Real Gases and Phase Transitions

So far, we’ve focused on ideal gas behavior, which works well at high temperatures and low pressures. However, real gases deviate from ideal behavior when molecules are packed closely together—typically at lower temperatures or higher pressures. Under these conditions, attractive intermolecular forces and the finite volume of molecules become important. As shown in Figure 97.1, a gas cools and condenses into a liquid, leading to a sharp drop in volume. Further cooling solidifies the liquid, though the total volume never reaches zero due to the actual size of the molecules.

These transitions are critical in many biological and technological applications. For example, sublimation and liquefaction are used in freeze-drying of foods and cryopreservation of tissues using [latex]{\text{LN}}_2[/latex] (liquid nitrogen).

PV Diagrams

A PV diagram plots pressure versus volume. For an ideal gas, the relationship follows the ideal gas law:

If the number of molecules and the temperature are held constant, this reduces to:

Plotting this equation on a graph produces a hyperbola, known as an isotherm, representing constant temperature. At lower temperatures, deviations from ideal behavior appear, especially when the gas condenses into a liquid. This behavior is depicted in Figure 97.2, where isotherms flatten out during the phase change from gas to liquid.

There is a specific critical temperature above which a gas cannot condense into a liquid, regardless of the pressure applied. Similarly, the critical pressure is the minimum pressure needed to liquefy a gas at that temperature. For instance, carbon dioxide cannot become a liquid above [latex]31.1^\circ \text{C}[/latex]. Critical points are important for understanding and designing pressure-based refrigeration and cryogenic systems. Table 97.1 lists representative critical temperatures and pressures.

Table 97.1: Critical Temperatures and Pressures

Substance	Critical temperature		Critical pressure
	[latex]\text{K}[/latex]	[latex]\text{º}\text{C}[/latex]	[latex]\text{Pa}[/latex]	[latex]\text{atm}[/latex]
Water	647.4	374.3	[latex]\text{22}\text{.}\text{12}×{\text{10}}^{6}[/latex]	219.0
Sulfur dioxide	430.7	157.6	[latex]7\text{.}\text{88}×{\text{10}}^{6}[/latex]	78.0
Ammonia	405.5	132.4	[latex]\text{11}\text{.}\text{28}×{\text{10}}^{6}[/latex]	111.7
Carbon dioxide	304.2	31.1	[latex]7\text{.}\text{39}×{\text{10}}^{6}[/latex]	73.2
Oxygen	154.8	−118.4	[latex]5\text{.}\text{08}×{\text{10}}^{6}[/latex]	50.3
Nitrogen	126.2	−146.9	[latex]3\text{.}\text{39}×{\text{10}}^{6}[/latex]	33.6
Hydrogen	33.3	−239.9	[latex]1\text{.}\text{30}×{\text{10}}^{6}[/latex]	12.9
Helium	5.3	−267.9	[latex]0\text{.}\text{229}×{\text{10}}^{6}[/latex]	2.2

Phase Diagrams

A phase diagram shows the phases of a substance as a function of temperature and pressure. These [latex]PT[/latex] graphs help us identify the conditions under which a substance exists as a solid, liquid, or gas. The boundaries between regions show where phases coexist in equilibrium.

Figure 97.3 illustrates the phase diagram for water. For instance, water boils at [latex]100^\circ \text{C}[/latex] under 1 atm of pressure, but the boiling temperature increases under higher pressures. This principle explains why food cooks faster in a pressure cooker.

At very low pressures, the liquid phase of some substances (like water or carbon dioxide) ceases to exist. The direct transition between solid and gas is called sublimation. This process is used in freeze-drying and occurs naturally with snowpacks.

The unique point where all three phases (solid, liquid, and gas) coexist in equilibrium is called the triple point. For water, this occurs at 273.16 K (0.01°C) and is used to define the Kelvin temperature scale. See Table 97.2 for triple point data of other substances.

Equilibrium Between Phases

At the boiling point, a liquid and its vapor are in equilibrium—meaning molecules evaporate and condense at the same rate. In a closed container, as shown in Figure 97.4, the liquid does not boil away as long as this balance exists.

If pressure and temperature increase together, the evaporation and condensation rates also increase, maintaining equilibrium. However, in an open container, like a pot of boiling water, steam escapes, preventing equilibrium and allowing the water to boil away.

Below the boiling point, evaporation can still occur if the vapor pressure of the substance is higher than the partial pressure of that vapor in the surrounding air. This explains why water slowly evaporates even at room temperature.

Vapor Pressure, Partial Pressure, and Dalton’s Law

Vapor pressure is the pressure exerted by a gas in equilibrium with its solid or liquid form. It depends on the temperature—higher temperatures increase vapor pressure.

In mixtures of gases, each gas exerts a partial pressure, which is the pressure it would exert if it occupied the entire volume alone. According to Dalton’s law of partial pressures, the total pressure is the sum of the partial pressures:

This law is important in understanding respiration, anesthesia delivery, and hyperbaric treatments. For example, oxygen’s partial pressure in the lungs affects gas exchange with blood.

Check Your Understanding

Q: Why does a drink with ice stay at [latex]0^\circ \text{C}[/latex] on a hot day?

A: As long as ice is present, it is in thermal equilibrium with the liquid water. The temperature remains constant at the freezing point until all the ice has melted. Only then does the water begin to warm up.

Q: Is energy involved in a phase change? Why is water sprayed on orange trees during freezing conditions?

A: Yes, energy is involved in every phase change. Going from solid to liquid or liquid to gas requires energy input to break molecular bonds. Conversely, gas condensing to liquid or liquid freezing to solid releases energy. Spraying water on trees releases latent heat during freezing, which protects the oranges from freezing damage.