Question

The vapor pressure for pure water and pure acetone is measured as a function of temperature. In each case, a graph of the log of the vapor pressure versus \(1 / T\) is found to be a straight line. The slope of the line for water is \(-4895 \mathrm{~K},\) and the slope of the line for acetone is \(-3765 \mathrm{~K}\). Determine \(\Delta H_{\text {vap }}\) for each substance. Account for the difference by discussing the molecular structure of the two molecules.

Short answer

The enthalpy of vaporization for water is \( 40,685.07 \text{ J/mol} \), and for acetone it is \( 31,302.99 \text{ J/mol} \). Water has a higher \(\text{ΔH}_{\text{vap}}\) due to stronger hydrogen bonding compared to acetone's dipole-dipole interactions and Van der Waals forces.

Step-by-step solution

Understanding the Clausius-Clapeyron Equation

First, recognize that the relationship between the natural logarithm of vapor pressure and the inverse of temperature for a pure substance is given by the Clausius-Clapeyron equation: \( \text{ln}(P) = -\frac{\text{\text{ΔH}}_{\text{vap}}}{R} \times \frac{1}{T} + C \) where \(P\) is the vapor pressure, \(T\) is the temperature in Kelvin, \(R\) is the gas constant, \(\text{\text{ΔH}}_{\text{vap}}\) is the enthalpy of vaporization, and \(C\) is a constant that depends on the chemical substance.

Extracting Enthalpies of Vaporization

The slope of the straight-line graph of \(\text{ln}(P)\) versus \(1/T\) corresponds to \(-\text{\text{ΔH}}_{\text{vap}}/R\) for each substance. By substituting the given slopes and the value of the gas constant \(R = 8.314 \text{ J/mol}\cdot\text{K}\), calculate the enthalpy of vaporization for water and acetone: \( \text{\text{ΔH}}_{\text{vap}} = -\text{slope}\times R\).

Calculate Enthalpies of Vaporization for Water and Acetone

Use the slopes from the graphs and the gas constant to find the enthalpies of vaporization. For water: \( \text{\text{ΔH}}_{\text{vap}} = -( -4895 \text{ K} ) \times 8.314 \text{ J/mol}\cdot\text{K} \). For acetone: \( \text{\text{ΔH}}_{\text{vap}} = -( -3765 \text{ K} ) \times 8.314 \text{ J/mol}\cdot\text{K} \).

Discussing Molecular Structure Differences

The difference in \(\text{\text{ΔH}}_{\text{vap}}\) is due to the different intermolecular forces present in water and acetone. Water has strong hydrogen bonds which require more energy to vaporize it compared to acetone which has weaker dipole-dipole interactions and Van der Waals forces due to its molecular structure lacking the extensive hydrogen bonding present in water.

Key concepts

Clausius-Clapeyron Equation

The Clausius-Clapeyron equation is a foundational concept in thermodynamics that relates the change in vapor pressure with temperature. It does so quite elegantly by creating an equation that allows for the calculation of the enthalpy of vaporization, \( \Delta H_{\text{vap}} \) - a measure of the energy required to turn a liquid into a vapor. According to its mathematical expression, \( \text{ln}(P) = -\frac{\text{\text{ΔH}}_{\text{vap}}}{R} \times \frac{1}{T} + C \), it elucidates that the natural logarithm of vapor pressure (\(\text{ln}(P)\)) plotted against the inverse of the temperature (\(1/T\)) produces a straight line whose slope is linked to the enthalpy of vaporization.

To understand it practically, let's apply this equation to an exercise where the slopes of the straight-line graphs for water and acetone are given, being \(-4895 \text{ K}\) and \(-3765 \text{ K}\) respectively. These values, when plugged into the Clausius-Clapeyron equation, yield the respective enthalpies of vaporization for these substances, showcasing the direct application of this pivotal equation in material and chemical science.

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in equilibrium with its condensed phases at a given temperature in a closed system. It is a critical property that indicates a liquid's propensity to vaporize. As temperature rises, a greater fraction of the molecules within the liquid have sufficient kinetic energy to escape the liquid's surface and form vapor, leading to an increase in vapor pressure. The exercise involving water and acetone demonstrates this relationship. Given that the vapor pressures form a straight line when graphed against \(1/T\), it indicates a consistent change in vapor pressure with temperature, which can be understood through the Clausius-Clapeyron equation, emphasizing the integral role of vapor pressure in studying phase transitions and calculating thermodynamic quantities.

Intermolecular Forces

Intermolecular forces are the forces of attraction or repulsion between neighboring particles such as atoms, molecules, or ions. They are significantly weaker than intramolecular bonds like covalent or ionic bonds, yet they play a crucial role in determining the physical state and properties of substances. Examples include London dispersion forces, dipole-dipole interactions, and hydrogen bonds. In the context of our exercise, the intermolecular forces influence the enthalpy of vaporization, as they must be overcome to convert a liquid into a gas. Water, with its strong hydrogen bonds, exhibits a higher \(\Delta H_{\text{vap}}\) compared to acetone, which primarily interacts through weaker dipole-dipole interactions. This highlights the significance of intermolecular forces in dictating the thermal energy required for the phase transition of substances.

Hydrogen Bonding

Hydrogen bonding is a special type of dipole-dipole interaction that occurs when a hydrogen atom, covalently bonded to a highly electronegative atom such as oxygen, nitrogen, or fluorine, experiences the electrostatic pull from another electronegative atom. This type of bond is particularly strong compared to other types of van der Waals forces and is critical in determining the properties of many substances, including water. Water's unique properties, like its relatively high boiling point and high heat of vaporization, are attributed to the presence of hydrogen bonds. In our exercise, water's deep negative slope in the Clausius-Clapeyron plot is indicative of its high \(\Delta H_ {\text{vap}}\), a direct consequence of its strong hydrogen bonding.

Molecular Structure

The molecular structure profoundly affects a substance's physical and chemical properties. It defines how atoms or molecules are arranged in space and what types of bonds and interactions can occur between them.

In the exercise provided, the comparison between water and acetone underscores the influence of molecular structure on enthalpy of vaporization. Water's molecular structure allows for extensive hydrogen bonding due to the presence of hydrogen atoms bonded to a highly electronegative oxygen, thus requiring more energy to vaporize. Acetone, on the other hand, has a structure that supports weaker dipole-dipole interactions, resulting in a lower enthalpy of vaporization. The molecular structure determines the strength and type of intermolecular forces present, which in turn influence a substance's vapor pressure and temperature response.