Question

Variation of the rate constant with temperature for the first-order reaction $$ 2 \mathrm{~N}_{2} \mathrm{O}_{5}(g) \longrightarrow 2 \mathrm{~N}_{2} \mathrm{O}_{4}(g)+\mathrm{O}_{2}(g) $$ is given in the following table. Determine graphically the activation energy for the reaction. $$ \begin{array}{lc} \mathrm{T}(\mathrm{K}) & \mathrm{k}\left(\mathrm{s}^{-1}\right) \\ \hline 273 & 7.87 \times 10^{3} \\ 298 & 3.46 \times 10^{5} \\ 318 & 4.98 \times 10^{6} \\ 338 & 4.87 \times 10^{7} \end{array} $$

Short answer

Determine the activation energy by plotting ln(k) against 1/T using the provided data in the table. The negative slope from the graph multiplied by the gas constant (-m*R) gives the activation energy Ea for the reaction.

Step-by-step solution

Convert Given Equation Form

The first step is to convert the Arrhenius equation into a form that allows us to use linear regression to calculate the activation energy. By using the natural logarithm on both sides, the equation can be transformed into: \(ln(k) = ln(A) - Ea/R * 1/T \). Now, this equation resembles the formula of a straight line \(y = mx + c\), where the slope \(m\) represents \(-Ea/R\). By plotting \(ln(k)\) against \(1/T\), the activation energy can be determined from the slope of the line.

Prepare Your Data for Plotting

Now, prepare your data. For each pair of T and k-values, calculate the 1/T and ln(k) value. For example, using the first temperature value (273K), 1/T would be approximately 0.00367 and using the first k value we obtain ln(k)=8.97.

Plot the Data

Next, plot ln(k) on the y-axis and 1/T on the x-axis. You should observe a negative linear trend, which complies with the transformed Arrhenius equation.

Determine the Slope of the Line and Calculate Activation Energy

Find the slope of the line which is the ratio of the vertical change (y-axis change) to the horizontal change (x-axis change). A crucial point to remember is that the slope should be negative as per the equation. Assuming that the slope is m, the activation energy can be calculated by rearranging the previous formula: \(Ea = -m * R\). R is the universal gas constant, which is approximately 8.31 J/(mol*K) in SI units.

Key concepts

Arrhenius Equation

The Arrhenius equation is a crucial concept in chemistry that describes how the rate constant of a reaction depends on temperature. It is expressed as:\[ k = A \times e^{-Ea/RT} \]Here, \(k\) represents the rate constant, \(A\) is the pre-exponential factor (also known as the frequency factor), \(Ea\) is the activation energy, \(R\) is the gas constant, and \(T\) is the temperature in Kelvin.

The equation can be transformed using natural logarithms to make it easier to analyze data:\[ \ln(k) = \ln(A) - \frac{Ea}{R} \times \frac{1}{T} \]This form is handy because it resembles a straight line equation \(y = mx + c\), where \(y\) corresponds to \( \ln(k)\), \(x\) corresponds to \(\frac{1}{T}\), and \(m\) is the slope equivalent to \(-\frac{Ea}{R}\).

Using this linear form allows for graphical determination of activation energy, which is vital for understanding the kinetics of a reaction. By plotting \(\ln(k)\) versus \(\frac{1}{T}\), the slope of the resulting line can provide insights into the activation energy of the reaction.

First-order Reaction

A first-order reaction is characterized by having a reaction rate that depends linearly on the concentration of one reactant. In mathematical terms, for a first-order reaction, the rate equation is:\[ rate = k [A] \]Where \([A]\) is the concentration of the reactant, and \(k\) is the rate constant.

For first-order reactions, the half-life (the time it takes for half of the reactant to be consumed) remains constant, and it does not depend on the initial concentration of the reactant. This property makes first-order reactions straightforward to analyze and predict over time.

In the context of our original problem involving the decomposition of \(\text{N}_2\text{O}_5\), this is a first-order reaction based on how the reaction rate is represented by the rate constant \(k\), which varies with temperature as indicated by the data provided.

Rate Constant

The rate constant \(k\) is an essential parameter in kinetic studies of chemical reactions. It provides information about how fast a reaction proceeds.

For a specific reaction, the rate constant is affected by factors such as the presence of a catalyst, the nature of the reactants, and most importantly, temperature. In the Arrhenius equation, \(k\) is expressed as a function of temperature, showing the exponential increase of the rate constant with an increase in temperature.

The unit of the rate constant varies depending on the order of the reaction. For first-order reactions, like the one in our exercise with \(\text{N}_2\text{O}_5\), the unit of \(k\) is \(\text{s}^{-1}\). Observing the table given, we see how \(k\) changes markedly with temperature, confirming the temperature dependence of reaction rates.

Temperature Dependence

Temperature dependence is a pivotal concept in reaction kinetics, significantly impacting the speed of chemical reactions. In general, as temperature increases, the rate of reaction and thus the rate constant \(k\) increases.

This is because higher temperatures provide kinetic energy to the reactant molecules, increasing their movement and likelihood of successful collisions that overcome the activation energy barrier. As a result, the frequency and energy of these collisions increase, accelerating the reaction.

• More successful collisions result from increased kinetic energy.
• The probability of surpassing the activation energy barrier increases.

In our sample problem, as seen from the provided data table, raising the temperature from 273 K to 338 K results in a substantial increase in the rate constant from \(7.87 \times 10^{-3}\) to \(4.87 \times 10^{-7}\), demonstrating the strong temperature dependence of the reaction rate for the decomposition of \(\text{N}_2\text{O}_5\).