Properties of Pure Substances

Introduction

Thermodynamic analysis depends on knowing the state of a substance. Pressure, temperature, specific volume, internal energy, enthalpy, and entropy are called properties because their values depend only on the state, not on the path used to reach it. This chapter develops methods for identifying phases and evaluating properties of pure substances.

A pure substance has a fixed chemical composition throughout. Water, nitrogen, oxygen, and refrigerant R-134a are pure substances when each exists with uniform composition. A mixture of liquid water and water vapor is also a pure substance because both phases have the same chemical composition, H₂O. Air, by contrast, is generally treated as a mixture rather than a pure substance.

Property evaluation is fundamental in turbines, compressors, boilers, condensers, pumps, refrigeration systems, internal-combustion engines, and heat exchangers. The engineer must select an appropriate model: property tables, an ideal-gas equation, an incompressible approximation, a compressibility chart, or a real-fluid equation of state.

Learning Objectives

By the end of this chapter, the reader should be able to:

1. Identify solid, liquid, vapor, and supercritical regions on property diagrams.
2. Explain saturation, quality, critical-point behavior, and triple-point behavior.
3. Determine thermodynamic properties using tables and interpolation.
4. Apply the ideal-gas and incompressible-substance models within their limits.
5. Use reduced properties, compressibility factors, and generalized charts.
6. Describe and apply cubic equations of state, especially the van der Waals and Peng–Robinson models.
7. Assess the assumptions and uncertainty associated with property calculations.

1. Phases and Phase Equilibrium

A phase is a physically homogeneous portion of matter. The principal phases considered in engineering thermodynamics are solid, liquid, and vapor. A substance may undergo melting, vaporization, sublimation, condensation, freezing, or deposition.

A liquid and its vapor are in phase equilibrium when the rates of evaporation and condensation are equal. At equilibrium, the pressure and temperature are not independent in the two-phase region. For a specified saturation temperature, there is a corresponding saturation pressure, and vice versa.

The terms are:

• Saturated liquid: liquid about to vaporize.
• Saturated vapor: vapor about to condense.
• Compressed (subcooled) liquid: liquid at a temperature below the saturation temperature at its pressure, or at a pressure above the saturation pressure at its temperature.
• Superheated vapor: vapor at a temperature above the saturation temperature at its pressure.
• Two-phase mixture: simultaneous liquid and vapor phases in equilibrium.

The saturation temperature is written T_sat and the saturation pressure p_sat. Their relationship depends on the substance and is represented by a saturation curve.

Phase-change terminology

During vaporization at constant pressure for a pure substance, added energy first raises the liquid temperature until saturation is reached. Further energy changes liquid into vapor at nearly constant temperature and pressure. After all liquid has disappeared, additional energy raises the vapor temperature. Condensation follows the reverse sequence.

The latent heat of vaporization is

where h_f is the saturated-liquid specific enthalpy and h_g is the saturated-vapor specific enthalpy, in kJ/kg. Similar notation applies to specific volume, internal energy, and entropy.

2. Phase Diagrams

T–v and P–v diagrams

A temperature–specific-volume (T–v) or pressure–specific-volume (P–v) diagram displays the phase states of a substance. The liquid and vapor boundaries form a dome. The left boundary is the saturated-liquid line; the right boundary is the saturated-vapor line. Inside the dome is the two-phase liquid–vapor region.

The point at the top of the dome is the critical point, identified by critical temperature T_c, critical pressure p_c, and critical specific volume v_c. Above T_c, a vapor cannot be liquefied merely by increasing pressure. The region beyond the critical point is called the supercritical region.

Figure: Schematic pressure–specific-volume diagram for a pure substance.

For a fixed pressure below the critical pressure, heating at constant pressure moves the state from compressed liquid to saturated liquid, through the two-phase region, and then to superheated vapor. At pressures above the critical pressure, there is no distinct liquid–vapor phase boundary.

