DAT General Chemistry: Reading Phase Diagrams

DAT High-Yield Concept

State Identification: Locate temperature on the x-axis and pressure on the y-axis; the bounded region containing that point dictates the thermodynamically stable phase.
Phase Boundaries: Points along a line separating two regions represent dynamic equilibrium between coexisting phases. Coexistence does not imply a fixed ratio of phase amounts.
Fusion Curve Slope: Most substances (e.g., $\ce{CO2}$ ) have a positive solid-liquid slope (denser solid), whereas $\ce{H2O}$ has a negative slope (denser liquid).
Process Paths: An isobaric path occurs at constant pressure (horizontal line), while an isothermal path occurs at constant temperature (vertical line).

Anatomy of Phase Diagrams and Stable Phases

A phase diagram is a graphical representation mapping the thermodynamically stable physical state of a substance as a function of pressure ( $P$ ) and temperature ( $T$ ). The diagram is partitioned into three main single-phase regions:

Solid: Stable at high pressure and low temperature (top-left region).
Liquid: Stable at intermediate temperature and high pressure (top-middle region).
Gas (Vapor): Stable at high temperature and low pressure (bottom-right region).

To identify the stable phase at a specific coordinate $(T, P)$ , locate the pressure on the vertical axis and temperature on the horizontal axis. The region in which the intersection lies corresponds to the phase with the lowest molar Gibbs free energy under those conditions.

A modification of File:Phase-diag.svg with an adjusted water solid-liquid coexistence: A typical phase diagram for a single-component material, exhibiting solid, liquid and gaseous phases. The solid green line shows the usual shape of the liquid-solid phase line. The dotted green line shows the anomalous behavior of water. (Source: me · CC BY-SA 3.0 · Wikimedia Commons)

Thermodynamic Pathways:

Isobaric path: Heating or cooling at constant pressure (horizontal line on the $P$ - $T$ plot).
Isothermal path: Compressing or decompressing at constant temperature (vertical line on the $P$ - $T$ plot).

Phase Boundary Lines and Two-Phase Equilibrium

The boundary lines separating physical regions mark conditions of two-phase equilibrium, where two phases coexist dynamically ( $\Delta G = 0$ ):

Sublimation Line: Boundary between solid and gas ( $\ce{solid <=> gas}$ ).
Fusion (Melting) Line: Boundary between solid and liquid ( $\ce{solid <=> liquid}$ ).
Vaporization Line: Boundary between liquid and gas ( $\ce{liquid <=> gas}$ ).

When a system lies directly on a boundary line, both coexisting phases are in dynamic equilibrium. Operating on a boundary line fixes the equilibrium pressure for a given temperature, but it does not dictate the relative molar amounts of the two phases present.

Slope of the Solid-Liquid Boundary

The slope of the fusion line ( $dP/dT$ ) depends on the relative density of the solid and liquid states:

Typical Substances (e.g., $\ce{CO2}$ ): Solid is denser than liquid ( $V_{\text{solid}} < V_{\text{liquid}}$ ). The fusion curve has a positive slope. Increasing pressure favors the denser solid state.
Anomalous Behavior ( $\ce{H2O}$ ): Solid ice is less dense than liquid water due to an open hydrogen-bonded crystal lattice ( $V_{\text{solid}} > V_{\text{liquid}}$ ). The fusion curve has a negative slope. Increasing pressure compresses ice into liquid water.

Identifying Stable Phases and Transitions: Worked Example

Problem: A sample of pure $\ce{CO2}$ is initially at $\pu{-80.0 degC}$ and $\pu{1.00 atm}$ . If the sample undergoes an isobaric process until its temperature reaches $\pu{25.0 degC}$ , determine the initial state, the final state, and the phase change that occurs. (Note: For $\ce{CO2}$ , the triple point pressure is $\pu{5.11 atm}$ at $\pu{-56.6 degC}$ ).

Step-by-Step Solution:

Identify Initial State: At $\pu{1.00 atm}$ and $\pu{-80.0 degC}$ , the pressure is below the triple point pressure ( $\pu{5.11 atm}$ ) and temperature is below $\pu{-56.6 degC}$ . The coordinate lies inside the solid region.
Trace the Pathway: An isobaric process holds pressure constant at $\pu{1.00 atm}$ while temperature increases horizontally from $\pu{-80.0 degC}$ to $\pu{25.0 degC}$ .
Locate Phase Boundary Crossing: Because $\pu{1.00 atm} < \pu{5.11 atm}$ , the horizontal line crosses the sublimation line ( $\ce{solid <=> gas}$ ) directly, bypassing the liquid region.
Identify Final State: At $\pu{25.0 degC}$ and $\pu{1.00 atm}$ , $\ce{CO2}$ resides in the gas region.

Answer: Initial state is solid, final state is gas, and the transition is sublimation ( $\ce{CO2(s) -> CO2(g)}$ ).

DAT Exam Trap: Do not assume every solid melts into a liquid upon heating! For substances like

\ce{CO2}

\pu{1 atm}

, ambient pressure is below the triple point pressure, causing solid dry ice to sublime directly into gas.

Phase Behavior Comparison

Property	Water ( $\ce{H2O}$ )	Carbon Dioxide ( $\ce{CO2}$ )
Fusion Curve Slope ( $dP/dT$ )	Negative ( $< 0$ )	Positive ( $> 0$ )
Relative Density	Liquid denser than solid	Solid denser than liquid
Isothermal Compression of Solid	Melts solid ice into liquid	Compresses solid into denser solid
Heating Solid at $\pu{1 atm}$	Melts to liquid, then boils to gas	Sublimes directly from solid to gas

Critical Thinking Questions

If a sample of ice at $\pu{-2.0 degC}$ and $\pu{1.0 atm}$ is subjected to an isothermal pressure increase to $\pu{500 atm}$ , what phase change occurs? How does this compare to $\ce{CO2}$ under identical isothermal compression?
Why does a two-phase boundary line fix the equilibrium pressure for a given temperature without specifying the relative ratio of the amounts of the two coexisting phases?
An unknown compound has a triple point at $\pu{2.5 atm}$ and $\pu{15 degC}$ . Describe the phase behavior observed when a solid sample at $\pu{-10 degC}$ is heated isobarically at $\pu{1.0 atm}$ to $\pu{30 degC}$ .

References

Sources returned by Google Search grounding during lesson verification.

Phase Diagrams - Chemistry LibreTexts — chem.libretexts.org
1.10: Phase Equilibria and Phase Diagrams - Chemistry LibreTexts / MIT OpenCourseWare — chem.libretexts.org

Reading Phase Diagrams

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