# The Vapor Pressure and Enthalpy of Vaporization

The Vapor Pressure and Enthalpy of Vaporization of                 a Liquid

Objectives

To calculate the vapor pressure of a liquid from data measured at different temperatures

To plot the vapor pressure of the liquid as a function of temperature

To use the vapor pressure data to construct a Clausius-Clapeyron plot

To determine the enthalpy of vaporization of the liquid using the Clausius-Clapeyron plot

To manipulate the Clausius-Clapeyron equation for materials other than “yours”

INTRODUCTION

Heat and Molecular Motion

On an atomic or molecular scale, heating a substance increases the motion of the atoms. If the atoms are bonded to form a molecule, attention can be focused on the molecule as a whole rather than on individual atoms.

When a molecule is heated it moves more rapidly. The energy put into a molecule is stored as translational, rotational, and vibrational motions of the molecules.

Motion of the center of mass of the molecule is called translational motion (meaning motion from place to place). Gaseous molecules have a substantial fraction of their total energy stored as translational kinetic energy. They move about rapidly, bouncing off one another and off the walls of their container.

Molecules can also rotate about their centers of mass. As a gas is heated the molecules rotate faster. Energy is stored in the rotating molecules just as energy is stored in a rotating flywheel.

Individual atoms in molecules are linked by chemical bonds that are “springy”. That is, bonded atoms are constrained to move like masses connected by springs. As the molecules are heated, the energies and amplitudes of these vibrations increase.

As a substance is cooled, energy is removed from it and molecular motions slow down.  Absolute zero is the lowest temperature on the kelvin scale. At that point, as much energy as possible has been removed from the substance. Molecular motion reaches a minimum and nearly, but not completely, ceases.

As energy is removed from gaseous molecules they will, at some temperature, condense into a liquid and eventually into a solid. This occurs because there are attractive forces between molecules. To separate the molecules in a solid to the distances characteristic of those in a gas, work must be done against these same attractive forces. This work is the energy required to warm and melt the solid, and to warm and vaporize the liquid.

Now imagine the following experiment. Begin by adding energy to ice at a uniform rate while measuring the temperature. For water a plot of temperature as a function of heat added (or as a function of time if heat is added uniformly) would look like Figure 1. As the solid warms, the molecules begin to move. The molecules in the solid are not on average “going” anywhere. They simply begin to bounce around in the cage formed by their neighboring molecules. In effect, they are like gas molecules confined to a very small container not much larger than the molecule itself. At the same time the amplitude of vibration of the bonded atoms increases and the molecules may begin to rotate in their small cages. A few molecules at the surface of the ice may acquire enough kinetic energy to break free from the solid to become gas molecules. The number of gas molecules may be large enough to produce a measurable pressure. This pressure is called the vapor pressure of the solid.

At the melting point the amplitude of the motions increases to the point where the attractive forces between neighboring molecules are no longer sufficient to hold them in fixed positions. The regular structure of the solid breaks down. The molecules move more freely. This produces a phase change – the solid becomes a liquid. The liquid has lower viscosity than the solid; the material assumes the shape of the container.

Overall, molecules have more energy in the liquid phase than in the solid phase. The energy required to accomplish the phase change is called the enthalpy of fusion.

Because liquid molecules have greater average energy than molecules in the solid, a greater number of them can escape into the gas phase, so that the vapor pressure of the liquid is greater than that of the solid. As the liquid is further warmed a greater number of molecules acquire sufficient energy to escape from the liquid so that the vapor pressure increases as the temperature of the liquid increases. At the normal boiling point, where the vapor pressure is equal to 1 atmosphere, all of the energy added goes into a second phase change, called vaporization. If the container is not closed, the vapor pressure cannot exceed atmospheric pressure. The liquid begins to boil, forming bubbles of vapor that escape the liquid. If sufficient energy is added, eventually all the liquid is converted to vapor.

