Chapter 3
Volumetric Properties
of Pure Fluids
Dr. Nahid Sanzida
nahidsanzida@che.buet.ac.bd
Outline
 Presents the phase rule, which relates the number of independent variables
required to fix the thermodynamic state of a system
 Describe qualitatively the general nature of PVT behavior of pure substances
 Provide a detailed treatment of the ideal-gas state
 Treat equations of state, which are mathematical formulations of the PVT behavior
of fluids
 Introduce generalized correlations that allow prediction of the PVT behavior of
fluids for which experimental data are lacking
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1
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Water phase diagram.
Source: Dembek, Mikolaj & Bocian, Szymon. (2019). Pure water as a mobile phase in liquid chromatography techniques. TrAC Trends in Analytical Chemistry. 123. 115793. 10.1016/j.trac.2019.115793.
3.1 THE PHASE RULE
For a multiphase system at equilibrium, the number of intensive variables that
must be arbitrarily fixed to establish its state is called the number of degrees of
freedom of the system.
This number is given by the phase rule of J. Willard Gibbs
F=2−π+N
Eq. (3.1)
where, F is the number of degrees of freedom, π is the number of phases, and N
is the number of chemical species present in the system.
A phase is a homogeneous region of matter.
As an example, the phase rule may be applied to an aqueous solution of ethanol in
equilibrium with its vapor. Here N = 2, π = 2, and
F=2−π+N=2−2+2=2
Phase Diagram of Water
The minimum number of degrees of freedom for any system is zero.
When F = 0, the system is invariant;
Eq. (3.1) becomes π = 2 + N. This value of
π is the maximum number of phases that
can coexist at equilibrium for a system
containing N chemical species.
When
N = 1, this limit is reached for π = 3,
characteristic of a triple point (where liquid,
vapor, and the common form of ice exist
together in equilibrium,
occurs at 0.01°C and 0.611657 kPa
PT diagram of water.
Example 3.1
How many phase-rule variables must be specified to fix the thermodynamic state of
each of the following systems?
(a) Liquid water in equilibrium with its vapor.
Ans:
(b) Liquid water in equilibrium with a mixture of water vapor and nitrogen.
Ans:
(c) A three-phase system of a saturated aqueous salt solution at its boiling
point with excess salt crystals present.
Ans:
Because of the high pressure and high kinetic energy, steam
turbines are mostly used. As steam contains thermal energy, high
pressure is obtained for running the turbines, to generate electricity.
3.2 PVT BEHAVIOR OF PURE SUBSTANCES
PT Diagram:
Lines 1-2 and 2-C represent the conditions at which solid
and liquid phases exist in equilibrium with a vapor phase.
Solid/liquid equilibrium is represented by line 2-3.
Line 1-2, the sublimation curve, separates the solid
and gas regions;
line 2-3, the fusion curve, separates the solid and
liquid regions;
line 2-C, the vaporization curve, separates the liquid
and gas regions
Point C is known as the critical point; its coordinates Pc
and Tc are the highest pressure and highest temperature
at which a pure chemical species is observed to exist in
vapor/liquid equilibrium.
Figure 3.1: PT diagram for a pure substance.
from A to B. The transition from liquid to gas is gradual
and does not include the usual vaporization step
PT Diagram:
The three lines meet at the triple point, where the
three phases coexist in equilibrium.
According to the phase rule:
- the triple point is invariant (F = 0).
- If the system exists along any of the two-phase
lines of Fig. 3.1, it is univariant (F = 1)
- in the single-phase regions it is divariant (F = 2).
A phase is generally considered a liquid if vaporization results from pressure reduction at constant temperature.
A phase is considered a gas if condensation results from temperature reduction at constant pressure.
Since neither process can be initiated in the region beyond the dashed lines, it is called the fluid region.
A fluid existing at a temperature greater than Tc is said to be supercritical. An example is atmospheric air (Tc, air =140.52 °C
2-6
P-T Diagram of Pure Substances
Constant-Pressure Phase-Change Process
TV Diagram:
TV Diagram:
TV Diagram: Steam as an Ideal Gas?
PV Diagram:
Figure 3.2: PV diagrams for a pure substance.
(a) Showing solid, liquid, and gas regions.
(b) Showing liquid, liquid/vapor, and vapor regions
with isotherms.
P-v-T Surface:
- All the points on the surface
represent equilibrium states.
- All states along the path of a quasiequilibrium process lie on the P-v-T
surface since such a process must
pass through equilibrium states.
- The single-phase regions appear as
curved surfaces on the P-v-T surface,
and the two-phase regions as
surfaces perpendicular to the P-T
plane.
P-v-T surface of a substance that
contracts on freezing.
P-v-T surface of a substance that
expands on freezing (like water).
- This is expected since the
projections of two-phase regions on
the P-T plane are lines.
P-v-T Surface:
PVT surface for carbon dioxide, with isotherms
shown in black and the vapor/liquid equilibrium
curve in white.
https://www.thingiverse.com/thing:1805581
Why a chemical engineer should study phase diagrams?
• By definition, a phase diagram is a graphical representation showing distinct phases
which are in thermodynamic equilibrium.
• Since these equilibrium relationships are dependent on the pressure, temperature,
and composition of the system, a phase diagram provides a graphical visualization of
the effects of these system variables on the equilibrium behavior between the phases.
• Phase diagrams are essential in the understanding and development of separation
processes (e.g. distillation, evaporation, chromatography, filtration, etc), especially in
the choice and design of separation unit operations, e.g. knowledge about high
pressure phase equilibria is essential not just in chemical processes and separation
operations, but is also important for the simulation of petroleum reservoirs, the
transportation of petroleum fluids, as well as in the refrigeration industry.
