Problem 6 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

The gas phase decomposition of dinitrogen pentoxide \(\left(\mathrm{N}_{2} \mathrm{O}_{5}\right)\) \\[ \mathrm{N}_{2} \mathrm{O}_{5} \rightarrow \mathrm{NO}_{2}^{*}+\mathrm{NO}_{3}^{*} \\] was studied in a large stainless steel cell, at \(294 \mathrm{K}\) and a pressure of 1 bar. The concentration of \(\mathrm{N}_{2} \mathrm{O}_{5}\) was monitored using infrared spectroscopy. The following results were obtained. (Section \(9.4)\) $$\begin{array}{lllllll} t / \mathrm{s} & 0 & 10 & 20 & 30 & 40 & 50 \\ {\left[\mathrm{N}_{2} \mathrm{O}_{8}\right] / 10^{-9} \mathrm{moldm}^{-3}} & 34.0 & 27.0 & 19.5 & 15.0 & 11.5 & 8.7 \\ t / \mathrm{s} & 60 & 70 & 80 & 90 & 100 & \\ {\left[\mathrm{N}_{2} \mathrm{O}_{8}\right] / 10^{-9} \mathrm{moldm}^{-3}} & 6.6 & 5.1 & 3.9 & 2.9 & 2.2 & \end{array}$$ Show that the reaction is first order and find a value for the rate constant at \(294 \mathrm{K}\)

Step-by-Step Solution

Answer

The reaction is first-order with a rate constant of \( k = 0.0274 \ ext{s}^{-1}\).

Step 1: Determine the First-Order Kinetics

In a first-order reaction, the rate of reaction depends linearly on the concentration of a single reactant. For the reaction \( \mathrm{N}_{2} \mathrm{O}_{5} \rightarrow \mathrm{NO}_{2}^{*} + \mathrm{NO}_{3}^{*} \), the rate law can be written as: \[ \text{rate} = k [\mathrm{N}_{2} \mathrm{O}_{5}] \]The integrated form of the first-order rate equation is given by:\[ \ln \left( \left[ \mathrm{N}_{2} \mathrm{O}_{5} \right]_t \right) = -kt + \ln \left( \left[ \mathrm{N}_{2} \mathrm{O}_{5} \right]_0 \right) \]Where:- \( [\mathrm{N}_{2} \mathrm{O}_{5}]_t \) is the concentration at time \( t \)- \( [\mathrm{N}_{2} \mathrm{O}_{5}]_0 \) is the initial concentration.- \( k \) is the rate constant.- \( t \) is time.We need to plot \( \ln \left( [\mathrm{N}_{2} \mathrm{O}_{5}] \right) \) against time \( t \) and check for linearity.

Step 2: Calculate the Natural Logarithm of Concentrations

For each concentration given in the table, calculate \( \ln \left( [\mathrm{N}_{2} \mathrm{O}_{5}] \right) \):- \( t = 0, [\mathrm{N}_{2} \mathrm{O}_{5}] = 34.0 \), \( \ln(34.0) = 3.53 \)- \( t = 10, [\mathrm{N}_{2} \mathrm{O}_{5}] = 27.0 \), \( \ln(27.0) = 3.30 \)- \( t = 20, [\mathrm{N}_{2} \mathrm{O}_{5}] = 19.5 \), \( \ln(19.5) = 2.97 \)- \( t = 30, [\mathrm{N}_{2} \mathrm{O}_{5}] = 15.0 \), \( \ln(15.0) = 2.71 \)- \( t = 40, [\mathrm{N}_{2} \mathrm{O}_{5}] = 11.5 \), \( \ln(11.5) = 2.44 \)- \( t = 50, [\mathrm{N}_{2} \mathrm{O}_{5}] = 8.7 \), \( \ln(8.7) = 2.16 \)- \( t = 60, [\mathrm{N}_{2} \mathrm{O}_{5}] = 6.6 \), \( \ln(6.6) = 1.89 \)- \( t = 70, [\mathrm{N}_{2} \mathrm{O}_{5}] = 5.1 \), \( \ln(5.1) = 1.63 \)- \( t = 80, [\mathrm{N}_{2} \mathrm{O}_{5}] = 3.9 \), \( \ln(3.9) = 1.36 \)- \( t = 90, [\mathrm{N}_{2} \mathrm{O}_{5}] = 2.9 \), \( \ln(2.9) = 1.06 \)- \( t = 100, [\mathrm{N}_{2} \mathrm{O}_{5}] = 2.2 \), \( \ln(2.2) = 0.79 \)

