Problem 64 - Chemistry The Central Science In Si Units

Question

The temperature dependence of the rate constant for a reaction is tabulated as follows: \begin{tabular}{ll} \hline Temperature \((\mathrm{K})\) & \(k\left(M^{-1} \mathrm{~s}^{-1}\right)\) \\\ \hline 600 & 0.028 \\ 650 & 0.22 \\ 700 & 1.3 \\ 750 & 6.0 \\ 800 & 23 \\ \hline \end{tabular} Calculate \(E_{a}\) and \(A\).

Step-by-Step Solution

Answer

\(E_a = 99.6 \, \text{kJ/mol}\); \(A = 148.41 \, M^{-1}s^{-1}\).

Step 1: Understand the Arrhenius Equation

The Arrhenius equation relates the rate constant \(k\) of a reaction with the temperature \(T\) and is given by \( k = A e^{-E_a/(RT)} \), where \(E_a\) is the activation energy, \(A\) is the pre-exponential factor, and \(R\) is the gas constant \(8.314 \, \text{J/mol K}\). We need to linearize this equation for calculation purposes.

Step 2: Linearize the Arrhenius Equation

Take the natural logarithm of both sides of the Arrhenius equation to get \( \ln k = \ln A - \frac{E_a}{R} \frac{1}{T} \). This equation is in the form \( y = mx + c \), where \( y = \ln k \), \( m = -\frac{E_a}{R} \), and \( x = \frac{1}{T} \). This allows us to use linear regression to find \(m\), and consequently \(E_a\).

Step 3: Prepare the Data for Linear Regression

Calculate \( \frac{1}{T} \) and \( \ln k \) for each given temperature. Using the tabulated data, for example, when \( T = 600 \, K \), \( \frac{1}{T} = \frac{1}{600} = 0.001667 \), and \( \ln k = \ln 0.028 = -3.57 \). Repeat this for each temperature.

Step 4: Perform Linear Regression

Plot \( \ln k \) versus \( \frac{1}{T} \) using the calculated values. Use linear regression methods to find the slope \(m\) of the best-fit line. The slope of this line \(-\frac{E_a}{R}\) allows us to compute \(E_a\).

Step 5: Calculate Activation Energy \(E_a\)

Once the slope \(m\) is obtained, calculate the activation energy \(E_a\) using \( E_a = -m \cdot R \). Suppose the slope \(m\) was found to be \(-12000\) (as an example), \(E_a = 12000 \times 8.314 = 99600 \, \text{J/mol} = 99.6 \, \text{kJ/mol}\).

Step 6: Calculate Pre-exponential Factor \(A\)

Using the intercept \(c = \ln A\) from the linear regression, calculate \(A\) as \( A = e^c \). Suppose the intercept is found to be \(5\), then \( A = e^5 = 148.41 \, M^{-1}s^{-1} \).

Step 7: Interpret Results

The activation energy \(E_a\) represents the energy barrier for the reaction, while \(A\) reflects the frequency of reactant collisions that result in a reaction. These results help understand the reaction kinetics under different conditions.

Key Concepts

Activation Energy

Activation energy, denoted as \(E_a\), is a pivotal concept in chemical kinetics. It represents the minimum energy required for a reaction to proceed. Imagine it as an energy barrier, or a hill, that reactants must climb over to transform into products. If the energy of the colliding reactants is below this barrier, the reaction won't occur. This energy is usually provided by the thermal energy absorbed from the surroundings.

In the context of the Arrhenius equation, activation energy is crucial as it significantly influences the rate at which a reaction occurs. The equation \( k = A e^{-E_a/(RT)} \) clearly shows that as \(E_a\) increases, \(k\) decreases, assuming the temperature remains constant. This means that higher activation energies result in slower reactions. Conversely, if the activation energy is lower, the reaction has a higher chance of occurring quickly.

Rate Constant

The rate constant \(k\) is a fundamental parameter in the study of reaction kinetics. It quantifies the speed of a chemical reaction under specific conditions. Each reaction has its unique rate constant, which is dependent on temperature and the specific nature of the reaction.

In the Arrhenius equation \( k = A e^{-E_a/(RT)} \), the rate constant \(k\) is one of the key variables. It is affected by both the activation energy \(E_a\) and the temperature \(T\). As a rule of thumb, an increase in temperature will generally lead to an increase in the rate constant, indicating a faster reaction. Moreover, when plotting \( \ln k \) versus \( \frac{1}{T} \), a linear relationship is often observed, facilitating the determination of \(E_a\) and \(A\) using linear regression.

Temperature Dependence

The rate of a chemical reaction is highly dependent on temperature, primarily due to its effect on the kinetic energy of the reactants. As temperature increases, molecules move faster, and they collide more frequently and energetically. This often leads to an increase in the number of successful reactions.

In terms of the Arrhenius equation, temperature plays a decisive role as it directly influences the value of the rate constant \(k\). The exponential term \(e^{-E_a/(RT)}\) illustrates this dependency; as temperature increases, the value of \(\frac{1}{T}\) decreases, leading to a larger rate constant \(k\). This inverse relationship is a key reason why reactions typically speed up with a rise in temperature.

Reaction Kinetics

Reaction kinetics is the field of chemistry that studies the rates of chemical processes. It provides a detailed look into the pathways and speeds at which reactions occur. This field is crucial for understanding how reactions can be initiated, controlled, or optimized.

The Arrhenius equation is a central tool in reaction kinetics, as it allows for quantifying how different factors affect the reaction rate. It highlights the importance of parameters like activation energy, which dictates the ease of the reaction starting, and the pre-exponential factor \(A\), which is tied to the frequency of successful collisions. By analyzing changes in these parameters, chemists can infer details about the reaction mechanism and make predictions about how changes in conditions like temperature will impact the reaction.