PV Diagrams

We can examine aspects of the behavior of a substance by plotting a graph of pressure versus volume, called a PV diagram. When the substance behaves like an ideal gas, the ideal gas law describes the relationship between its pressure and volume. That is,

Now, assuming the number of molecules and the temperature are fixed,

For example, the volume of the gas will decrease as the pressure increases. If you plot the relationship [latex]\text{PV}=\text{constant}[/latex] on a [latex]\text{PV}[/latex] diagram, you find a hyperbola. Figure 97.2 shows a graph of pressure versus volume. The hyperbolas represent ideal-gas behavior at various fixed temperatures, and are called isotherms. At lower temperatures, the curves begin to look less like hyperbolas—the gas is not behaving ideally and may even contain liquid. There is a critical point—that is, a critical temperature—above which liquid cannot exist. At sufficiently high pressure above the critical point, the gas will have the density of a liquid but will not condense. Carbon dioxide, for example, cannot be liquefied at a temperature above [latex]\text{31}\text{.}0\text{º}\text{C}[/latex]. Critical pressure is the minimum pressure needed for liquid to exist at the critical temperature. Table 97.1 lists representative critical temperatures and pressures.

[latex]\text{PV}[/latex] diagrams. (a) Each curve (isotherm) represents the relationship between [latex]P[/latex] and [latex]V[/latex] at a fixed temperature; the upper curves are at higher temperatures. The lower curves are not hyperbolas, because the gas is no longer an ideal gas. (b) An expanded portion of the [latex]\text{PV}[/latex] diagram for low temperatures, where the phase can change from a gas to a liquid. The term “vapor” refers to the gas phase when it exists at a temperature below the boiling temperature.

Phase Diagrams

The plots of pressure versus temperatures provide considerable insight into thermal properties of substances. There are well-defined regions on these graphs that correspond to various phases of matter, so [latex]\text{PT}[/latex] graphs are called phase diagrams. Figure 97.3 shows the phase diagram for water. Using the graph, if you know the pressure and temperature you can determine the phase of water. The solid lines—boundaries between phases—indicate temperatures and pressures at which the phases coexist (that is, they exist together in ratios, depending on pressure and temperature). For example, the boiling point of water is [latex]\text{100}\text{º}\text{C}[/latex] at 1.00 atm. As the pressure increases, the boiling temperature rises steadily to [latex]\text{374}\text{º}\text{C}[/latex] at a pressure of 218 atm. A pressure cooker (or even a covered pot) will cook food faster because the water can exist as a liquid at temperatures greater than [latex]\text{100}\text{º}\text{C}[/latex] without all boiling away. The curve ends at a point called the critical point, because at higher temperatures the liquid phase does not exist at any pressure. The critical point occurs at the critical temperature, as you can see for water from Table 97.1. The critical temperature for oxygen is [latex]–\text{118}\text{º}\text{C}[/latex], so oxygen cannot be liquefied above this temperature.

Similarly, the curve between the solid and liquid regions in Figure 97.3 gives the melting temperature at various pressures. For example, the melting point is [latex]0\text{º}\text{C}[/latex] at 1.00 atm, as expected. Note that, at a fixed temperature, you can change the phase from solid (ice) to liquid (water) by increasing the pressure. Ice melts from pressure in the hands of a snowball maker. From the phase diagram, we can also say that the melting temperature of ice rises with increased pressure. When a car is driven over snow, the increased pressure from the tires melts the snowflakes; afterwards the water refreezes and forms an ice layer.

At sufficiently low pressures there is no liquid phase, but the substance can exist as either gas or solid. For water, there is no liquid phase at pressures below 0.00600 atm. The phase change from solid to gas is called sublimation. It accounts for large losses of snow pack that never make it into a river, the routine automatic defrosting of a freezer, and the freeze-drying process applied to many foods. Carbon dioxide, on the other hand, sublimates at standard atmospheric pressure of 1 atm. (The solid form of [latex]{\text{CO}}_{2}[/latex] is known as dry ice because it does not melt. Instead, it moves directly from the solid to the gas state.)

All three curves on the phase diagram meet at a single point, the triple point, where all three phases exist in equilibrium. For water, the triple point occurs at 273.16 K [latex]\left(0\text{.}\text{01}\text{º}\text{C}\right)[/latex], and is a more accurate calibration temperature than the melting point of water at 1.00 atm, or 273.15 K [latex]\left(0\text{.}0\text{º}\text{C}\right)[/latex]. See Table 97.2 for the triple point values of other substances.

Equilibrium

Liquid and gas phases are in equilibrium at the boiling temperature. (See Figure 97.4) If a substance is in a closed container at the boiling point, then the liquid is boiling and the gas is condensing at the same rate without net change in their relative amount. Molecules in the liquid escape as a gas at the same rate at which gas molecules stick to the liquid, or form droplets and become part of the liquid phase. The combination of temperature and pressure has to be “just right”; if the temperature and pressure are increased, equilibrium is maintained by the same increase of boiling and condensation rates.

One example of equilibrium between liquid and gas is that of water and steam at [latex]\text{100}\text{º}\text{C}[/latex] and 1.00 atm. This temperature is the boiling point at that pressure, so they should exist in equilibrium. Why does an open pot of water at [latex]\text{100}\text{º}\text{C}[/latex] boil completely away? The gas surrounding an open pot is not pure water: it is mixed with air. If pure water and steam are in a closed container at [latex]\text{100}\text{º}\text{C}[/latex] and 1.00 atm, they would coexist—but with air over the pot, there are fewer water molecules to condense, and water boils. What about water at [latex]\text{20}\text{.}0\text{º}\text{C}[/latex] and 1.00 atm? This temperature and pressure correspond to the liquid region, yet an open glass of water at this temperature will completely evaporate. Again, the gas around it is air and not pure water vapor, so that the reduced evaporation rate is greater than the condensation rate of water from dry air. If the glass is sealed, then the liquid phase remains. We call the gas phase a vapor when it exists, as it does for water at [latex]\text{20}\text{.}0\text{º}\text{C}[/latex], at a temperature below the boiling temperature.

Vapor Pressure, Partial Pressure, and Dalton’s Law

Vapor pressure is defined as the pressure at which a gas coexists with its solid or liquid phase. Vapor pressure is created by faster molecules that break away from the liquid or solid and enter the gas phase. The vapor pressure of a substance depends on both the substance and its temperature—an increase in temperature increases the vapor pressure.

Partial pressure is defined as the pressure a gas would create if it occupied the total volume available. In a mixture of gases, the total pressure is the sum of partial pressures of the component gases, assuming ideal gas behavior and no chemical reactions between the components. This law is known as Dalton’s law of partial pressures, after the English scientist John Dalton (1766–1844), who proposed it. Dalton’s law is based on kinetic theory, where each gas creates its pressure by molecular collisions, independent of other gases present. It is consistent with the fact that pressures add according to Pascal’s Principle. Thus water evaporates and ice sublimates when their vapor pressures exceed the partial pressure of water vapor in the surrounding mixture of gases. If their vapor pressures are less than the partial pressure of water vapor in the surrounding gas, liquid droplets or ice crystals (frost) form.

PhET Explorations: States of Matter—Basics

Heat, cool, and compress atoms and molecules and watch as they change between solid, liquid, and gas phases.