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Pure Substances and Phases

A pure substance is homogeneous and invariable in chemical composition. It may exist in more than one phase, but the chemical composition is the same in all phases.
Examples: water, nitrogen, helium, and carbon dioxide.
Notes from slides:
- Nitrogen and gaseous air are considered pure substances.
- A mixture of liquid and gaseous water is a pure substance, but a mixture of liquid and gaseous air is not due to absence of CO₂.
Source: Cengel & Boles, Thermodynamics - An Engineering Approach, 8th Edition.

Phase Concepts and Phase Identification

At room temperature and pressure, copper is a solid, mercury is a liquid, and nitrogen is a gas. Under other conditions, each may appear in a different phase.
A phase is identified as having a distinct molecular arrangement that is homogeneous throughout and separated from others by boundary surfaces.
Phases of a Pure Substance
- Pure substance may exist in multiple phases with the same chemical composition.
- The arrangement of atoms/molecules differs between phases (solid, liquid, vapor).

Phase Change Processes and Coexistence of Phases

Many practical situations involve two phases of a pure substance coexisting in equilibrium:
- Water exists as a mixture of liquid and vapor in boiler and condenser of a steam power plant.
- Refrigerant in a freezer undergoes liquid↔vapor changes.
In a piston–cylinder device containing liquid water at 20 °C and 1 atm, heating can drive the system through the following states:
1) Compressed/sub-cooled liquid; 2) Saturated liquid; 3) Saturated liquid–vapor mixture; 4) Saturated vapor; 5) Superheated vapor.
Phase change processes are represented on phase diagrams with boundaries between regions of different phases.

Saturation Concepts

For a given pressure, the saturation temperature Tsat is the temperature at which a pure substance changes phase.
For a given temperature, the saturation pressure Psat is the pressure at which a pure substance changes phase.
A state at which a phase change begins or ends is called a saturation state.
Saturation states terminology:
- Compressed or sub-cooled liquid: liquid not about to vaporize.
- Saturated liquid: liquid about to vaporize.
- Saturated liquid–vapor mixture: liquid and vapor coexist in equilibrium.
- Saturated vapor: vapor about to condense.
- Superheated vapor: vapor at a temperature higher than the saturation temperature for a given pressure.
When a liquid is about to vaporize, it lies on the saturated liquid line; when a vapor is about to condense, it lies on the saturated vapor line.
Example: Water boils at 100 °C at 1 atm (~100 kPa). If the pressure is raised to 500 kPa, boiling occurs at 151.8 °C.

Temperature–Pressure (Tsat–Psat) and Phase Boundaries

The Tsat–Psat relationship forms the saturated line on a P–T diagram for a substance.
The P–v diagram for a pure substance has a saturated liquid line and a saturated vapor line; the region between these lines is the two‑phase (saturated liquid–vapor) region.
A vapor–liquid mixture lies inside the saturation curve between the saturated liquid line and the saturated vapor line.
Sublimation and sublimation lines exist for solid–vapor transitions (e.g., dry ice CO₂ subliming to CO₂ gas).
Ice at −20 °C, 100 kPa and ice at 0 °C, 100 kPa illustrate solid–vapor and solid–liquid boundaries on the P–T diagram.

T–v and P–v Diagrams for Pure Substances

The general shape of the P–v diagram for a pure substance is similar to the T–v diagram, but lines of constant temperature in the P–v diagram slope downward.
T–v Diagram for Water shows a point of inflection with zero slope on the saturation curve.
At supercritical pressures (P > Pcr), there is no distinct phase-change process (no separated liquid and vapor phases).

Saturation Diagram Features: Triple Point and Critical Point

Under certain conditions, all three phases (solid, liquid, vapor) coexist in equilibrium. On P–v or T–v diagrams, such triple-phase states form a line called the triple line; on a P–T diagram, this point is the triple point.
States on the triple line have the same temperature and pressure but different specific volumes.
The point at which saturated liquid and saturated vapor states are identical is called the critical point.
Summary:
- Triple line/point = three phases coexistence at same P and T.
- Critical point = liquid and vapor phases become indistinguishable.

Saturation States and Phase-Change Boundaries (Quiz-style Examples)

Determine the phase of water at:
- (a) T = 260 °C, P = 5 MPa
- (b) T = −20 °C, P = 100 kPa
Answers (from slides):
- (a) Compressed liquid
- (b) Compressed solid
Important data: Saturation values (example)
- Psat at 0 °C ≈ 798.7 kPa
- Psat at −10 °C ≈ 290.9 kPa
- Psat at 100 °C ≈ 101.3 kPa
Additional data: At 10 MPa, vf ≈ 0.001452 m³/kg and vg ≈ 0.01803 m³/kg (for certain substances in superheated/vapor regions).

Mixtures, Quality, and Properties of Saturated Mixtures

During vaporization, a substance exists as a mixture of saturated liquid and saturated vapor.
The quality x is defined as the ratio of the mass of vapor to the total mass of the mixture:
- $x = \frac{m{ ext{vapor}}}{m{ ext{total}}}$
- $mf = (1 - x)\, m{ ext{total}}$
- $mg = x\, m{ ext{total}}$
The specific volumes of the saturated liquid and saturated vapor are vf and vg, respectively. The total volume is V = Vf + Vg, with
- $Vf = mf v_f$
- $Vg = mg v_g$
The volume fraction of vapor is:
- $ext{Volume fraction of vapor} = \frac{Vg}{V} = \frac{Vg}{Vf+Vg} = \frac{x vg}{(1 - x) vf + x v_g}$
Quality is meaningful only in the saturated liquid–vapor region; it has no meaning in compressed liquid or superheated vapor regions.
In saturated mixtures, quality can serve as one of two independent intensive properties needed to describe a state. The properties of the saturated liquid are the same whether isolated or in a mixture with saturated vapor.
During vaporization, only the amount of saturated liquid changes; the properties of the liquid remain the same; similarly for the saturated vapor.
A saturated mixture can be treated as the combination of two subsystems: saturated liquid and saturated vapor. The mixture properties are the weighted averages:
- $\bar{z} = (1 - x) zf + x zg$
- $\bar{v} = (1 - x) vf + x vg$
- Additional equivalent forms:
- $\bar{z} = zf + x\bigl(zg - z_f\bigr)$
- $\bar{v} = vf + x\bigl(vg - v_f\bigr)$

Thermodynamic Tables and Data Representation

For most substances, thermodynamic properties are too complex for simple equations, so properties are presented in tables.
Some properties are easy to measure; others require relations between measurable and non-measurable properties.
Tables are extensive and region-specific: separate tables exist for:
- Superheated vapor
- Compressed liquid
- Saturated (mixture) region
Source: Moran et al., Fundamentals of Engineering Thermodynamics, 8th Edition, and related texts.

Saturated and Phase-Behavior Tables (Saturation Data and Examples)

Saturated Water Tables
- Table B.1.1 – Saturated Water (data at various Psat)
- Table B.1.2 – Saturated Water (Pressure table for different Tsat)
- Table B.1.3 – Superheated Vapor (Water)
- Table B.1.4 – Compressed Liquid Water
Example from Table B.1.2 (Saturated Water, Pressure):
- Compute specific volume of saturated steam at 200 kPa with quality x = 70% (0.70):
- Given: vf = 0.001061 m³/kg, vg = 0.88467 m³/kg, x = 0.70
- The saturated mixture average volume is:
- v{ ext{avg}} = (1 - x) vf + x v_g = (1 - 0.70)(0.001061) + 0.70(0.88467) \
  = 0.0003183 + 0.620269 \
= 0.6206\ ext{m}^3/ ext{kg}

Ideal Gas Law and Compressibility

The ideal-gas equation of state is:
- $P v = R T$
- where R is the specific gas constant.
Not all substances behave as ideal gases; to quantify deviations, we use the compressibility factor Z:
- $Z = \frac{P v}{R T}$
- For an ideal gas, $Z = 1$ .
Practical notes:
- At all temperatures, Z → 1 as P → 0 (low-density limit).
- At temperatures around 300 K and above, Z is near unity up to about 10 MPa for many gases.
- At lower temperatures or very high pressures, Z deviates significantly from 1, especially near the critical point and the saturated vapor line.
Practical usage: ideal gas approximation is often accurate for psychrometric and air-conditioning calculations at moderate pressures.
Some terms:
- Gas vs vapor: Gas is typically used for substances above their critical temperature; Vapor refers to a gas-like phase near condensation.
- The ideal-gas relation is the basis for many simple thermodynamic calculations, but corrections are needed for high pressures or low temperatures.