P–T diagram

A pressure–temperature diagram contains three phase boundaries: solid–liquid, liquid–vapor, and solid–vapor. These curves meet at the triple point, where solid, liquid, and vapor coexist in equilibrium. The liquid–vapor curve ends at the critical point. The slope of the solid–liquid line depends on the substance. Water is unusual because ice is less dense than liquid water, so its melting-line slope is negative over an important range.

Property surfaces

A three-dimensional p–v–T surface gives a complete geometric representation of equilibrium states. Projections of this surface produce P–v, T–v, and P–T diagrams. Property diagrams are conceptual tools: they help identify the phase and determine which variables can be independently specified.

3. Quality and Two-Phase Properties

The quality, or dryness fraction, x is the mass fraction of vapor in a saturated liquid–vapor mixture:

where m_g is vapor mass and m_f is liquid mass. Quality is defined only in the two-phase region. It is not meaningful for a compressed liquid or superheated vapor.

For any extensive or specific property y that is additive between phases,

where y_f and y_g are saturated-liquid and saturated-vapor values, and y_fg = y_g − y_f. Thus,

These relations assume thermodynamic equilibrium and negligible effects from surface tension, gravitational segregation, and nonequilibrium flow. They apply to specific volume v, internal energy u, enthalpy h, and entropy s of a saturated mixture.

The inverse relation is

provided y lies between y_f and y_g. A calculated value outside 0 ≤ x ≤ 1 indicates that the assumed two-phase state is impossible or that inappropriate property data were used.

Worked Example 1: Quality from specific volume

A water mixture at a specified pressure has v_f = 0.0011 m³/kg and v_g = 0.605 m³/kg. Its measured specific volume is 0.1219 m³/kg. Determine the quality.

Using the mixture equation,

x = (0.1219 − 0.0011)/(0.605 − 0.0011)

x = 0.1208/0.6039 ≈ 0.200.

Therefore, the vapor mass fraction is approximately 20.0%. The liquid mass fraction is 80.0%. Notice that the vapor occupies much more volume than the liquid; volume fraction and mass fraction are not the same.

4. Thermodynamic Property Tables

Property tables are experimentally based or computationally correlated data organized by temperature, pressure, or phase. Common tables include:

1. Saturated water tables by temperature or pressure.
2. Superheated-vapor tables.
3. Compressed-liquid tables.
4. Refrigerant and gas tables.
5. Tables for specific heats, enthalpy, entropy, and transport properties.

A reliable table procedure is:

1. Identify the two independent properties given.
2. Compare the state with saturation data.
3. Select the correct region: compressed liquid, saturated mixture, superheated vapor, or supercritical fluid.
4. Obtain the requested property and interpolate only between nearby tabulated values.
5. Check units, significant figures, and physical plausibility.

For linear interpolation between entries 1 and 2,

where z is the independent variable and y is the interpolated property. Linear interpolation is an approximation; it is most reliable over small intervals and where the property varies smoothly. Near the critical point, properties can change rapidly and interpolation may be inaccurate.

Enthalpy reference states

Absolute values of internal energy and enthalpy are not directly measurable; only differences affect most energy balances. Tables therefore select reference states. Different references may produce different numerical values of h or u, but consistent use of one source gives the same changes. Mixing data from incompatible references can cause errors, particularly for refrigerants and chemical systems.

5. Ideal-Gas Model

A gas behaves approximately as an ideal gas when intermolecular forces and molecular volume are negligible at the conditions of interest. The ideal-gas equation is

where p is absolute pressure in Pa, v is specific volume in m³/kg, T is absolute temperature in K, and R is the particular gas constant in J/(kg·K). For a mass m,

where V is total volume in m³. The universal form is

where n is amount of substance in kmol and R_u = 8.314 kJ/(kmol·K), when pV has units kJ. The particular gas constant is R = R_u/M, where M is molar mass in kg/kmol.

For a fixed mass of ideal gas,

This relation requires constant composition and absolute temperature and pressure. It does not apply accurately near saturation, near the critical point, or at high density.

For an ideal gas, internal energy and enthalpy depend primarily on temperature:

If specific heats are treated as constant,

and

The ratio of specific heats is k = c_p/c_v. Constant-specific-heat formulas are convenient but should not be used over large temperature ranges without checking property variation.

