Thermodynamics · Chapter 3
Thermodynamics · Chapter 3

Properties of Pure Substances

To apply the energy balance we need property values. This chapter shows how a pure substance moves between phases, how that behavior is mapped on property diagrams, and how to read tables and the ideal-gas law to get the numbers.

Learning objectives

  • Identify pure-substance phases from independent state data.
  • Navigate saturation, superheated, and compressed-liquid property sources.
  • Use quality only inside the equilibrium two-phase region.
  • Interpolate tabulated properties without crossing a phase boundary.
  • Compare ideal-gas and real-gas equations on mass or mole bases.
  • Verify phase bounds, cycle closure, conservation, and model uncertainty.
3–1

Pure Substance

Fixed chemical composition

A pure substance has the same chemical composition throughout. Separate liquid and vapor phases of water still form one pure substance; air remains a useful pure-substance model while its composition is fixed and no constituent separates.

Physical meaning

“Pure” describes composition, not the number of phases. This distinction lets one set of equilibrium property data represent coexisting liquid and vapor.

3–2

Phases of a Pure Substance

Molecular structure and phase

Solids resist shape change, liquids retain nearly fixed volume, and gases expand to fill a container. Engineering phase labels come from equilibrium property behavior; metastable or nonequilibrium states require additional judgment and cannot be read blindly from equilibrium tables.

3–3

Phase-Change Processes of Pure Substances

Saturation links pressure and temperature

At liquid–vapor equilibrium, specifying saturation pressure fixes saturation temperature. Compressed liquid lies on the liquid side, superheated vapor on the vapor side, and latent energy changes phase at saturation without requiring a temperature rise.

Saturation dependence
\[P_{sat}=P_{sat}(T),\qquad T_{sat}=T_{sat}(P)\]
EXAMPLE 1Superheated refrigerant table interpolation
A sealed sample of R-134a is at 800 kPa absolute and 45.0 °C. Supplied authoritative table data give \(T_{sat}=31.3^\circ\text{C}\), \(h=275.3\ \text{kJ/kg}\) at 40 °C, and \(h=284.1\ \text{kJ/kg}\) at 50 °C, all at 800 kPa. Identify the phase and estimate enthalpy.

Recognition strategy

Pressure and temperature first classify the phase. Because temperature exceeds saturation at the same absolute pressure, use the superheated table and interpolate between bracketing temperatures rather than applying quality.

Given information

  • Closed fixed-mass sample at one equilibrium state; no process or cycle.
  • \(P=800\ \text{kPa absolute}\), \(T=45.0^\circ\text{C}=318.15\ \text{K}\), and supplied \(T_{sat}=31.3^\circ\text{C}\).
  • Mass-specific basis; supplied superheated-table enthalpies at 40 and 50 °C; no ideal-gas assumption.

Model and interpretation

Compare temperature at fixed pressure, then linearly interpolate \(h\). Table interpolation is approximate between tabulated states; the source data are givens, not invented. Heat, work, KE, PE, entropy generation, exergy reference, and efficiencies do not apply to this single state.

Step-by-step solution
1. Classify the state: \(45.0>31.3^\circ\text{C}\), so the sample is superheated vapor, not a saturated mixture.
2. Form the interpolation fraction \(f=(45-40)/(50-40)=0.500\).
3. Interpolate \(h=275.3+f(284.1-275.3)=279.7\ \text{kJ/kg}\).
4. Confirm quality is not calculated: \(x\) is undefined in this single-phase superheated region.

Verification

Units: The interpolation fraction is dimensionless and both endpoint properties are kJ/kg, so the result retains kJ/kg.

Signs and direction: Enthalpy rises with temperature at fixed pressure, and the positive interpolation increment follows that physical direction.

Plausibility: At the midpoint temperature, 279.7 kJ/kg is exactly between 275.3 and 284.1, providing an independent interpolation check.

Final result: Superheated vapor; \(h\approx279.7\ \text{kJ/kg}\) from the supplied table interpolation.

Engineering interpretation

Phase classification controls the property source. For release work, cite the refrigerant formulation and version, avoid interpolation across phase boundaries, and propagate pressure and temperature uncertainty because a state near saturation can change region.

3–4

Property Diagrams for Phase-Change Processes

Read regions before values

The saturation dome separates compressed liquid, two-phase mixture, and superheated vapor. Saturated liquid and vapor form its boundaries; they merge at the critical point. Above the critical point there is no liquid–vapor boundary, so quality is meaningless.

vTcomp.liquidwet mixturesuperheatedcritical pointfg
Figure 3–1  A T–v diagram: the dome separates compressed liquid (left), the wet liquid–vapor mixture (under the dome), and superheated vapor (right).
3–5

Property Tables

Two-phase lever rule
\[y=y_f+x(y_g-y_f)=y_f+xy_{fg},\qquad 0\le x\le1\]

A disciplined lookup sequence

Write the substance, absolute pressure, absolute temperature, and known second property. Classify phase, select the matching table, bracket the state, and interpolate only within one region. Record the source, edition, endpoint values, and interpolation fraction.

Physical meaning

Quality is vapor mass divided by total mass, not vapor volume fraction. Because \(v_g\) often greatly exceeds \(v_f\), a small vapor mass may occupy most of a vessel.

EXAMPLE 2Two-phase water inventory in a rigid receiver
A rigid 0.900 m³ receiver contains 3.00 kg of equilibrium water at 200 kPa absolute. Supplied saturated-water data are \(v_f=0.001061\ \text{m}^3/\text{kg}\), \(v_g=0.8857\ \text{m}^3/\text{kg}\), \(u_f=504.5\ \text{kJ/kg}\), and \(u_g=2529.1\ \text{kJ/kg}\). Find quality and total internal energy.

Recognition strategy

A rigid volume and mass give specific volume. If that value lies between saturated endpoints at the stated pressure, the state is a mixture and the lever rule determines quality and internal energy.

Given information

  • Closed receiver at one equilibrium state; \(V=0.900\ \text{m}^3\), \(m=3.00\ \text{kg}\), and \(P=200\ \text{kPa absolute}\).
  • Authoritative saturated-table values are explicitly supplied as givens at the same pressure.
  • Mass basis; exact volume inventory with approximate tabular values; KE and PE neglected because the receiver is stationary.

Model and interpretation

The tank wall is the boundary. Compute \(v=V/m\), classify against \(v_f,v_g\), then apply \(y=y_f+x(y_g-y_f)\). Quality is a mass fraction valid only under the saturation dome. This state calculation includes no heat, work, entropy generation, exergy datum, or efficiency.

Step-by-step solution
1. Compute the measured specific volume: \(v=0.900/3.00=0.3000\ \text{m}^3/\text{kg}\).
2. Since \(v_f
3. Find specific internal energy: \(u=504.5+0.3379(2529.1-504.5)=1188.6\ \text{kJ/kg}\).
4. Convert to the extensive inventory: \(U=mu=(3.00)(1188.6)=3565.8\ \text{kJ}\).

Verification

Units: The volume lever rule uses only m³/kg, quality is dimensionless, and kg times kJ/kg correctly yields total kJ.

Signs and direction: Quality is positive and below one; positive vapor mass and an internal energy above \(u_f\) follow the correct direction.

Plausibility: Both \(v\) and \(u\) fall between saturated endpoints; vapor mass \(xm=1.014\ \text{kg}\) is independently plausible.

Final result: \(x=0.338\) and \(U\approx3.57\times10^3\ \text{kJ}\).

Engineering interpretation

Although vapor occupies most of the receiver, it is only about one-third of the mass because vapor specific volume is much larger. Vessel decisions require traceable steam tables, equilibrium justification, volume calibration, and relief-code review.

3–6

The Ideal-Gas Equation of State

Mass and mole forms
\[PV=mRT,\qquad P\bar V=N R_uT,\qquad R=\frac{R_u}{M}\]

Model rather than identity

The ideal-gas law is accurate when intermolecular effects are small, generally at low reduced pressure and sufficiently high reduced temperature. Use absolute pressure and temperature and do not apply it through a condensation region. Keep mass and mole bases consistent.

3–7

Compressibility Factor

Real-gas correction and reduced variables
\[Z=\frac{Pv}{RT},\qquad P_r=\frac{P}{P_c},\qquad T_r=\frac{T}{T_c}\]

Quantify ideal-model departure

A value of \(Z\) near one supports the ideal-gas density model at that state; it does not guarantee accurate enthalpy or entropy. Obtain \(Z\) from an authoritative chart or equation of state for the actual composition, interpolating or iterating consistently.

