• An equation of state is a relation between state variables
• It is a thermodynamic equation describing the state of matter under a given set of physical conditions.
• It is a constitutive equation which provides a mathematical relationship between two or more state functions associated with the matter, such as its temperature, pressure, volume, or internal energy.
• Equations of state are useful in describing the properties of fluids, mixtures of fluids, solids, and even the interior of stars.
EQUATION OF STATE
Assumption:
1) the gas consists of a large number of molecules, which are in random motion and obey Newton's laws of motion;
2) the volume of the molecules is negligibly small compared to the volume occupied by the gas; and
3) no forces act on the molecules except during elastic collisions of negligible duration.
CLASSICAL IDEAL GAS LAW
PV = RT
CUBIC EQUATIONS OF STATE
1. Van der Waals equation of state
2. Redlich–Kwong equation of state
3. Soave modification of Redlich–Kwong
4. Peng–Robinson equation of state
VAN DER WAALS EQUATION OF STATE
The van der Waals equation may be considered as the ideal gas law, “improved” due to two independent reasons: Molecules are thought as particles with volume, not
material points. Thus V cannot be too little, less than some constant. So we get (V – b) instead of V.
We consider molecules attracting others within a distance of several molecules' radii affects pressure we get dengan (P + a/V2) instead of P.
RTbVVa
P 2
where V is molar volume
The substance-specific constants a and b can be calculated from the critical properties Pc, Tc, and Vc as
c
2c
2
c
2c
2
PTR
6427
PTR
6427
a
c
c
c
c
PTR
81
PTR
81
b
Cubic form of vdW eos
2Va
bVRT
P
0P
abV
Pa
VP
RTbV 23
0ABZAZ1BZ 23
2r
r22 T
P6427
TRaP
A
r
r
TR
81
RTbP
B
Principle of Corresponding States (PCS)
The principle of Corresponding States (PCS) was stated by van der Waals and reads: “Substances behave alike at the same reduced states. Substances at same reduced states are at corresponding states.”
Reduced properties provide a measure of the “departure” of the conditions of the substance from its own critical conditions and are defined as follows
cr T
TT
cr P
PP
cr V
VV
• The PCS says that all gases behave alike at the same reduced conditions.
• That is, if two gases have the same “relative departure” from criticality (i.e., they are at the same reduced conditions), the corresponding state principle demands that they behave alike.
• In this case, the two conditions “correspond” to one another, and we are to expect those gases to have the same properties.
Reduced form of vdW EOS:
rr2r
r T81V3V3
P
• This equation is “universal”.
• It does not care about which fluids we are talking about.
• Just give it the reduced conditions “Pr, Tr” and it will give you back Vr — regardless of the fluid.
• As long as two gases are at corresponding states (same reduced conditions), it does not matter what components you are talking about, or what is the nature of the substances you are talking about; they will behave alike.
The compressibility factor at the critical point, which is defined as
c
ccc RT
VPZ
Zc is predicted to be a constant independent of substance by many equations of state; the Van der Waals equation e.g. predicts a value of 0.375
REDLICH-KWONG EOS
The Redlich–Kwong equation is adequate for calculation of gas phase properties when:
bVVa
bVRT
P
c
2c
2
PTR
42748.0a
c
c
PTR
08662.0b
cc T2T
PP
21rT
SOAVE-REDLICH-KWONG EOS
bVVa
bVRT
P
c
2c
2
PTR
42748.0a c
c
PTR
08662.0b
25.0r
2 T115613.055171.148508.01
r2 T30288.0exp202.1:HorF
PENG-ROBINSON EOS
22 bbV2Va
bVRT
P
c
2c
2
PTR
45724.0a
c
c
PTR
07780.0b
25.0r
2 T12699.054226.137464.01
Cubic form of PR eos
0BBABZB3B2AZB1Z 32223
2r
r22 T
P45724.0
TRPa
A
r
r
TP
07780.0RTbP
B
22 bbV2Va
bVRT
P
SOLVING CUBIC EQUATION
0cZcZcZ 012
23
eos c2 c1 c0
vdW – B – 1 A – ABRK – 1 A – B – B2 – ABSRK – 1 A – B – B2 – ABPR B – 1 A – 2B – 3B2 AB – B2 – B3
0cZcZcZ 012
23
27c
cK21
2
01232 c27cc9c2
221
L
4L
27K
D23
(determinant)
Calculate:
31
D2L
M
31
D2L
N
Case 1: D > 01 real root and 2 imaginary roots
3c
NMZ 21
Case 2: D = 0 three real roots and at least two are equal
3c
NMZ 21
3c
NM21
ZZ 232
Case 3: D < 0three, distinct, real roots
3c
k120cos3K
2Z 2i
Where k = 0 for i = 1k = 1 for i = 2k = 2 for i = 3
27K4L
cos 3
21
The minus sign applies when B > 0, The plus sign applies when B < 0.
NON CUBIC EQUATIONS OF STATE
VIRIAL EOS
...VD
VC
VB
1Z 32
2P'CP'B1RTPV
Z
RTB
'B 2
2
RTBC
'C
3
3
RTB2BC3D
'D