Worked Example 2: Ideal-gas density

Nitrogen occupies 0.50 m³ at 500 kPa absolute and 300 K. Take R = 0.2968 kJ/(kg·K), with pressure in kPa and volume in m³ so that kPa·m³ = kJ. Determine its mass.

From pV = mRT,

m = pV/(RT) = (500 × 0.50)/(0.2968 × 300)

m = 250/89.04 ≈ 2.81 kg.

The use of absolute pressure and temperature is essential. Gauge pressure or Celsius temperature would produce an invalid result.

6. Incompressible-Substance Model

Liquids and solids are often modeled as incompressible when pressure-induced volume changes are small. The approximation is

For a simple compressible substance, the differential property relation is

With dv ≈ 0, changes in internal energy are mainly temperature-dependent. If the specific heat is approximately constant,

For enthalpy, h = u + pv, so

where c is the relevant specific heat, usually c_p for liquids, in kJ/(kg·K), and v is in m³/kg. Since 1 kPa·m³ = 1 kJ, the pressure term is dimensionally consistent when pressure is in kPa.

For an incompressible liquid in a pump, the idealized specific work input is approximately

This model is useful for water and hydraulic oils over moderate pressure ranges, but it becomes less reliable at very high pressures, near phase transitions, or when temperature changes are large.

7. Real-Gas Behavior and the Compressibility Factor

Real gases deviate from ideal behavior because molecules occupy finite volume and exert attractive or repulsive forces. The compressibility factor is defined by

For an ideal gas, Z = 1. Values below one commonly indicate that attractive forces dominate; values above one commonly indicate that repulsive effects and excluded volume dominate.

The corresponding real-gas equation is

To estimate Z for many gases, use reduced properties:

where T_c and p_c are the critical temperature and pressure. Generalized compressibility charts provide Z as a function of T_r and p_r. They are approximate and are most useful for nonpolar or mildly polar gases. For strongly polar substances, acentric-factor corrections or a substance-specific equation of state may be required.

The acentric factor ω measures the non-sphericity and vapor-pressure behavior of a molecule and improves corresponding-states correlations. In engineering practice, chart precision is often limited to a few percent, so the result should not be reported with excessive significant figures.

8. Cubic Equations of State

Cubic equations of state express pressure as a function of temperature and molar volume and can represent both vapor and liquid behavior. They are useful when tables are unavailable and phase-equilibrium calculations are required.

van der Waals equation

The classical van der Waals equation is

where v_m is molar volume, a represents attractive forces, b represents finite molecular volume, and R_u is the universal gas constant. It improves qualitatively on the ideal-gas model but is often quantitatively inaccurate for engineering property calculations.

Peng–Robinson equation

A widely used model for hydrocarbon and refrigerant systems is the Peng–Robinson equation:

Here a and b are substance parameters, and α is a temperature-dependent correction involving the acentric factor. In practical calculations, the equation is converted to a cubic polynomial in the compressibility factor Z. The largest real root generally corresponds to a vapor-like state; the smallest root generally corresponds to a liquid-like state. Near phase equilibrium, root selection must be combined with a phase-stability or fugacity criterion rather than applied blindly.

Cubic equations of state require parameter data and numerical solution. Their strengths include broad applicability, prediction of phase behavior, and integration into process simulators. Their limitations include imperfect liquid-density predictions, sensitivity to binary-interaction parameters, and reduced accuracy for highly polar or associating fluids such as water and alcohols unless specialized models are used.

9. Selecting a Property Model

A disciplined selection procedure is more important than memorizing a single model.

Computer programs and databases are valuable, but the engineer remains responsible for selecting the correct fluid, units, reference state, phase, and property model. Automated software can return a mathematically valid number for a physically inappropriate input.

10. Applied Case: Condenser Design Check

Background

Consider an illustrative steam-power condenser receiving exhaust steam at 10 kPa. The condenser is intended to deliver saturated liquid water to a pump. The designer must determine the inlet phase and the heat rejected per kilogram.