The data showing the equilibria between the three phases is displayed in a phase diagram.  A two dimensional phase diagram shows pressure as a function of temperature. The lines on the diagram represent the temperature and pressure combinations at which two phases can coexist in equilibrium. A point where three phases can coexist is called a triple point. A portion of the phase diagram for water is shown in Figure 2.

Latent Heat

When a substance is heated, will a thermometer immersed in the substance always indicate a temperature increase? (Refer to Figure 1.)  Not for a pure substance. Figure 1 shows that while the solid is melting or the liquid is vaporizing, the temperature remains constant. Heat that produces a phase change such as the melting of ice or the boiling of water without producing a temperature change is called latent heat. A thermometer can be used to measure temperature changes but cannot always be used to measure energy changes.

Vapor Pressure and the Clausius – Clapeyron Equation

Refer to the portion of Figure 2 showing the equilibrium between liquid and vapor. Note that the vapor pressure of liquid does NOT increase linearly with temperature. It increases more rapidly, in an exponential manner. The same type of increase is observed for the equilibrium between solid and vapor.  An equation which fits the data for the observed equilibria is the Clausius – Clapeyron equation:

or

In either of these equations DH is the enthalpy change for a phase transition: either the enthalpy of sublimation for the solid-gas equilibrium or the enthalpy of vaporization for the liquid-gas equilibrium. R is the gas constant (8.314 J/ mol.K), T is the temperature in kelvin and A is a constant with units of pressure.

By taking the logarithm of both sides of the above equation the equation can be converted to a linear form:

A plot with log P on the y-axis and 1/T on the x-axis, gives a straight line whose slope is equal to  -DH/2.303R. This plot thus allows us to determine the enthalpy of vaporization of a liquid for which we have vapor pressure data.

Figure 1:  Warming Curve: The plot presents the data for 1 kg of water as it passes from

ice at -50.0°C to steam at temperatures above 100.0°C.

A:  Warming solid; no phase change

B:  Melting solid; DT = 0

C:  Warming liquid; no phase change

D:  Boiling liquid; DT = 0

E:   Warming vapor; no phase change

In regions B and D of the warming curve, the total heat required to accomplish

the phase change is a function of  DHfus and of  DHvap for the material.

Figure 2: Phase Diagram for Water

Processing the Data

If we assume that the vapor pressure of our liquid below 5°C is negligible in comparison to atmospheric pressure, we can calculate the number of moles of air trapped in our system by assuming the air behaves as an ideal gas:

Vcorr = Vexpt’l – 0.20 mL (for the meniscus)

and P  is the atmospheric pressure (torr), V is the volume (mL) at the lowest temperature, R is the gas constant (6.237 x 104 mL.torr/mol.K), and T is the temperature of the ice-water mixture in kelvins.

For each of the temperature-volume measurements made, the partial pressure (torr) of air in the graduate can be calculated using the number of moles of air calculated above and assuming ideal gas behavior. Using Dalton’s law of partial pressures we obtain the vapor pressure of water:

where Patm  is barometric pressure.

The remainder of the calculations are described in the report. The physical procedure to collect data is described on the next page.

Procedure:

Typically, the set up for this                                                                                                   experiment is quite simple. A large                                                                                        test tube filled with the solvent is                                                                                           clamped vertically. A graduate is

filled with solvent to a level that will                                                                                                 leave an air bubble of between 4 and                                                                                     5 mL in size when the graduate is                                                                                          inserted into the test tube. The                                                                                                solvent is then warmed up to a                                                                                               high temperature and readings of the                                                                                             volume of the air bubble recorded at                                                                                          intervals as the solvent is cooled                                                                                                 back down. For non-aqueous                                                                                                  solvents the apparatus would have                                                                                         to be modified to alleviate safety                                                                                           concerns. The final reading is at a                                                                                          low enough temperature that the                                                                                       vapor pressure of the solvent is zero.                                                                                                That allows calculation of the                                                                                                 number of moles of air trapped in the                                                                                    graduate.

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