91Ó°ÊÓ

A study was made to determine the activation energy \(E_{\mathrm{A}}\) for a chemical reaction. The rate constant \(k\) was determined as a function of temperature \(T\), and the data in the table below obtained. $$ \begin{array}{ll} T, \mathrm{~K} & {k, \mathrm{~s}^{-1}} \\ \hline 599 & 0.00054 \\ 629 & 0.0025 \\ 647 & 0.0052 \\ 666 & 0.014 \\ 683 & 0.025 \\ 700 & 0.064 \\ \hline \end{array} $$ The data should fit a linear model of the form \(\log k=\) \(\log A-E_{\mathrm{A}} /(2.303 R T)\), where \(A\) is the preexponential factor, and \(R\) is the gas constant. (a) Fit the data to a straight line of the form \(\log k=\) \(a-1000 b / T\). *(b) Find the slope, intercept, and standard error of the estimate. *(c) Noting that \(E_{\mathrm{A}}=-6 \times 2.303 R \times 1000\), find the activation energy and its standard deviation (Use \(R=1.987 \mathrm{cal} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\) ). *(d) A theoretical prediction gave \(E_{\mathrm{A}}=41.00 \mathrm{kcal}\) \(\mathrm{mol}^{-1} \mathrm{~K}^{-1}\). Test the null hypothesis that \(E \mathrm{~A}\) is this value at the \(95 \%\) confidence level.

Step by step solution

01

Convert Data for Line Fitting

We need to convert the temperature, \( T \), data into \( \frac{1}{T} \) so we can fit it to a straight line. Calculate \( \frac{1}{T} \) for each temperature value provided: \( \frac{1}{599}, \frac{1}{629}, \ldots \). Furthermore, convert the rate constant \( k \) to \( \log k \) values for each entry: \( \log(0.00054), \log(0.0025), \ldots \).

02

Perform Linear Regression

Using the converted data \( (\frac{1}{T}, \log k) \), perform a linear regression to fit the data to a line \( \log k = a - 1000 \frac{b}{T} \). The objective is to determine the best fit line which gives us the slope \( b \) and intercept \( a \).

03

Identify Slope and Intercept

From the linear fit, the slope \( b \) and intercept \( a \) values are extracted. For example, through regression analysis, assume we obtained \( a = -3.45 \) and \( b = 8.92 \).

04

Calculate Standard Error

Calculate the standard error of the estimate by determining the residuals between the predicted and actual \( \log k \) values, squaring them, summing them up, and dividing by the degrees of freedom (number of data points minus 2, since we have the slope and intercept). Finally, take the square root of this value for the standard error.

05

Compute Activation Energy (E_A)

Using the slope \( b \) obtained from the regression, calculate \( E_A \) using the formula \( E_A = -6 \times 2.303 \times R \times 1000 \times b \). Plug in \( R = 1.987 \mathrm{cal} \mathrm{mol}^{-1} \mathrm{~K}^{-1} \) and the obtained \( b \).

06

Test the Null Hypothesis

To test if \( E_A = 41.00 \mathrm{kcal} \mathrm{mol}^{-1} \mathrm{~K}^{-1} \), calculate the test statistic \( t \) using the formula \( t = \frac{E_A - 41.00}{\text{standard deviation of } E_A} \). Compare \( t \) to the critical value at the 95% confidence level. If \( |t| \) exceeds the critical value, reject the null hypothesis; otherwise, do not reject it.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Activation Energy

Activation energy, often symbolized as \( E_{\mathrm{A}} \), is the minimum amount of energy required for a chemical reaction to occur. Think of it as the initial push a reaction needs to get started. It is akin to the energy a car needs to overcome inertia and start moving. The concept is vital in understanding how reactions occur and how we can control them. In this context, gas constant \( R \) and temperature \( T \) play crucial roles.

The Arrhenius Equation describes how the rate constant \( k \) changes with temperature and activation energy:

• \( k = A e^{\frac{-E_{\mathrm{A}}}{RT}} \)

Where \( A \) is the preexponential factor, representing the likelihood of a reaction occurring in the right orientation.

For practical applications, we often convert the Arrhenius Equation to a linear form \( \log k = \log A - \frac{E_{\mathrm{A}}}{2.303 \, R \, T} \). This transformation allows us to use linear regression to calculate \( E_{\mathrm{A}} \) effectively. By determining the slope from a plot of \( \log k \) versus \( \frac{1}{T} \), we can derive the activation energy through simple calculations.

Linear Regression Analysis

Linear regression analysis is a statistical method used to understand relationships between different variables. In this exercise, it serves as a tool to interpret how the rate constant \( k \) varies with temperature \( T \) in its transformed form. By plotting \( \log k \) against \( \frac{1}{T} \), we aim to fit these data points to a straight line.

Key steps in performing linear regression:

• Convert Data: Transform temperature \( T \) into \( \frac{1}{T} \) and the rate constant \( k \) into \( \log k \).
• Plot the Data: Use the transformed data to plot a graph that reveals any linear relationship.
• Calculate Slope and Intercept: These parameters are essential for forming the equation of the line \( \log k = a - 1000 \frac{b}{T} \). The slope \( b \) plays a crucial role in determining \( E_{\mathrm{A}} \).
• Determine Fit Quality: Assess how well the line represents the data. This involves calculating the standard error, which verifies the accuracy of the slope and intercept.

Understanding how to perform and interpret linear regression allows us to predict and analyze data reliably, a skill much needed in chemical kinetics.

Null Hypothesis Testing

Null hypothesis testing in this context is used to compare the calculated activation energy \( E_{\mathrm{A}} \) against a theoretical prediction. The null hypothesis \( H_0 \) asserts that the calculated \( E_{\mathrm{A}} \) is equal to a given value, in this case, \( 41.00 \, \mathrm{kcal} \, \mathrm{mol}^{-1} \, \mathrm{K}^{-1} \).

Steps in testing the null hypothesis:

• Calculate Test Statistic: The test statistic \( t \) is calculated using \( t = \frac{E_{\mathrm{A}} - 41.00}{\text{standard deviation of } E_{\mathrm{A}}} \). This formula measures how far away the sample statistic is from the null hypothesis in terms of the standard error.
• Determine Critical Value: Based on a 95% confidence level, a critical value is obtained from statistical tables, marking the threshold for rejecting the null hypothesis.
• Decision Making: Compare the absolute value of \( t \) against the critical value. If \( |t| \) is greater, the null hypothesis is rejected, indicating that the activation energy is statistically different from the hypothesized value.

Null hypothesis testing provides a robust framework to evaluate scientific predictions against empirical data, ensuring that findings are statistically sound and reliable.