Other Equations of State (Overview)

While the ideal-gas equation is simple and useful, several more accurate equations of state exist for broader ranges:
- Van der Waals equation of state (accounts for intermolecular attraction and finite molecular size).
- Beattie– Bridgeman equation of state (parameters tabulated for various substances; shown in Table 3–4).
- Benedict–Webb–Rubin equation of state (parameters tabulated for various substances; shown in Table 3–4).
- Virial equation of state (coefficients a(T), b(T), c(T), … are functions of temperature).
Note: The van der Waals model includes two effects not in the ideal-gas model (attractive forces and molecular finite size), but its accuracy is often limited.
Table 3–4 provides constants for Beattie– Bridgeman and Benedict–Webb– Rubin equations for various gases (e.g., Air, Ar, CO₂, He, H₂, N₂, O₂). These constants enable calculation of P–v–T behavior over larger regions than an ideal gas model.
In addition to these, other commonly used equations of state exist for chemical engineering calculations.

Practical Data and Examples from the Slides

Saturated and compressed liquid/vapor data are provided in tables (e.g., Table B.1.x in Borgnakke & Sonntag / Moran et al.).
Example data points:
- Water at 100 kPa with a quality of 10%:
- vf = 0.001043 m³/kg, vg = 1.6940 m³/kg
- Volume fraction of vapor computed as:
- ext{Volume fraction of vapor} = rac{Vg}{Vf+Vg} = rac{x vg}{(1 - x) vf + x vg} = rac{0.1 imes 1.694}{(1-0.1) imes 0.001043 + 0.1 imes 1.694} \
  = rac{0.1694}{0.17034} \
≈ 0.9945
- Thus, about 99.45% of the total volume is vapor; liquid volume is about 0.5%.
For water at 100 kPa, Psat ≈ 101.3 kPa; for 0 °C, Psat ≈ 798.7 kPa; for −10 °C, Psat ≈ 290.9 kPa; for 45.81 °C, Psat ≈ 10 kPa (values given in slides).

Quick Practice and Concept Check

Define a pure substance and explain when a mixture can still be a pure substance.
List the five states of a pure substance in the two-phase region and describe each.
Explain what the saturation temperature Tsat and saturation pressure Psat mean.
Define quality x and give its physical meaning in a saturated mixture.
Derive the formula for the mixture property:
- $\bar{z} = (1 - x) zf + x zg$
- And for the mixture volume:
- $\bar{v} = (1 - x) vf + x vg$
Explain why quality is not defined in compressed liquid or superheated vapor regions.
What is the critical point and how does it differ from the triple point?
What is the compressibility factor Z and what does Z ≈ 1 imply for a gas?
Briefly describe the Van der Waals equation and why it was introduced.

Data References (Indicative)

Saturated water tables: Table B.1.1, Table B.1.2, Table B.1.3, Table B.1.4.
Data examples: vf and vg values at specific Psat (e.g., vf ≈ 0.001061 m³/kg, vg ≈ 0.88467 m³/kg at 200 kPa; vf ≈ 0.001043 m³/kg, vg ≈ 1.6940 m³/kg at 100 kPa).
Critical, triple points and phase boundaries are discussed in standard texts (Moran et al.; Borgnakke & Sonntag; Cengel & Boles).

References and Sources Mentioned in the Slides

Cengel & Boles, Thermodynamics - An Engineering Approach, 8th Edition.
Moran et al., Fundamentals of Engineering Thermodynamics, 8th Edition.
Borgnakke & Sonntag, Fundamentals of Thermodynamics, 8th Edition.
Beattie-Bridgeman and Benedict-Webb-Rubin equations of state (Table 3–4 references).
Beattie-Bridgeman constants table and Benedict-Webb-Rubin constants table provide gas-specific constants for P–v–T calculations.