration at time \( t \), and
• \( t \) is the time elapsed.

For example, consider the reaction where initially \( [A] = 0.816 \text{ M} \) and after 16.0 minutes, \( [A] = 0.632 \text{ M} \). Plugging these values into the formula gives the rate constant. It's important to remember that a larger \( k \) value implies a quicker reaction. Calculating \( k \) accurately helps understand the dynamics of the reaction at different stages.

Half-life of a reaction

The half-life (\( t_{1/2} \)) of a reaction indicates the period required for the concentration of a reactant to reduce to half its original value. For first-order reactions, the half-life is independent of the initial concentration and can be determined by the formula:
\[t_{1/2} = \frac{\ln(2)}{k}\]Here:

• \( \ln(2) \) is the natural logarithm of 2, approximately equal to 0.693.

Once the rate constant (\( k \)) is known, this formula makes it straightforward to find the half-life. Understanding \( t_{1/2} \) is vital in predicting how long a reaction will take to reach a certain completion level, especially when planning experimental timelines or chemical processes.

Reaction concentration change over time

In first-order reactions, the change in reactant concentration over time can be modeled using exponential functions. This is key in understanding how concentrations shift as the reaction moves forward. The general formula is:
\[ [A(t)] = [A_0] \cdot e^{-kt} \]which outlines the concentration \( [A(t)] \) at time \( t \), where:

• \( A_0 \) is the starting concentration,
• \( e \) is the base of the natural logarithm, and
• \( k \) is the rate constant.

For practical purposes, if you want to know the concentration of a substance at a future time, simply plug the values of \( A_0 \), \( k \), and \( t \) into the formula. For example, if you already know \( k \) and wish to find out when \( [A] = 0.235 \text{ M} \), rearrange and solve the formula for time. Or, find the concentration after they give you a certain amount of time, like 2.5 hours. This approach gives a deep understanding of how concentration dwindles over time as the reaction progresses.