Pure Substance
Properties of Pure Substances & Equations of State
A pure substance is one that has a homogeneous and invariable
chemical composition. It may exist in more than one phase, but
the chemical composition is the same in all phases.
Dr. Md. Zahurul Haq
Liquid water, a mixture of liquid water and water vapour (steam),
and a mixture of ice and liquid water are all pure substances; every
phase has the same chemical composition.
A mixture of liquid air and gaseous air is not a pure substance
because the composition of the liquid phase is different from that of
the vapour phase.
Professor
Department of Mechanical Engineering
Bangladesh University of Engineering & Technology (BUET)
Dhaka-1000, Bangladesh
zahurul@me.buet.ac.bd
http://teacher.buet.ac.bd/zahurul/
Sometimes a mixture of gases, such as air, is considered a pure
substance as long as there is no change of phase. Strictly speaking,
this is not true.
ME 201: Basic Thermodynamics
c Dr. Md. Zahurul Haq
(BUET) Properties of Pure Substances & Equations of State
ME 201 (2012)
1 / 19
c Dr. Md. Zahurul Haq
Phase Equilibrium in a Pure Substance
(BUET) Properties of Pure Substances & Equations of State
ME 201 (2012)
2 / 19
Phase Equilibrium in a Pure Substance
Vapour-Liquid-Solid Phase Equilibrium
T086
0 ≤ T < 100o C: as T ↑⇒ v ↑ & P = const.
T = 100o C: P & T remains constant, v ↑ with heating upto last
drop of liquid.
T >
100o C:
c Dr. Md. Zahurul Haq
T087
as T ↑⇒ v ↑ & P = const.
(BUET) Properties of Pure Substances & Equations of State
ME 201 (2012)
Temperature-Volume Diagram for Water
3 / 19
c Dr. Md. Zahurul Haq
(BUET) Properties of Pure Substances & Equations of State
ME 201 (2012)
4 / 19
Phase Equilibrium in a Pure Substance
Phase Equilibrium in a Pure Substance
T096
T050
T089
T048
⇐= Some Triple Point Data
T095
Some Critical Point Data
c Dr. Md. Zahurul Haq
T088
(BUET) Properties of Pure Substances & Equations of State
ME 201 (2012)
5 / 19
c Dr. Md. Zahurul Haq
Phase Equilibrium in a Pure Substance
Phase Equilibrium: Gibb’s Phase Rule
Saturation temperature designates the temperature at which
vaporization takes place at a given pressure. This pressure is
called the saturation pressure for the given temperature.
If a substance exists as liquid at the saturation temperature and
pressure, it is called a saturated liquid. If the temperature of
the liquid is lower than the saturation temperature for the existing
pressure, it is called either a sub-cooled liquid (implying
T > Tsat (P) or a compressed liquid (implying P > Tsat (P)).
When a substance exists as part liquid and part vapour at the
saturation temperature, its quality, x is defined as the ratio of
the mass of vapour to the total mass.
If a substance exists as vapour at the saturation temperature, it is
called saturated vapour. When the vapour is at a temperature
greater than the saturation temperature, it is said to exist as
superheated vapour.
(BUET) Properties of Pure Substances & Equations of State
ME 201 (2012)
6 / 19
Phase Equilibrium in a Pure Substance
Physical Meaning of Some Terminology
c Dr. Md. Zahurul Haq
(BUET) Properties of Pure Substances & Equations of State
ME 201 (2012)
7 / 19
Gibb’s Phase Rule
The number of degrees of freedom within a heterogeneous mixture of
pure substances is given by Gibb’s phase rule as
f = C −P +2
f
C
P
≡ number of degrees of freedom
≡ number of components (pure substances) in the mixture
≡ number of phases
A homogeneous (P = 1) pure substance (C =1) requires
f = 1 − 1 + 2 = 2 intensive properties to fix its state.
A homogeneous (P = 1) mixture of two pure substances (C = 2)
requires f = 2 − 1 + 2 = 3 intensive properties to fix its state.
A two-phase (P = 2) pure substance (C = 1) ⇒ f = 1 − 2 + 2 = 1:
Each phase requires one intensive property to fix its state & we
also need to know how much of each phase is present.
c Dr. Md. Zahurul Haq
(BUET) Properties of Pure Substances & Equations of State
ME 201 (2012)
8 / 19
Phase Equilibrium in a Pure Substance
Thermodynamic Surfaces
Phase Equilibrium in a Pure Substance
Thermodynamic Surfaces
Thermodynamic Surfaces
T017
P-T phase diagram is composed of three unique curves:
1 Fusion line
region of 2-phase solid-liquid equilibrium,
2 Vaporization line
region of 2-phase liquid-vapour equilibrium,
3 Sublimation line
region of 2-phase solid-vapour equilibrium.
These three lines intersect at one point, called the triple point, which
is the only point where all three phases can be in equilibrium
c simultaneously.
Dr. Md. Zahurul Haq (BUET) Properties of Pure Substances & Equations of State
ME 201 (2012)
9 / 19
Phase Equilibrium in a Pure Substance
T090
P-v-T diagram for a substance that expands on freezing (e.g. water)
c Dr. Md. Zahurul Haq
Thermodynamic Surfaces
(BUET) Properties of Pure Substances & Equations of ME
State
201 (2012)
Phase Equilibrium in a Pure Substance
10 / 19
Thermodynamic Surfaces
T018
At triple point of a pure substance, C = 1, P = 3, and the
number of degrees of freedom are f = 1 − 3 + 2 = 0; i.e., there is
no flexibility in the thermodynamic state & none of the properties
can be varied & still keep the system at the triple point.