Step 3: Plot Data and Verify Linearity

Plot \( \ln \left( [\mathrm{N}_{2} \mathrm{O}_{5}] \right) \) against time \( t \) to check for linearity. The data follows a straight line, confirming that the reaction is first-order.

Step 4: Determine the Slope (Rate Constant)

Calculate the slope of the line using any two points from the plotted data, for instance at \( t = 0 \) and \( t = 100 \):\[ \text{slope} = \frac{0.79 - 3.53}{100 - 0} = -0.0274 \]The negative slope corresponds to the rate constant \( k \). So, \( k = 0.0274 \ ext{s}^{-1}\).

Key Concepts

First-Order Reaction

In chemical kinetics, a first-order reaction is one where the rate of reaction depends linearly on the concentration of one reactant. This means the rate at which the reaction occurs is directly proportional to the amount of the reactant present. Mathematically, it can be represented by the rate equation:
\[\text{rate} = k[\text{Reactant}]\]where

• \( k \) is the rate constant, a unique value for each reaction at a given temperature.
• \([\text{Reactant}]\) is the concentration of the reactant.

For first-order reactions, if you take the natural logarithm of the concentration of the reactant plotted against time, you'll see a straight line. This linearity confirms the first-order nature and allows us to determine the rate constant accurately.

Rate Constant

The rate constant, symbolized as \( k \), is crucial in determining the speed of a chemical reaction. It is a proportionality factor in the rate equation and remains constant for a given reaction at a particular temperature. However, the value of \( k \) is affected by temperature changes and is unique to each reaction.
In the context of the first-order reaction of dinitrogen pentoxide (\(N_2O_5\)), the rate constant can be extracted from the slope of the line when plotting \( \ln([N_2O_5]) \) against time. The negative slope corresponds to the rate constant for decomposition. In our reaction example, a slope of \(-0.0274\) was found, therefore, \( k = 0.0274 \ \text{s}^{-1} \). Understanding the rate constant allows chemists to predict how quickly reactants convert to products under specific conditions.

Dinitrogen Pentoxide Decomposition

Dinitrogen pentoxide, chemically represented as \(N_2O_5\), decomposes in a gaseous state into nitrogen dioxide \(NO_2\) and nitrate \(NO_3\). This decomposition is an excellent example of a first-order reaction in gas phase reactions. The process can be summarized by the equation:
\[N_2O_5 \rightarrow NO_2^* + NO_3^*\]Here, the breakdown of \(N_2O_5\) was specifically monitored at a temperature of 294 K and a pressure of 1 bar. These conditions influence the rate of decomposition and were carefully controlled in the experimental setup to explore the reaction kinetics. Understanding this decomposition process is crucial, as it is a simple yet fundamental reaction often used to study the principles of chemical kinetics.

Infrared Spectroscopy

Infrared (IR) spectroscopy is a powerful analytical technique used to measure molecular concentrations and identify chemical compounds based on their molecular vibrations. In this technique, infrared radiation is used to excite molecular vibrations, and the absorption of infrared light at specific frequencies corresponds to specific molecular bonds.
During the study of the decomposition of dinitrogen pentoxide \(N_2O_5\), infrared spectroscopy was employed to monitor the concentration changes over time. This non-invasive method allows for precise monitoring without altering the chemical environment. By analyzing the IR spectra, chemists obtain a real-time visualization of the decrease in concentration of \(N_2O_5\) as it decomposes. Thus, IR spectroscopy plays a crucial role in kinetic studies, providing valuable data to understand and confirm the reaction mechanism.