Question

$$ \begin{aligned} &\begin{aligned} &\text { The following table gives the vapor pressure of hexaflu- } \\ &\text { orobenzene }\left(\mathrm{C}_{6} \mathrm{~F}_{6}\right) \text { as a function of temperature: } \end{aligned}\\\ &\begin{aligned} &4 \end{aligned}\\\ &\begin{array}{lc} \hline \text { Temperature (K) } & \text { Vapor Pressure (torr) } \\ \hline 280.0 & 32.42 \\ 300.0 & 9247 \\ 320.0 & 225.1 \\ 330.0 & 334.4 \\ 340.0 & 482.9 \\ \hline \end{array} \end{aligned} $$ (a) By plotting these data in a suitable fashion, determine whether the Clausius-Clapeyron equation is obeyed. If it is obeyed, use your plot to determine \(\Delta H_{\text {vap }}\) for \(C_{6} F_{6}\) (b) Use these data to determine the boiling point of the compound.

Short answer

The exercise asks to determine whether the Clausius-Clapeyron equation is obeyed and find the heat of vaporization and boiling point of hexafluorobenzene. By plotting the natural logarithm of vapor pressure against the inverse of temperature, we can check for linearity. If the graph is linear, we can calculate the slope and use it to find the heat of vaporization through the formula: \(\Delta H_{vap} = -slope \times R\). Finally, to find the boiling point, rearrange the Clausius-Clapeyron equation and use one of the original data points to solve for the constant C and then the temperature at which the vapor pressure equals 760 torr (1 atm).

Step-by-step solution

Plotting the data

First, transform the given data to more appropriate values by taking the natural logarithm of the vapor pressure values and the inverse of the temperature values: \(\) \begin{array}{c|c|c} \hline \text{Temperature (K)} & \text{1/T} \left(\frac{1}{K}\right) & \text{ln(Vapor Pressure)} \\ \hline 280.0 & \frac{1}{280} & \ln(32.42) \\ 300.0 & \frac{1}{300} & \ln(92.47) \\ 320.0 & \frac{1}{320} & \ln(225.1) \\ 330.0 & \frac{1}{330} & \ln(334.4) \\ 340.0 & \frac{1}{340} & \ln(482.9) \\ \hline \end{array} \(\) After calculating the values, plot them on graph paper or use graphing software such as Excel or Desmos.

Determining linearity

Inspect the graph and determine if the data points form a straight line or a close approximation of one. If the graph is indeed linear or close to linear, it indicates that the Clausius-Clapeyron equation is obeyed.

Calculating ΔHvap

To find the heat of vaporization, calculate the slope of the line using the linear regression equation from the graph or using any slope calculation method from the data points. The slope can be found using the relation: $$ slope = -\frac{\Delta H_{vap}}{R} $$ To find the ΔHvap, we rearrange the equation as follows: $$ \Delta H_{vap} = -slope \times R $$ Use your calculated slope and R value (8.314 J/mol*K) to find ΔHvap.

Finding the boiling point

To determine the boiling point of the compound under standard atmospheric pressure, we need to find the temperature at which the vapor pressure equals 760 torr (1 atm). We can do this by solving the Clausius-Clapeyron equation: $$ \ln (760) = -\frac{\Delta H_{vap}}{R}\frac{1}{T} + C $$ First, rearrange the equation and solve for \(1/T\): $$ \frac{1}{T} = \frac{-R}{\Delta H_{vap}} (\ln(760) - C) $$ To find the constant C, use one of the original data points from the table, such as Temperature = 280 K and Vapor Pressure = 32.42 torr. $$ C = \ln(32.42) + \frac{\Delta H_{vap}}{R}\frac{1}{280} $$ Substitute the found value for C into the equation for \(1/T\) and solve for T, which represents the boiling point of the compound.

Key concepts

Vapor Pressure

Understanding vapor pressure is crucial in grasping the behavior of liquids and how they transition to gases. Vapor pressure is defined as the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases (solid or liquid) at a given temperature in a closed system. The equilibrium is when the rate of evaporation of liquid equals the rate of condensation of vapor.

In simpler terms, this pressure develops when molecules escape from the surface of the liquid and enter the gas phase. The ability of a liquid to change into vapor is not only inherent to its nature but is also influenced by temperature – higher temperatures generally lead to higher vapor pressures as more molecules have the energy to break free from the liquid.

Key Factors Affecting Vapor Pressure

• Substance identity: Different substances have different inherent vapor pressures at the same temperature.
• Temperature: As previously mentioned, increasing the temperature will usually increase the vapor pressure.
• Intermolecular Forces: Substances with stronger intermolecular forces tend to have lower vapor pressures.

Heat of Vaporization (ΔHvap)

The heat of vaporization, often represented as \(\Delta H_{vap}\), is the amount of energy required to convert a given amount of a substance from a liquid into a gas at a constant temperature and pressure. This is a crucial concept in chemical thermodynamics because it quantifies the energy changes during phase transitions.

For water, for instance, its high \(\Delta H_{vap}\) reflects the strong hydrogen bonds that need to be overcome for it to vaporize. The determination of \(\Delta H_{vap}\) can be experimentally achieved through various methods, including the Clausius-Clapeyron equation, which links it to vapor pressure and temperature.

Importance of ΔHvap

• Indicative of intermolecular forces: The larger the \(\Delta H_{vap}\), the stronger the intermolecular forces within the liquid.
• Energy Calculations: Understanding \(\Delta H_{vap}\) is essential for calculating energy requirements in industrial processes involving phase changes.

Boiling Point Determination

The boiling point of a liquid is the temperature at which its vapor pressure equals the surrounding pressure. For pure substances, the normal boiling point is determined at one standard atmosphere of pressure (760 torr). To determine an unknown boiling point, we can utilize the already mentioned Clausius-Clapeyron equation, which requires knowledge of the vapor pressure at various temperatures.

By graphing the natural logarithm of vapor pressure against the inverse temperature (1/T), we can obtain a straight line whose slope is related to \(\Delta H_{vap}\). With this data, we can predict the temperature at which the vapor pressure will be equal to one atmosphere, thus determining the boiling point.

Understanding how to determine boiling points is essential for the characterization of substances and is also relevant in the design of equipment for distillation and other separation processes.

Chemical Thermodynamics

Chemical thermodynamics is the study of the interrelation of heat and work with chemical reactions or with the physical changes of state within the confines of the laws of thermodynamics. It involves topics like internal energy, entropy, and the thermodynamic potentials, which are helpful for understanding chemical processes and predicting the direction and extent of chemical reactions.

In this realm, the Clausius-Clapeyron equation is a fundamental relationship derived from these principles. It provides a way to quantify the relationship between vapor pressure and temperature, and as we've seen, the enthalpy of phase changes such as vaporization.

The study of chemical thermodynamics not only informs us about the energy requirements of reactions but also guides industries in optimizing reactions, using energy more efficiently, and even drives the innovation of new technologies that are rooted in the laws of thermodynamics.