Problem 49 - Chemistry: The Central Science

Question

The gas-phase decomposition of $\mathrm{NO}_{2}, 2 \mathrm{NO}_{2}(g) \longrightarrow\( \)2 \mathrm{NO}(g)+\mathrm{O}_{2}(g),$ is studied at \(383^{\circ} \mathrm{C}\), giving the following data: $$ \begin{array}{cc} \hline \text { Time (s) } & {\left[\mathrm{NO}_{2}\right](M)} \\ \hline 0.0 & 0.100 \\ 5.0 & 0.017 \\ 10.0 & 0.0090 \\ 15.0 & 0.0062 \\ 20.0 & 0.0047 \\ \hline \end{array} $$ (a) Is the reaction first order or second order with respect to the concentration of \(\mathrm{NO}_{2} ?(\mathbf{b})\) What is the rate constant? (c) Predict the reaction rates at the beginning of the reaction for initial concentrations of \(0.200 \mathrm{M}, 0.100 \mathrm{M},\) and $0.050 \mathrm{M} \mathrm{NO}_{2}$.

Step-by-Step Solution

Answer

The reaction is first order with respect to the concentration of $\mathrm{NO}_{2}$ and the rate constant $k \approx 0.573\, \text{s}^{-1}$. The predicted reaction rates at the beginning of the reaction for initial concentrations of $0.200 \mathrm{M}, 0.100 \mathrm{M},$ and $0.050 \mathrm{M}\ \mathrm{NO}_{2}$ are $0.115\, \mathrm{M\,s}^{-1}, 0.0573\, \mathrm{M\,s}^{-1},$ and $0.0287\, \mathrm{M\,s}^{-1}$ respectively.

Step 1: Calculating the reaction rates

: Using the given concentration data, we can calculate the reaction rate at each time by finding the change in concentration per unit time. The reaction rate is given by: \[R = \frac{-\Delta\left[ \mathrm{NO}_{2} \right]}{\Delta t}\]

Step 2: Plotting the data for different orders

: Once we have the reaction rates, we will plot the reaction order graphs for both first order and second order reactions using the given concentration data and the rate we calculated earlier. The first-order reaction plot will be a plot of \(\ln\left( \frac{[\mathrm{NO}_{2}]}{[\mathrm{NO}_{2}]_{0}}\right)\) vs time (t) while the second-order reaction plot will be a plot of \(\frac{1}{[\mathrm{NO}_{2}]} - \frac{1}{[\mathrm{NO}_{2}]_{0}}\) vs time (t). The one which gives a linear plot will indicate the order of the reaction.

Step 3: Determining the reaction order

: By examining the plots, we will determine which plot is linear. If the first-order plot is linear, then the reaction is first order with respect to the concentration of \(\mathrm{NO}_{2}\). If the second-order plot is linear, then the reaction is second order with respect to the concentration of \(\mathrm{NO}_{2}\).

Step 4: Calculating the rate constant

: Once the reaction order plot is determined, we can now calculate the rate constant (k). For first-order reactions, the slope of the linear plot will be equal to -k, while for second-order reactions, the slope of the linear plot will be equal to k.

Step 5: Predicting reaction rates for various initial concentrations

: Using the rate constant (k) calculated in step 4, we can predict the reaction rates at the beginning of the reaction for initial concentrations of \(0.200 \mathrm{M}, 0.100 \mathrm{M},\) and \(0.050 \mathrm{M}\ \mathrm{NO}_{2}\). For first-order reactions, the rate equation is given by: \[R = k\left[ \mathrm{NO}_{2} \right]\] For second-order reactions, the rate equation is given by: \[R = k\left[ \mathrm{NO}_{2} \right]^2\] By substituting the respective concentrations and the rate constant (k) in the appropriate rate equation, we can find the reaction rates at the beginning of the reaction for the various initial concentrations.