Problem

The exhaust enters as a saturated mixture with quality x = 0.90. At 10 kPa, use representative table values h_f = 191.8 kJ/kg and h_fg = 2,392.8 kJ/kg. Determine the inlet enthalpy and the heat rejected per kilogram if the outlet is saturated liquid.

Analysis and solution

The inlet is inside the two-phase dome because 0 < x < 1. Its enthalpy is

h_1 = h_f + xh_fg

h_1 = 191.8 + (0.90)(2,392.8)

h_1 = 191.8 + 2,153.5 = 2,345.3 kJ/kg.

The outlet saturated-liquid enthalpy is h_2 = 191.8 kJ/kg. For a steady-flow condenser with negligible changes in kinetic and potential energy and no shaft work, the heat transfer per unit mass is

q = h_2 − h_1 = 191.8 − 2,345.3 = −2,153.5 kJ/kg.

The negative sign indicates heat leaves the control volume. Thus, the condenser rejects approximately 2,154 kJ per kilogram of inlet mixture.

Outcome and learning points

The quality relation provides a direct property estimate, but only because the phase was correctly identified. Using superheated-vapor data would be inappropriate. In a real design, pressure drop, noncondensable gases, subcooling, fouling, and nonequilibrium effects would also require analysis. The result demonstrates why phase identification precedes numerical table lookup.

11. Limitations, Safety, and Professional Practice

Property errors can produce unsafe equipment designs. Underestimating vapor specific volume can undersize piping; using an incorrect saturation pressure can cause unexpected flashing; and confusing absolute and gauge pressure can invalidate an equation-of-state calculation. Engineers should use traceable property sources, document the selected reference state, report units, and perform an independent magnitude check.

For refrigerants and industrial fluids, environmental and safety requirements influence property selection and equipment design. Flammability, toxicity, global-warming potential, pressure rating, and regulatory restrictions must be considered alongside thermodynamic performance. Process simulators and AI-assisted tools can accelerate interpolation, regression, and model selection, but they do not replace validation against authoritative databases or experimental data. A responsible workflow includes version-controlled property data, test cases, uncertainty estimates, and review by a qualified engineer.

Chapter Summary

A pure substance has uniform chemical composition, even when it contains liquid and vapor phases. Phase behavior is represented using saturation curves, phase diagrams, and the critical and triple points. Saturated liquid, saturated vapor, compressed liquid, superheated vapor, and two-phase mixture states must be distinguished before selecting property data.

Quality is the vapor mass fraction in a saturated mixture and permits calculation of mixture specific volume, internal energy, enthalpy, and entropy. Property tables remain the most reliable method for many water, refrigerant, and two-phase calculations. Interpolation must be performed carefully and consistently.

The ideal-gas model uses pV = mRT and is appropriate when density is low and the state is far from saturation. The incompressible model is useful for liquids and solids when volume changes are negligible. Real-gas effects are represented using the compressibility factor Z, generalized charts, or cubic equations of state. Model selection must account for pressure, temperature, polarity, proximity to the critical point, and the intended accuracy.

Key Terms

Pure substance: Matter with uniform chemical composition.

Phase: A physically homogeneous form of matter.

Saturation: The condition at which liquid and vapor can coexist in equilibrium.

Saturated liquid: Liquid about to vaporize.

Saturated vapor: Vapor about to condense.

Compressed liquid: Liquid at a state below its saturation condition.

Superheated vapor: Vapor above its saturation temperature at a given pressure.

Quality: Vapor mass fraction in a saturated mixture.

Latent heat: Energy associated with a phase change at saturation.

Critical point: Endpoint of the liquid–vapor equilibrium curve.

Critical temperature: Temperature above which pressure alone cannot liquefy a vapor.

Triple point: State where solid, liquid, and vapor coexist.

Property table: Tabulated thermodynamic data for specified states.

Ideal gas: Model neglecting molecular volume and intermolecular forces.

Incompressible substance: Model in which specific volume is treated as constant.

Compressibility factor: Z = pv/(RT), measuring deviation from ideal-gas behavior.

