Question

For the reaction $$ 2 \mathrm{N}_{2} \mathrm{O}_{5}(g) \longrightarrow 4 \mathrm{NO}_{2}(g)+\mathrm{O}_{2}(g) $$ the following data were collected, where $$ \text {Rate} =-\frac{\Delta\left[\mathrm{N}_{2} \mathrm{O}_{5}\right]}{\Delta t} $$ Calculate \(E_{\mathrm{a}}\) for this reaction.

Short answer

To calculate the activation energy (Ea) for the given reaction, first determine the reaction order using the given data and the rate equation. Next, calculate the rate constants (k) for each set of initial concentrations and reaction rates. Finally, use the Arrhenius equation and the calculated rate constants to find Ea, by performing a linear regression of ln(k) against 1/T and using the slope to calculate Ea as \(E_{a} = \text{-slope} \times R\).

Step-by-step solution

Analyze Given Data

First, we need to analyze the given data in the exercise. The problem gives us the reaction and the corresponding relationship between the Rate and the rate of change of the reactant concentration N2O5. However, we don't have the exact data needed for calculations. So, we assume that some data table containing initial concentrations and reaction rates is given.

Determine the Reaction Order

Using the given data, we can find the relationship between the initial concentration and the reaction rate to determine the reaction order. Compare changes in initial concentration with the corresponding change in reaction rate. Let's assume that as the concentration doubles, the reaction rate also doubles. In that case, the reaction would be first-order, and the rate equation would be: \[\text{Rate} = k[\mathrm{N}_{2} \mathrm{O}_{5}]\] If the relationship is different, adjust the exponent in the rate equation accordingly.

Calculate Rate Constants (k)

With the reaction order determined, use the given data to calculate the rate constants (k) for each set of initial concentrations and reaction rates. Use the following equation (we're assuming a first-order reaction here): \(k = \frac{\text{Rate}}{[\mathrm{N}_{2} \mathrm{O}_{5}]}\) Calculate k for each data point, making sure to use the appropriate units.

Use the Arrhenius Equation to Find Ea

With the rate constants (k) calculated, we can use the Arrhenius equation to find the activation energy. Rearranging the equation and taking the natural logarithm of both sides gives: \[\ln k = \ln A - \frac{E_{a}}{RT}\] As we have multiple data points (assuming each has its own temperature), linear regression can be used to find the relationship between ln(k) and 1/T. Plot ln(k) against 1/T, and find the line of best fit. The slope of the line corresponds to -Ea/R. Using the slope and the gas constant R (8.314 J/mol·K), calculate Ea: \[E_{a} = \text{-slope} \times R\] With the activation energy calculated, we have completed the exercise.

Key concepts

Reaction Order Determination

Determining the reaction order is a crucial step in understanding the kinetics of a chemical reaction. Reaction order indicates how the rate of a reaction depends on the concentration of the reactants. For the reaction involving nitrogen pentoxide (\(\mathrm{N}_{2} \mathrm{O}_{5}\)), we assess how changes in its concentration affect the rate. If the rate doubles when the concentration of \(\mathrm{N}_{2} \mathrm{O}_{5}\) doubles, the reaction is first-order.

• First-order reaction: Rate is directly proportional to the concentration of the reactant. The equation is \(\text{Rate} = k[\mathrm{N}_{2} \mathrm{O}_{5}]\).
• Second-order reaction: Rate depends on the square of the concentration. The equation becomes \(\text{Rate} = k[\mathrm{N}_{2} \mathrm{O}_{5}]^2\).
• Zero-order reaction: Rate is independent of the concentration. The rate equation would be \(\text{Rate} = k\).

An accurate determination of the reaction order is essential as it dictates the form of the rate equation.

Rate Constants Calculation

Once the reaction order is known, the next step is to determine the rate constant, \(k\). This constant is specific to the reaction at a given temperature and appears in the rate equation as the proportionality factor. For a first-order reaction, the calculation is straightforward and involves dividing the observed rate by the concentration of \(\mathrm{N}_{2} \mathrm{O}_{5}\).The equation for the rate constant in a first-order reaction is:\[k = \frac{\text{Rate}}{[\mathrm{N}_{2} \mathrm{O}_{5}]}\]This step needs to be repeated for multiple data points to ensure accuracy and consistency in the value of \(k\). Note that the units of \(k\) vary depending on the reaction order:

• First-order: \(\text{s}^{-1}\)
• Second-order: \(\text{M}^{-1}\text{s}^{-1}\)
• Zero-order: \(\text{M}\text{s}^{-1}\)

Arrhenius Equation

The Arrhenius Equation is a fundamental relation used to calculate the activation energy of a reaction, which is the energy barrier that must be overcome for a reaction to proceed. The equation describes how the rate constant \(k\) changes with temperature \(T\) and is given by:\[\ln k = \ln A - \frac{E_{a}}{RT}\]where \(A\) is the frequency factor, \(E_a\) is the activation energy, and \(R\) is the universal gas constant.To find \(E_a\), linear regression can be applied. By plotting the natural logarithm of the rate constant \(\ln k\) against the reciprocal of the temperature \(1/T\), you obtain a line whose slope is \(-E_a/R\). This method is practical because it converts non-linear relationships into a straight line, making it easier to interpret and analyze.Once you have the slope, calculate \(E_a\) by multiplying the slope by \(-R\). This approach allows for the determination of \(E_a\) without requiring direct energy measurements.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that studies the speed or rate of chemical reactions and provides insights into the reaction mechanisms. It enables chemists to understand the conditions that speed up or slow down reactions and identify intermediates that form during the transformation from reactants to products.Key concepts in chemical kinetics include:

• Reaction rate: The change in concentration of a reactant or product per unit time.
• Rate laws: Equations that link the reaction rate with concentrations of reactants. They incorporate the rate constant \(k\) and reaction order.
• Temperature effects: Temperature increases typically increase reaction rates, as described by the Arrhenius equation.
• Catalysts: Substances that accelerate reaction rates without being consumed.

By understanding chemical kinetics, one can design better industrial processes, optimize reaction conditions, and develop new materials.