At critical point, the densities of the liquid & the vapour phases
become equal and, consequently, where the physical interface
between the liquid & the vapour disappears.
c Dr. Md. Zahurul Haq
(BUET) Properties of Pure Substances & Equations of ME
State
201 (2012)
11 / 19
T092
P-v-T diagram for a substance that contracts on freezing (e.g. CO2 )
c Dr. Md. Zahurul Haq
(BUET) Properties of Pure Substances & Equations of ME
State
201 (2012)
12 / 19
Phase Equilibrium in a Pure Substance
Thermodynamic Surfaces
Equation of State
Equations of State
For simple compressible substances, any intensive property is solely a
function of two other independent intensive properties. Equations of
State (EOS) have the following form:
f (P, v , T ) = 0
Ideal-gas EOS:
PV = nRu T
Ru
PV = mRT ←→ Pv =
T = RT ←→ P = ρRT
M
T047
T046
Contracts on freezing (e.g. CO2 )
Expands on freezing (e.g. H2 O)
c Dr. Md. Zahurul Haq
(BUET) Properties of Pure Substances & Equations of ME
State
201 (2012)
13 / 19
Equation of State
14 / 19
Equation of State
van der Waals’ Equation
P=
P
≡ pressure [kPa]
V
≡ volume [m3 ]
T
≡ temperature [K]
Ru ≡ universal gas constant, 8.314 kJ/kmol.K
R
≡ specific gas constant, [kJ/kg.K]
c Dr.
Zahurul Haq (BUET) Properties of Pure Substances & Equations of ME
State
201 (2012)
M Md. ≡
molecular weight [kg/kmol]
Redlich-Kwong Equation
RT
v −b
−
a
v2
P=
b =⇒ account for the volume occupied by gas molecules
a =⇒ account for intermolecular forces of attraction.
a = 0.42748
a
RT
√
−
v − b v (v + b) T
R 2 Tc2.5
Pc
b = 0.08664
RTc
Pc
Example: Pressure exerted by 3.7 kg CO in a 0.030 m 3 container at 215 K
At critical point:
∂P
∂v T =
27 R 2 Tc2
64 Pc ,
∂2 P
∂v 2
=⇒ vc = 3b, a =
b=
Pc vc
3
⇒ Zc = RTc = 8 = 0.375
T
RTc
8Pc
=0
T097
c Dr. Md. Zahurul Haq
(BUET) Properties of Pure Substances & Equations of ME
State
201 (2012)
15 / 19
1
Using ideal gas law: P = 78.7 bar.
2
Using van der Waals’ Eq.: P = 66.9 bar.
given, a = 1.463 bar .m 6 /kmol 2 & b = 0.0394 m 3 /kmol
3
Using Redlich-Kwong Eq.: P = 69.2 bar.
given, a = 17.26 bar .m 6 .K 1/2 /kmol 2 & b = 0.02743 m 3 /kmol
4
NIST Table −→ P = 69.13 bar.
c Dr. Md. Zahurul Haq
(BUET) Properties of Pure Substances & Equations of ME
State
201 (2012)
16 / 19
Equation of State
Equation of State
Other Equations of State
Generalized Compressibility Chart
Departure from ideal gas
Reduced properties, PR ≡
Viral Equation
Z =
B
C
D
Pv
=1+
+ 2 + 3 + ···
RT
v
v
v
Pv
Z ≡ RT
, for ideal gas, Z = 1
P
T
Pc & TR ≡ Tc
Beattie-Bridgeman Equation
RT
A
(1 − e)(v + B) − 2
2
v
v
where, A = A0 1 − av , B = B0 1 − vb , e = vTc 3
Benedict-Webb-Rubin (BWR) Equation
P=
P=
RT
Bo RT − Ao − Co /T 2 bRT − a
aα
+
+
+ 6
v
v2
v3
v
γ
−γ
c + 3 2 1 + 2 exp
v T
v
v2
c Dr. Md. Zahurul Haq
T094
(BUET) Properties of Pure Substances & Equations of ME
State
201 (2012)
17 / 19
Equation of State
Example: Sp. volume of Water at 20.0 MPa & 520o C
Ideal Gas Law: R =
videal =
Ru
M
= 8314/18 = 461.89 J/kg-K
RT
(461.89)(273 + 520)
=
= 0.0183 m 3 /kg
P
20000000
Compressibility Chart: PR =
From chart: Z = 0.83.
20
22.09
= 0.905, TR =
793
647.3
= 1.23
v = Zvideal = (0.83)(0.183) = 0.0152 m 3 /kg
Tabulated value, based on experimental data:
7→ v = 0.01551 m 3 /kg
c Dr. Md. Zahurul Haq
(BUET) Properties of Pure Substances & Equations of ME
State
201 (2012)
19 / 19
c Dr. Md. Zahurul Haq
(BUET) Properties of Pure Substances & Equations of ME
State
201 (2012)
18 / 19