91Ó°ÊÓ

The vapor pressure of ethylene glycol at several temperatures is given below:$$\begin{array}{|l|r|r|r|r|r|r|}\hline T\left(^{\circ} \mathrm{C}\right) & 79.7 & 105.8 & 120.0 & 141.8 & 178.5 & 197.3 \\\\\hline p^{*}(\mathrm{mm} \mathrm{Hg}) & 5.0 & 20.0 & 40.0 & 100.0 & 400.0 & 760.0 \\\\\hline\end{array}$$ a semilog plot e vapor-pressure data and determine a linear expression for \(\ln p^{*}\) function of \(1 / T(\mathrm{K}) .\) Use the results to estimate the heat of vaporization \((\mathrm{kJ} / \mathrm{mol})\) of ethylene glycol, and then use that value in the Clausius-Clapeyron equation to estimate the vapor pressures at each of the temperatures given in the table.(b) Repeat Part (a) using the Slope and Intercept functions of APEx to obtain the expression for \(\ln p^{*}\) vs. \(1 / T(\mathrm{K})\).(c) Use the results from Part (b) to estimate vapor pressures of ethylene glycol at \(50^{\circ} \mathrm{C}, 80^{\circ} \mathrm{C},\) and \(110^{\circ} \mathrm{C} .\) Also estimate the boiling point of this substance at system pressures of \(760 \mathrm{mm} \mathrm{Hg}\) and 2000 mm Hg. Compare all five results with those obtained directly using APEx functions. In which of the estimates at the given temperatures and pressures would you have the least confidence? Explain your reasoning.

Step by step solution

01

Create a semilog plot

Plot the given data in the form of a semilog graph with \( \ln p^{*} \) as the vertical axis and \( 1/T(K) \) as the horizontal axis. Use temperature data in Kelvin.

02

Determine the linear expression

Using the least squares method, obtain the equation of the line which best fits the data points. This equation will be of the form \( \ln p^{*} = -\Delta H_{vap}/R * 1/T + C \) where R is the gas constant (R=8.314 J/(mol*K)), \( \Delta H_{vap} \) is the heat of vaporization and C is constant.

03

Estimate the heat of vaporization

From the slope of the linear equation obtained in step 2, estimate the heat of vaporization \(\Delta H_{vap} \) using the formula \( \Delta H_{vap} = -Slope * R \).

04

Estimate the vapor pressure

Substitute the heat of vaporization into the Clausius-Clapeyron equation to estimate the vapor pressures at each of the given temperatures.

05

Repeat steps 1 to 4 with APEX functions

Use the APEX functions to obtain the linear expression and estimate the vapor pressures. Compare these results with the calculated results.

06

Calculate the vapor pressures and boiling point

Use the obtained linear equation to estimate the vapor pressures of ethylene glycol at temperatures 50°C, 80°C, and 110°C. Also, estimate the boiling point of the substance at system pressures of 760 mm Hg and 2000 mm Hg.

07

Analyze the results

Compare all the results obtained and identify which estimates at the given temperatures and pressures provide the least confidence. Provide reasoning based on the data analysed.

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Key Concepts

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

Clausius-Clapeyron Equation

The Clausius-Clapeyron equation is a fundamental principle in physical chemistry that relates the change in pressure with the change in temperature of a substance as it moves between two phases, such as liquid and gas. This relationship is particularly useful for estimating the vapor pressure of a liquid at various temperatures. The equation is expressed as:
\[\ln(p^{*}) = -\frac{\Delta H_{vap}}{R} \cdot \frac{1}{T} + C\] where \( p^{*} \) is the vapor pressure, \( T \) is the temperature in Kelvin, \( R \) is the universal gas constant, \( \Delta H_{vap} \) is the heat of vaporization, and \( C \) is a constant that can be determined through experimental data.
To apply this equation in practice, one must obtain a linear expression of \( \ln p^{*} \) versus \( 1/T \), which is done by plotting experimental vapor pressure data against the inverse of temperature. The slope of this line provides the heat of vaporization, and the intercept gives the constant \( C \). This allows for the calculation of vapor pressures at other temperatures by substituting these values back into the Clausius-Clapeyron equation.

Heat of Vaporization

The heat of vaporization, \( \Delta H_{vap} \), is a critical component in understanding phase transitions, specific to the process of a substance transitioning from liquid to gas. It represents the amount of energy required to vaporize one mole of a liquid at its boiling point under standard atmospheric pressure. This value is essential for the Clausius-Clapeyron equation, as it directly influences the curve of the vapor pressure versus temperature plot.

Estimating the heat of vaporization typically involves analyzing a graph of vapor pressure against temperature. Once the linear representation of \( \ln p^{*} \) versus \( 1/T \) is determined, the negative of the slope multiplied by the gas constant \( R \) yields the heat of vaporization:
\[ \Delta H_{vap} = -(\text{Slope}) \times R \] In the context of an educational exercise, students are often tasked with finding this slope through linear regression analysis on their semilog plot. This calculation allows them to understand the thermodynamic principles governing phase changes while providing a practical approach to real-world chemical problems.

Semilog Plot

A semilog plot is a type of graph used to visualize data where one axis (usually the y-axis) is on a logarithmic scale and the other axis (x-axis) is on a linear scale. This kind of plot is particularly useful when the data covers a large range of values, as it can show the smaller and larger values on the same graph without losing detail.

For estimating vapor pressures, a semilog plot helps to linearize the exponential relationship expressed by the Clausius-Clapeyron equation. In the plot, the natural logarithm of the vapor pressure (\( \ln p^{*} \)) is graphed against the reciprocal of the temperature in Kelvin (\( 1/T \)). The result is a straight line when the assumptions of the Clausius-Clapeyron equation hold true, which greatly simplifies the process of determining the heat of vaporization and subsequent calculations of vapor pressures at various temperatures.

Linear Regression Analysis

Linear regression analysis is a statistical method used to model the relationship between a dependent variable and one or more independent variables. In the context of vapor pressure estimation, linear regression is applied to determine the best-fit line through a set of data points on the semilog plot of natural log vapor pressure versus reciprocal temperature.

The equation of this best-fit line is of the form:
\[ y = mx + b \] where \( y \) corresponds to \( \ln p^{*} \), \( x \) is \( 1/T \), \( m \) is the slope of the line, and \( b \) is the y-intercept. In the Clausius-Clapeyron equation, the slope (\( m \)) is related to the negative heat of vaporization (\( -\Delta H_{vap} \)) and the intercept (\( b \)) corresponds to the constant \( C \).

Using linear regression, one can find the values of \( m \) and \( b \) that minimize the sum of the squared differences between the observed data points and the values predicted by the model. This ensures that the derived heat of vaporization is as accurate as possible with respect to the given data, resulting in reliable vapor pressure estimations.