EXAMPLE 3Real-gas nitrogen storage estimate
A 12.0 m³ nitrogen buffer is at 8.00 MPa absolute and 320 K. A validated vendor equation of state supplies \(Z=0.910\) and \(R=0.2968\ \text{kJ/(kg\!\cdot\!K)}\) at this state. Determine real-gas mass, compare the ideal-gas estimate, and report ideal-model error.

Recognition strategy

The high absolute pressure signals a real-gas check. Use the supplied compressibility factor in \(PV=ZmRT\), then set \(Z=1\) only for the comparison model.

Given information

  • Closed rigid buffer at one state; \(V=12.0\ \text{m}^3\), \(P=8000\ \text{kPa absolute}\), and \(T=320\ \text{K absolute}\).
  • Mass basis; validated vendor-EOS givens \(Z=0.910\) and \(R=0.2968\ \text{kJ/(kg\!\cdot\!K)}\).
  • No process or cycle; stationary storage makes KE and PE irrelevant.

Model and interpretation

Use the real-gas equation as the primary approximate property model and ideal gas as a screening comparison. The control mass is the nitrogen inside the wall. No property is interpolated here. Heat, work, entropy generation, exergy environment, and component efficiency require a process and are not assigned.

Step-by-step solution
1. Solve the real-gas relation symbolically: \(m_{real}=PV/(ZRT)\).
2. Evaluate \(m_{real}=(8000)(12.0)/[(0.910)(0.2968)(320)]=1110.7\ \text{kg}\).
3. Set \(Z=1\) for the ideal comparison: \(m_{ideal}=96000/[(0.2968)(320)]=1010.8\ \text{kg}\).
4. Compute signed relative error \((m_{ideal}-m_{real})/m_{real}=-0.0900\), an underprediction of 9.00%.

Verification

Units: With kPa·m³ equal to kJ, division by kJ/kg leaves kg; the relative model error is dimensionless.

Signs and direction: Because \(Z<1\), the denominator is smaller and real-gas mass exceeds ideal mass, matching the negative ideal-model error sign.

Plausibility: The ratio \(m_{ideal}/m_{real}=Z=0.910\) independently confirms exactly 9.00% underprediction.

Final result: \(m_{real}=1.111\times10^3\ \text{kg}\); ideal gas underpredicts by \(9.00\%\).

Engineering interpretation

A nine-percent inventory error is operationally significant. Certified storage accounting must use the approved composition-specific EOS, instrument calibration and uncertainty, vessel volume tolerance, and pressure-vessel requirements; the supplied \(Z\) cannot be reused at another state.

3–8

Other Equations of State

Representative cubic model
\[\left(P+\frac{a}{v^2}\right)(v-b)=RT\]

Selection and iteration

Real-fluid equations may have multiple volume roots below the critical temperature. Phase stability determines whether a liquid root, vapor root, or two-phase equilibrium calculation is physical. Iterate with stated tolerances and reject roots that violate phase bounds or continuity.

Problem-solving strategy

  1. Identify substance and composition, then express pressure and temperature on absolute bases.
  2. Locate the state region before choosing a saturation, compressed-liquid, superheated, ideal-gas, or real-gas source.
  3. Choose mass or mole basis and list every supplied or traceable property value with units.
  4. Interpolate only between bracketing states in one region; iterate an EOS with root and phase checks.
  5. Audit quality bounds, table trends, conservation, entropy generation, uncertainty, and cycle closure.

Chapter recap

  • Classify phase before opening a table or equation of state.
  • Saturation pressure and temperature are linked; quality is valid only in a two-phase equilibrium mixture.
  • Interpolation requires bracketing states from the same phase region and source.
  • Ideal gas is an approximation; a supplied real-gas model quantifies departure at a stated condition.
  • For processes, conserve mass and energy, require nonnegative entropy generation, and verify cycle closure.

Before moving on

You should be able to identify a state, choose and document its property source, interpolate transparently, distinguish \(u\) from \(h\), calculate mixture quality, compare ideal and real models, and reject physically invalid roots or phase assumptions.

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