Christophe Coquelet, Deresh Ramjugernath. Phase Diagrams in Chemical Engineering: Application to Distillation and Solvent Extraction. Zeeshan Nawaz and Shahid
Naveed. Advances in Chemical Engineering, Intech, 23 p., 2012, 978-953-51-0392-9. ff10.5772/33632ff. ffhal-00877007f
PVT equation of state
Single Phase Region (F=2)
An equation of state may be solved for any one of the three quantities P, V, or T, given
values for other two.
For example, if V is considered a function of T and P, then
This equation is, however, mostly applicable for liquid phase calculation.
PVT equation of state
For small change of T and P, a reasonable approximation can be
made if β and κ is assumed constant,
so after integration…
Example 3.2
Note: If we assume incompressible liquid,
β and κ are 0
ΔV or Δρ are 0,
i.e. V and ρ are constant.
Problems 3.6
Problems 3.9
For liquid water the isothermal compressibility is given by:
where c and b are functions of temperature only. If 1 kg of water is compressed
isothermally and reversibly from 1 to 500 bar at 60°C, how much work is required? At
60°C, b = 2700 bar and c = 0.125 cm3・g-1.
Hint: use
3.3 IDEAL GAS AND IDEAL-GAS STATE
The equation of state:
P Vig = RT
Assumptions
1. For gas phase only
2. Valid at low pressure
3. Assume no interaction among molecules.
4. From degree of freedom for one component gas system (F=2).
5. Pressure has no effect on U and H of an ideal gas so we write
U(T) and H(T)
Remember this for pure species ideal gas!
Problem 3.14 (7th)/3.24(8th)
Property Relations for the Ideal-Gas State
Process Calculations for the Ideal-Gas State
Closed and mechanically reversible process
dU = dQ + dW
Process Calculations for the Ideal-Gas State
Isothermal Process
Isobaric Process
Isochoric (Constant-V) Process
Isothermal processes can occur in any kind of system, including highly
structured machines, and even living cells. Various parts of the cycles
of some heat engines are carried out isothermally and may be
approximated by a Carnot cycle. Phase changes, such as melting or
evaporation, are also isothermal processes.
A simple Isobaric process is boiling water. You have a pot of
water on the stove, it is at atmospheric pressure. As the water
boils, the steam coming off expands (roughly 1600x the volume
of water). There is nothing to contain the steam so it stays at
atmospheric pressure. Energy is changed, pressure is constant.
In engineering of internal combustion engines, isochoric
processes are very important for their thermodynamic cycles
(Otto and Diesel cycle), therefore the study of this process is
crucial for automotive engineering.
Process Calculations for the Ideal-Gas State
Adiabatic Process; Constant Heat Capacities
dQ = 0.
See page 81 of your book for the details of these derivations
These equations may also be expressed as:
by definition
(3.27)
Equation (3.27) is valid only for the ideal-gas state, for constant heat capacities, and for
adiabatic, mechanically reversible, closed-system processes.
Example 3.4
Problem 3.10, 3.19, 3.28, 3.29, 3.33
An ideal gas, initially at 30°C and 100 kPa, undergoes the following cyclic processes in a
closed system:
(a) In mechanically reversible processes, it is first compressed adiabatically to 500 kPa, then
cooled at a constant pressure of 500 kPa to 30°C, and finally expanded isothermally to its
original state.
Calculate Q, W, ΔU, and ΔH for each step of the process and for the cycle. Take CP = (7/2)R
and CV = (5/2)R.
cooled at a constant pressure of 500 kPa to 30°C, and finally expanded isothermally to its original state.
Irreversible Processes
The work of an irreversible process is usually calculated by a two-step procedure.
First, W is determined for a mechanically reversible process that accomplishes the same change of state as
the actual irreversible process.
Second, this result is multiplied or divided by an efficiency to give the actual work.
Problem 3.28
(b) The cycle traverses exactly the same changes of state, but each step is irreversible with an efficiency of
80% compared with the corresponding mechanically reversible process. Note: The initial step can no longer be
adiabatic.
Calculate Q, W, ΔU, and ΔH for each step of the process and for the cycle. Take CP =(7/2)R and CV = (5/2)R.
A tank of 0.1 m3 volume contains air at
25°C and 101.33 kPa. The tank is
connected to a compressed-air line
which supplies air at the constant
conditions of 45°C and 1500 kPa. A
valve in the line is cracked so that air
flows slowly into the tank until the
pressure equals the line pressure.
If the process occurs slowly enough
that the temperature in the tank
remains at 25°C, how much heat is lost
from the tank?
Assume air to be an ideal gas for which
CP = (7/2)R and CV = (5/2)R.
Virial Equation: Two Forms of the Virial Equation
Compressibility factor (Z). is a measure
of the deviation of the real-gas molar
volume from its ideal-gas value.
Virial Equation: Two Forms of the Virial Equation
Figure: Generalized
compressibility chart for
various gases
Application of Virial Equation:
Figure: Compressibility-factor graph for methane. Shown
are isotherms of the compressibility factor Z, as
calculated from PVT data for methane by the defining
equation Z = PV/RT. They are plotted vs. pressure for a
number of constant temperatures, and they show
graphically what the virial expansion in P represents
analytically.
This equation expresses direct linearity between Z and
P and is often applied to vapors at subcritical
temperatures up to their saturation pressures. At higher
temperatures it often provides a reasonable
approximation for gases up to a pressure of several
bars, with the pressure range increasing as the
temperature increases.
For more simpler form, use up to second term only and solve according to
Ans for c) 3497 cm3 mol-1
A Generic Cubic Equation of State: