Phase Diagrams
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You should be comfortable with the definitions of temperature, pressure, and the three fundamental states of matter (solid, liquid, gas). Familiarity with the concept of equilibrium between phases is also assumed.
Why this matters
Phase diagrams are the maps that tell us where matter exists in its different forms.
In the previous lecture, we examined the internal structure of solids, distinguishing between the ordered lattices of crystalline materials and the disordered networks of amorphous ones. However, structure is only one aspect of a material's identity; its state is equally critical. A substance can be a solid, a liquid, or a gas, and which state it adopts depends entirely on the external conditions of temperature and pressure. Phase diagrams provide a comprehensive visual summary of these conditions, allowing chemists and engineers to predict phase transitions without performing experiments. They are indispensable tools in fields ranging from metallurgy to atmospheric science, serving as the bridge between microscopic molecular interactions and macroscopic bulk behaviour.
The Landscape of States
Imagine a map with temperature running from left to right and pressure from bottom to top. On this map there are three regions: at low temperature and high pressure, particles are tightly packed and vibrate in place (the solid region); at moderate temperature and pressure, particles flow past one another (the liquid region); and at high temperature or low pressure, particles are far apart and move freely (the gas region). The boundaries between these regions are not arbitrary lines but specific conditions where two phases coexist in equilibrium. For example, along the boundary between the solid and liquid regions, the rate of melting exactly equals the rate of freezing. If you move your conditions across this boundary, you force the system to change its phase. The most critical point on this map is the triple point, a unique location where all three phases—solid, liquid, and gas—can coexist simultaneously. This is a state of perfect balance, where the chemical potentials of all three phases are identical. Moving away from this point in any direction breaks the balance, causing one phase to disappear as the others adjust. This conceptual map is the foundation of the phase diagram, translating abstract thermodynamic principles into a practical graphical tool that we can use to navigate the behaviour of matter under varying conditions.
Anatomy of a Phase Diagram
A standard phase diagram plots pressure on the vertical axis and temperature on the horizontal axis. The diagram is divided into three main regions by three curves: the sublimation curve, the fusion (melting) curve, and the vaporisation curve. The sublimation curve separates the solid and gas regions. Along this line, the solid and gas phases are in equilibrium, meaning the rate of sublimation equals the rate of deposition. The fusion curve separates the solid and liquid regions. Here, melting and freezing occur at the same rate. The vaporisation curve separates the liquid and gas regions, representing the boiling point at a given pressure. These curves meet at the triple point, a unique set of conditions where solid, liquid, and gas coexist. For water, this occurs at \( 273.16 \ \mathrm{K} \) and \( 611.657 \ \mathrm{Pa} \). Above the triple point, the vaporisation curve continues until it reaches the critical point. At the critical point, the distinction between liquid and gas disappears, and the substance becomes a supercritical fluid. The critical temperature and pressure are specific to each substance. For water, the critical point is at \( 647.1 \ \mathrm{K} \) and \( 22.06 \ \mathrm{MPa} \). Above the critical temperature, no amount of pressure can liquefy the gas. The slopes of these curves provide insight into the relative densities of the phases. For most substances, the fusion curve has a positive slope, indicating that the solid is denser than the liquid. However, water is an exception; its fusion curve has a negative slope, meaning ice is less dense than liquid water. This anomaly is crucial for the survival of aquatic life in cold climates, as ice floats and insulates the water below.
Interpreting Phase Transitions
To understand how a substance changes phase, we must consider how changes in temperature and pressure move the system across the diagram. Suppose we start with a gas at high temperature and low pressure. If we increase the pressure at constant temperature, we move vertically upwards on the diagram. Eventually, we cross the vaporisation curve, and the gas condenses into a liquid. If instead the temperature were above the critical temperature, compressing the gas would never produce a separate liquid phase: the fluid would simply become denser and denser. Alternatively, if we lower the temperature at constant pressure, we move horizontally to the left. We first cross the vaporisation curve, condensing the gas into a liquid. Further cooling crosses the fusion curve, freezing the liquid into a solid. The path taken determines the sequence of phases observed. For example, if the pressure is below the triple point pressure, cooling a gas will lead directly to a solid via sublimation, skipping the liquid phase entirely. This is why dry ice (solid carbon dioxide) sublimes at atmospheric pressure rather than melting. The phase diagram thus allows us to predict the outcome of any combination of heating, cooling, compression, or expansion. It is a powerful tool for designing industrial processes, such as fractional distillation, where different components are separated based on their different vaporisation curves. By carefully controlling temperature and pressure, we can selectively condense or vaporise specific components, enabling the purification of complex mixtures.
The Clapeyron Equation
The slope of the phase boundary curves is not arbitrary; it is governed by the Clapeyron equation, which relates the slope to the enthalpy change of the phase transition and the volume change. The equation is expressed as \( \frac{\mathrm{d}P}{\mathrm{d}T} = \frac{\Delta H_{\mathrm{trans}}}{T \Delta V_{\mathrm{trans}}} \), where \( \Delta H_{\mathrm{trans}} \) is the molar enthalpy change for the transition, \( T \) is the absolute temperature, and \( \Delta V_{\mathrm{trans}} \) is the molar volume change. For the fusion curve, \( \Delta H_{\mathrm{trans}} \) is the enthalpy of fusion, and \( \Delta V_{\mathrm{trans}} \) is the difference in molar volume between the liquid and solid. For most substances, the liquid is less dense than the solid, so \( \Delta V_{\mathrm{trans}} \) is positive, resulting in a positive slope. This means that increasing pressure raises the melting point. However, for water, the solid (ice) is less dense than the liquid, so \( \Delta V_{\mathrm{trans}} \) is negative, resulting in a negative slope. This means that increasing pressure lowers the melting point of ice. This phenomenon is known as regelation, where ice melts under pressure and refreezes when the pressure is released. The Clapeyron equation provides a quantitative link between the microscopic properties of the substance (enthalpy and volume) and the macroscopic behaviour observed in the phase diagram. It allows us to calculate the slope of the phase boundary if we know the thermodynamic properties of the phases. This is essential for understanding how external conditions affect phase stability and for predicting the behaviour of materials under extreme conditions.
A common error is to treat the triple point and the normal melting point as the same thing. For water, the triple point is 273.16 K at only 611.657 Pa, where ice, liquid and vapour coexist, while the normal melting point is 273.15 K at 101.325 kPa, where ice and liquid coexist under air. The triple point of a pure substance is fixed by nature and is used as a fixed point in thermometry; impurities change the phase diagram itself, so a mixture has no single triple point. C
Worked example
Problem. Estimate the change in the melting point of ice when the pressure is raised by 10.0 MPa. Data: \(\Delta H_{\mathrm{fus}} = 6.01\ \mathrm{kJ\,mol^{-1}}\); molar volume of ice \(19.7\ \mathrm{cm^3\,mol^{-1}}\); molar volume of liquid water \(18.0\ \mathrm{cm^3\,mol^{-1}}\); normal melting point 273.15 K.
- Volume change on melting: \(\Delta V_{\mathrm{fus}} = 18.0 - 19.7 = -1.7\ \mathrm{cm^3\,mol^{-1}} = -1.7 \times 10^{-6}\ \mathrm{m^3\,mol^{-1}}\).
- Clapeyron slope: \(\dfrac{\mathrm{d}P}{\mathrm{d}T} = \dfrac{\Delta H_{\mathrm{fus}}}{T\,\Delta V_{\mathrm{fus}}} = \dfrac{6010}{273.15 \times (-1.7 \times 10^{-6})} = -1.29 \times 10^{7}\ \mathrm{Pa\,K^{-1}}\).
- Invert: \(\dfrac{\mathrm{d}T}{\mathrm{d}P} = -7.73 \times 10^{-8}\ \mathrm{K\,Pa^{-1}}\), that is, about \(-0.077\ \mathrm{K}\) per MPa.
- For \(\Delta P = 1.00 \times 10^{7}\ \mathrm{Pa}\): \(\Delta T = -7.73 \times 10^{-8} \times 1.00 \times 10^{7} = -0.77\ \mathrm{K}\).
- Interpret: even 100 atmospheres lowers the melting point by less than 1 K. (This is why pressure melting cannot by itself explain ice skating: a skater's blade produces far less than the hundreds of atmospheres needed for a few kelvin.)
Units check. \(\mathrm{K} \times \mathrm{m^3\,mol^{-1}} / \mathrm{J\,mol^{-1}} = \mathrm{K\,m^3\,J^{-1}} = \mathrm{K\,Pa^{-1}}\); multiplied by Pa this gives K.
Three quick tests
- What is the significance of the critical point on a phase diagram?
Answer
The critical point is the temperature and pressure above which the liquid and gas phases are indistinguishable, forming a supercritical fluid. - Why does the fusion curve of water have a negative slope?
Answer
Because ice is less dense than liquid water, so \( \Delta V_{\mathrm{fus}} \) is negative, leading to a negative slope according to the Clapeyron equation. - What happens if you cool a gas at a pressure below the triple point pressure?
Answer
The gas will sublime directly into a solid, skipping the liquid phase entirely.
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Every lecture and every result in this unit, with prerequisites and threads.
Lecture notes checked line by line by the editor; every number in the worked example was recomputed.