Question

The temperature dependence of the rate constant for the reaction is tabulated as follows: $$\overline{\begin{array}{ll}\text{ Temperature (K) }& k\left({\mathbf{M}}^{-1}{\mathbf{s}}^{-1}\right)\\ 600& 0.028\\ 650& 0.22\\ 700& 1.3\\ 750& 6.0\\ 800& 23\end{array}}$$ Calculate $E_g$ and $A$.

Short answer

The short version of the answer is as follows: First, rewrite the Arrhenius equation in a linear form: $ln(k)=-\frac{E_g}{R}\frac{1}{T}+ln(A)$. Then, use the given temperature-rate constant pairs to find the corresponding sets of points $\frac{1}{T}$ and $ln(k)$. Perform linear regression to find the slope ($m$) and the intercept ($b$) of the dataset. Finally, determine the activation energy ($E_g$) and the pre-exponential factor ($A$) using the relationships: $E_g=-m\times R$ and $A=e^b$.

Step-by-step solution

Re-write the Arrhenius equation in terms of natural logarithm

First, we will take the natural logarithm of both sides of the Arrhenius Equation to achieve a linear equation: $$ln(k)=ln(A)-\frac{E_g}{RT}$$ Now, rewrite the equation in the form of a straight line ($y=mx+b$): $$ln(k)=-\frac{E_g}{R}\frac{1}{T}+ln(A)$$ Here, $y=ln(k)$, $m=-\frac{E_g}{R}$, $x=\frac{1}{T}$, and $b=ln(A)$.

Utilize the given data points to linearize the dataset

Using the given data points, we can develop the equation in a linear form: $$\frac{1}{T}=[\frac{1}{600},\frac{1}{650},\frac{1}{700},\frac{1}{750},\frac{1}{800}]$$ $$ln(k)=ln[0.028,0.22,1.3,6.0,23]$$ Now, we have two corresponding sets of points, $\frac{1}{T}$ and $ln(k)$, which can be used to determine the slope and intercept of the linearized equation.

Calculate the slope and intercept of the straight line

For this step, we will use linear regression to calculate the slope ($m$) and the intercept ($b$), based on the data points: $\frac{1}{T}$ and $ln(k)$: Plugging these values into any linear regression tool or using the formulas of linear regression: $m=-\frac{E_g}{R}$ and $b=ln(A)$

Determine $E_g$ and $A$ from the slope and intercept

Now that we have the slope ($m$) and the intercept ($b$), we can determine the activation energy ($E_g$) and the pre-exponential factor ($A$): $$E_g=-m\times R$$ $$A=e^b$$ By calculating the $E_g$ and $A$ values, we will have the desired parameters for the temperature dependence of the rate constant for the given reaction.

Key concepts

Arrhenius Equation

The Arrhenius Equation is a fundamental formula in chemistry that helps us understand how reaction rates change with temperature. This equation is represented as:
$$k=A{e}^{-\frac{E_a}{RT}}$$ where:

• $k$ is the rate constant, which varies with temperature.
• $A$ is the pre-exponential factor, also known as the frequency factor.
• $E_a$ is the activation energy, the minimum energy required for the reaction to occur.
• $R$ is the universal gas constant $(8.314{\text{ J mol}}^{-1}{\text{K}}^{-1})$.
• $T$ is the absolute temperature measured in Kelvin.

By applying the natural logarithm to both sides, we transform it to a linear form:
$$\ln (k)=\ln (A)-\frac{E_a}{RT}$$ This transformation allows us to view the equation as a straight line $(y=mx+b)$, which is useful for determining the activation energy and pre-exponential factor when experimental data is available.

Temperature Dependence

Temperature has a significant effect on the rate of chemical reactions. Generally, increasing the temperature speeds up a reaction, while decreasing it slows the reaction down. This behavior is quantified through the Arrhenius Equation, demonstrating how reaction rates depend on temperature.
The key idea is that as temperature increases, more molecules have the energy needed to surpass the activation energy barrier, leading to a higher reaction rate constant $(k)$.
For example, the relationship between temperature and the rate constant can be observed by plotting $\ln (k)$ against $1/T$. The plot should ideally form a straight line, with the slope related to the activation energy $E_a$.

• A steep slope indicates a high sensitivity of the rate constant to temperature changes, meaning the activation energy is large.
• A gentle slope indicates lower sensitivity, meaning the activation energy is small.

Thus, understanding the temperature dependence of reaction rates allows chemists to predict and control the speed of chemical processes under various conditions.

Activation Energy

Activation energy $(E_a)$ is a crucial concept in chemical kinetics, representing the minimum energy required for reactants to convert into products during a reaction. It serves as an energy barrier that must be overcome for a reaction to proceed.
In the context of the Arrhenius Equation, the activation energy helps determine the sensitivity of the reaction rate to changes in temperature. Higher activation energy means that the rate of reaction will increase substantially with a small increase in temperature, while lower $E_a$ implies less sensitivity.
Activation energy can be calculated from the slope of a plot of $\ln (k)$ against $1/T$ as part of linear regression. By knowing the activation energy:

• We can anticipate how the rate of reaction would change with variations in temperature.
• We can deduce the kinetic behavior of the reaction and understand how different conditions affect reaction speed.

Understanding activation energy is essential for designing and optimizing chemical reactions, particularly in industrial and laboratory settings. It allows chemists to predict the impacts of temperature changes, ensuring efficient and controlled chemical processes.