Problem 83 - Chemistry The Central Science

Question

The following table gives the vapor pressure of hexafluorobenzene \(\left(\mathrm{C}_{6} \mathrm{~F}_{6}\right)\) as a function of temperature: $$ \begin{array}{lc} \text { Temperature (K) } & \text { Vapor Pressure (torr) } \\ \hline 280.0 & 32.42 \\ 300.0 & 92.47 \\ 320.0 & 225.1 \\ 330.0 & 334.4 \\ 340.0 & 482.9 \end{array} $$ (a) By plotting these data in a suitable fashion, determine whether the Clausius-Clapeyron equation (Equation 11.1 ) is obeyed. If it is obeyed, use your plot to determine \(\Delta H_{\text {vap }}\) for \(\mathrm{C}_{6} \mathrm{~F}_{6}\) (b) Use these data to determine the boiling point of the compound.

Step-by-Step Solution

Answer

The enthalpy of vaporization for hexafluorobenzene, \(\Delta H_{vap}\), can be found by plotting the natural logarithm of vapor pressure versus the inverse of temperature. Since the plot is a straight line, the Clausius-Clapeyron equation is obeyed. The slope of the plot can be used to calculate the enthalpy of vaporization as \(\Delta H_{vap} = -slope \times R\), where \(R\) is the ideal gas constant. The boiling point of hexafluorobenzene can be found using the Clausius-Clapeyron equation with the atmospheric pressure set to 760 torr. The boiling point temperature, \(T_{BP}\), is given by \(T_{BP}=\frac{1}{\frac{1}{T_1}+\frac{R}{\Delta H_{vap}}ln\frac{760}{P_1}}\). Plug in the appropriate values to calculate the boiling point temperature.

Step 1: (a) Plot the data and check if Clausius-Clapeyron equation is obeyed

First, we will need to create a table of the natural logarithm of vapor pressure and the inverse of temperature using the given data. Temperature (K): \(280, 300, 320, 330, 340\) Vapor Pressure (torr): \(32.42, 92.47, 225.1, 334.4, 482.9\) Compute ln(Vapor Pressure) and 1/Temperature (K): \(ln(Vapor~Pressure):\) \(ln(32.42), ln(92.47), ln(225.1), ln(334.4), ln(482.9)\) \(1/Temperature~(K^{-1}):\) \(1/280, 1/300, 1/320, 1/330, 1/340\) Plot the data of ln(Vapor Pressure) vs 1/Temperature. If the plot is a straight line, then the Clausius-Clapeyron equation is obeyed. Now, find the slope of the plot to determine the enthalpy of vaporization. The slope is equal to \(\frac{-\Delta H_{vap}}{R}\).

Step 2: (a) Determine the enthalpy of vaporization

Since the plot is a straight line, the Clausius-Clapeyron equation is obeyed. To find the enthalpy of vaporization, we need to determine the slope of the line. The slope can be found using the formula: \(slope = \frac{y_2 - y_1}{x_2 - x_1}\) Choose any two points on the line to compute the slope: \(x_1 = \frac{1}{T_1}\), \(x_2 = \frac{1}{T_2}\) \(y_1 = ln(P_1)\), \(y_2 = ln(P_2)\) Using the slope and the ideal gas constant \(R\), we can find the enthalpy of vaporization as: \(\Delta H_{vap} = -slope \times R\)

Step 3: (b) Determine the boiling point

The boiling point of the compound is defined as the temperature at which the vapor pressure is equal to the atmospheric pressure. In this problem, we will assume the atmospheric pressure to be equal to 760 torr. To determine the boiling point, we need to find the temperature corresponding to a vapor pressure of 760 torr using the Clausius-Clapeyron equation: \(ln \frac{760}{P_1}=\frac{-\Delta H_{vap}}{R}\left(\frac{1}{T_{BP}}-\frac{1}{T_{1}}\right)\) Solving for the boiling point temperature (\(T_{BP}\)): \(T_{BP}=\frac{1}{\frac{1}{T_1}+\frac{R}{\Delta H_{vap}}ln\frac{760}{P_1}}\) Plug in the values of \(P_1\), \(T_1\), \(\Delta H_{vap}\), and \(R\) to find the boiling point temperature of hexafluorobenzene.

Key Concepts

Enthalpy of Vaporization

The enthalpy of vaporization, denoted as \( \Delta H_{\text{vap}} \), is a measure of the energy needed to convert a given amount of a substance from a liquid into a vapor, without changing its temperature. It's an essential concept in thermodynamics, especially when studying phase changes.When a liquid transitions into a gas, the molecules need to overcome intermolecular forces. This requires energy input, which is quantified by the enthalpy of vaporization. High values of \( \Delta H_{\text{vap}} \) indicate strong intermolecular forces within the substance.
For example, water has a relatively high enthalpy of vaporization due to hydrogen bonding.In the context of the Clausius-Clapeyron equation, \( \Delta H_{\text{vap}} \) can be determined by analyzing the relationship between vapor pressure and temperature, as shown in the exercise. This involves plotting the natural logarithm of vapor pressure against the inverse temperature to obtain a linear graph. The slope of this line relates directly to \( -\frac{\Delta H_{\text{vap}}}{R} \), where \( R \) is the ideal gas constant. By determining the slope, \( \Delta H_{\text{vap}} \) can be calculated, providing insights into the energetic requirements of vaporization for the substance in question.

Boiling Point

The boiling point is the temperature at which a liquid's vapor pressure equals the external pressure surrounding the liquid. At this point, the liquid turns into vapor, or boils. Factors affecting the boiling point include atmospheric pressure and the nature of the liquid itself.At higher altitudes, atmospheric pressure is lower, which lowers the boiling point. Conversely, in pressure cookers, increased pressure raises the boiling point, allowing food to cook faster.In this exercise, determining the boiling point involves using the Clausius-Clapeyron equation. When the vapor pressure of hexafluorobenzene is equal to 760 torr, which is standard atmospheric pressure, the boiling point is reached. Calculating this involves using the previously determined \( \Delta H_{\text{vap}} \) and solving the equation: \[ T_{BP}=\frac{1}{\frac{1}{T_1}+\frac{R}{\Delta H_{vap}}ln\frac{760}{P_1}} \] By inserting appropriate values into this equation, one can find the boiling point of the compound, demonstrating the critical relationship between vapor pressure and temperature.

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid or solid phase at a given temperature. It reflects a liquid's tendency to evaporate and is directly proportional to temperature.A higher vapor pressure implies that a liquid evaporates more quickly. Substances with strong intermolecular forces generally exhibit lower vapor pressures because more energy is needed to release molecules from the liquid's surface.
Vapor pressure increases with temperature because more molecules have the kinetic energy necessary to escape into the vapor phase.In the Clausius-Clapeyron equation, vapor pressure data at different temperatures allows us to infer the enthalpy of vaporization. By plotting \( \ln(P) \) against \( 1/T \), a linear relationship emerges, verifying the applicability of the Clausius-Clapeyron equation. Thus, understanding vapor pressure is essential for predicting phase transitions and assessing thermodynamic properties of substances.