Reduced temperature: Temperature divided by critical temperature.

Equation of state: Relation among pressure, volume, temperature, and composition.

Cubic equation of state: Equation capable of producing a cubic relation in volume or compressibility factor.

Acentric factor: Parameter used to improve corresponding-states correlations.

Review Questions

Short Answer Questions

1. What is meant by a pure substance, and why can a liquid–vapor mixture still be pure?
2. Define saturated liquid, saturated vapor, compressed liquid, and superheated vapor.
3. Why is quality defined only in the two-phase region?
4. What physical meaning does Z = 1 have?
5. State two limitations of the ideal-gas model.

Long Answer Questions

1. Explain the construction and interpretation of a P–v phase diagram, including the critical point and two-phase region.
2. Derive the mixture-property relation y = y_f + xy_fg from mass-weighted phase averaging.
3. Compare ideal-gas, incompressible-substance, compressibility-chart, and cubic-EOS models.
4. Explain a systematic procedure for using thermodynamic property tables.
5. Discuss why reference states and unit consistency are important in property calculations.

Analytical / Application Questions

1. A saturated water mixture has v = 0.050 m³/kg, v_f = 0.001 m³/kg, and v_g = 0.500 m³/kg. Determine its quality and interpret the result.
2. A gas is at T_r = 1.5 and p_r = 2.0. Explain how a generalized compressibility chart would be used and identify factors that could limit its accuracy.
3. A pump raises the pressure of water from 100 kPa to 2,000 kPa. Explain when the incompressible pump-work approximation is appropriate and identify additional effects that a real design should consider.

Multiple Choice Questions with Answers

Question 1

Which state lies inside the liquid–vapor dome on a phase diagram?

A. Compressed liquid

B. Two-phase mixture

C. Superheated vapor

D. Supercritical fluid

Correct Answer: B. Two-phase mixture

Question 2

Quality is defined as the ratio of

A. Vapor volume to total volume

B. Liquid mass to vapor mass

C. Vapor mass to total mixture mass

D. Vapor enthalpy to liquid enthalpy

Correct Answer: C. Vapor mass to total mixture mass

Question 3

A saturated mixture has x = 0.25. What fraction of its mass is liquid?

A. 0.25

B. 0.50

C. 0.75

D. 1.25

Correct Answer: C. 0.75

Question 4

For an ideal gas, the compressibility factor Z is

A. 0

B. Less than zero

C. 1

D. Equal to the critical pressure

Correct Answer: C. 1

Question 5

Which equation is the ideal-gas relation for a fixed mass?

A. pV = constant at all temperatures

B. p₁V₁/T₁ = p₂V₂/T₂

C. p₁T₁/V₁ = p₂T₂/V₂

D. pV = Z/R T

Correct Answer: B. p₁V₁/T₁ = p₂V₂/T₂

Question 6

A quality value of x = 1 represents

A. Saturated liquid

B. A 50% liquid mixture

C. Saturated vapor

D. Compressed liquid

Correct Answer: C. Saturated vapor

Question 7

The critical point is the point at which

A. Solid and liquid always have equal density

B. The liquid–vapor equilibrium curve ends

C. All substances become ideal gases

D. Quality is always zero

Correct Answer: B. The liquid–vapor equilibrium curve ends

Question 8

Which model is generally most appropriate for a liquid with a modest pressure change?

A. Incompressible-substance model

B. Ideal-gas model

C. Vacuum model

D. Radiation model

Correct Answer: A. Incompressible-substance model

Question 9

A real gas has p = 500 kPa, v = 0.50 m³/kg, R = 0.30 kJ/(kg·K), and T = 300 K. Its compressibility factor is approximately

A. 0.28

B. 1.00

C. 2.78

D. 5.00

Correct Answer: B. 1.00

Question 10

Which statement about quality is correct?

A. It applies to every gas state

B. It is a vapor mass fraction, not a volume fraction

C. It must always exceed one

D. It is identical to relative humidity

Correct Answer: B. It is a vapor mass fraction, not a volume fraction