Problem 73 - Chemistry Matter and Change

Question

Dinitrogen pentoxide decomposes in chloroform at a rate of \(2.48 \times 10^{-4} \mathrm{mol} /(\mathrm{L} \cdot \mathrm{min})\) at a particular tempera- ture according to the equation \(2 \mathrm{N}_{2} \mathrm{O}_{5} \rightarrow 4 \mathrm{NO}_{2}+\mathrm{O}_{2}\) The reaction is first order in \(\mathrm{N}_{2} \mathrm{O}_{5}\) . Given an initial concentration \(0.400 \mathrm{mol} / \mathrm{L},\) what is the rate constant for the reaction? What is the approximate \(\left[\mathrm{N}_{2} \mathrm{O}_{5}\right]\) after the reaction proceeds for 1.30 \(\mathrm{h} ?\)

Step-by-Step Solution

Answer

Rate constant \(k\) is \(6.20 \times 10^{-4} \text{min}^{-1}\); \([\mathrm{N}_2\mathrm{O}_5]\) after 1.30 hours is approximately \(0.381 \text{mol/L}\).

Step 1: Identify the Parameter to Calculate

We need to calculate the rate constant for the decomposition of \(\mathrm{N}_2\mathrm{O}_5\) and find the concentration of \([\mathrm{N}_2\mathrm{O}_5]\) after 1.30 hours. Since the reaction is first order, the integrated rate law for a first-order reaction is used.

Step 2: Write the Integrated Rate Law for First-Order Reaction

The integrated rate law for a first-order reaction is given by the equation:\[\ln [\mathrm{A}]_t = -kt + \ln [\mathrm{A}]_0\]where \([\mathrm{A}]_t\) is the concentration at time \(t\), \([\mathrm{A}]_0\) is the initial concentration, and \(k\) is the rate constant.

Step 3: Use Rate Law to Find Rate Constant

The rate of reaction is given as \(2.48 \times 10^{-4} \, \mathrm{mol/L\cdot min}\) and the initial concentration \([\mathrm{N}_2\mathrm{O}_5]_0\) is \(0.400 \, \mathrm{mol/L}\). Since this is a first-order reaction, the rate \(r\) is also equal to \(k[\mathrm{N}_2\mathrm{O}_5]_0\).So, we have:\[k = \frac{\text{rate}}{[\mathrm{N}_2\mathrm{O}_5]_0} = \frac{2.48 \times 10^{-4} \, \mathrm{mol/L\cdot min}}{0.400 \, \mathrm{mol/L}} = 6.20 \times 10^{-4} \, \text{min}^{-1}\]

Step 4: Convert Time from Hours to Minutes

The time given is 1.30 hours. Convert this to minutes for consistency with the units of the rate constant:\[1.30 \, \mathrm{hours} = 1.30 \times 60 \, \mathrm{minutes} = 78 \, \mathrm{minutes}\]

Step 5: Use Integrated Rate Law to Find Final Concentration

Now plug the values into the integrated rate law:\[\ln [\mathrm{N}_2\mathrm{O}_5]_t = - (6.20 \times 10^{-4} \, \text{min}^{-1})(78 \, \text{min}) + \ln(0.400 \, \text{mol/L})\]Calculate:\[= - (6.20 \times 10^{-4})(78) + \ln(0.400)\]\[= -0.04836 + (-0.916)\]\[\ln [\mathrm{N}_2\mathrm{O}_5]_t = -0.96436\]Calculate the exponential to find \([\mathrm{N}_2\mathrm{O}_5]_t\):\[[\mathrm{N}_2\mathrm{O}_5]_t = e^{-0.96436} \approx 0.381 \, \text{mol/L}\]

Step 6: Final Step: Verify Units and Reasonableness

Ensure that all dimensions match and that each calculation step is consistent. The calculated final concentration fits the expectation of a decrease from the initial value.

Key Concepts

First-Order Reaction

First-order reactions are a fundamental concept in chemical kinetics, where the reaction rate depends linearly on the concentration of one reactant. In these reactions, a single reactant influences the rate directly, offering a predictable pattern of change over time.
For a first-order reaction, if you double the concentration of the reactant, the reaction rate also doubles. It's a direct relationship
and can often be observed in simple decomposition reactions. The general form of a first-order reaction can be expressed as:\[ A \rightarrow \, Products \]where the rate is given by \( \text{Rate} = k[A] \). Here, \( k \) is the rate constant, and \( [A] \) is the concentration of the reactant \( A \).

This straightforward dependence makes it easy to model and predict. First-order reactions are often used as a stepping stone for understanding more complex reaction mechanisms.

Integrated Rate Law

The integrated rate law for a first-order reaction provides a mathematical tool to determine the concentration of reactants over time. It is derived from the basic rate equation and is given by:
\[ \ln [A]_t = -kt + \ln [A]_0 \]
Here, \([A]_t\) is the concentration at time \( t \), \([A]_0\) represents the initial concentration, and \( k \) is the rate constant. This formula allows you to calculate how much of the reactant remains as time progresses.

The integrated rate law is crucial for plotting concentration versus time data in studies. It's particularly useful because:

• It allows prediction of how long a reaction will take to reach a certain level.
• You can calculate the half-life of a reactant, the time needed for half of the reactant to decompose, using \( t_{1/2} = \frac{0.693}{k} \).

Thus, the integrated rate law is a versatile tool in kinetics that transforms reaction rate calculations into manageable arithmetic.

Rate Constant

The rate constant, \( k \), is a central figure in chemical kinetics. It is a parameter that links the rate of reaction to the concentrations of reactants in any rate equation.
The value of the rate constant is specific to each reaction and depends on factors like temperature and the presence of a catalyst.

In first-order reactions, the rate constant possesses units of \( \text{time}^{-1} \), such as \( \text{min}^{-1} \) in this case. This tells you how quickly a reaction proceeds without necessarily knowing the specific concentrations
at any given moment.

Calculating the rate constant involves dividing the rate of the reaction by the initial concentration of the reactant:
\[ k = \frac{\text{rate}}{[\text{A}]_0} \]
Understanding the rate constant allows you to predict how changes to temperature or the reaction milieu affect the speed of a reaction. As a rule of thumb, a larger rate constant signals a faster reaction.

Reaction Rate Calculation

Calculating reaction rates in chemical kinetics lets you quantify how fast reactants turn into products. In the realm of first-order reactions, this calculation becomes more straightforward due to the linear dependency on reactant concentration.
To perform a reaction rate calculation, you typically use the formula:
\[ \text{Rate} = k[A] \]
where \( k \) is the rate constant, and \([A]\) is the concentration of the reactant.

This calculation helps predict:

• How different conditions, like a change in reactant concentration, influence the speed of the reaction.
• The dynamics of reaction pathways, especially plotting concentration against time for visual analysis.

If you're using the integrated rate law, you can determine the concentration of reactants at any time after the reaction starts. This is critical in laboratory settings where precise measures dictate reaction outcomes and efficiency. Understanding the demands of reaction rate calculation fosters accurate control over chemical processes.