EQUATION OF STATE
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EQUATION OF STATE
CHAPTER 2
• 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 state2. Redlich–Kwong equation of state3. Soave modification of Redlich–Kwong4. 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.
RTbVVaP 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
6427a
c
c
c
c
PTR
81
PTR
81b
Cubic form of vdW eos
2Va
bVRTP
0P
abVPaV
PRTbV 23
0ABZAZ1BZ 23
2r
r22 T
P6427
TRaPA
r
r
TR
81
RTbPB
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 c
r PPP
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 T81V3V3P
• 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
Substance ValueH2O 0.23He 0.30H2 0.30Ne 0.29N2 0.29Ar 0.29
Zc of various substances
Standing-Katz Compressibility Factor Chart
Application of PCS
REDLICH-KWONG EOS
The Redlich–Kwong equation is adequate for calculation of gas phase properties when:
bVVa
bVRTP
c
2c
2
PTR42748.0a
c
c
PTR08662.0b
cc T2T
PP
21rT
Cubic form of RK eos
bVVa
bVRTP
0ABZBBAZZ 223
2r
r22 T
P42748.0TRPaA
r
r
TP08662.0
RTbPB
SOAVE-REDLICH-KWONG EOS
bVVa
bVRTP
c
2c
2
PTR42748.0a
c
c
PTR08662.0b
25.0r
2 T115613.055171.148508.01
r2 T30288.0exp202.1:HorF
Cubic form of SRK eos
bVVa
bVRTP
0ABZBBAZZ 223
2r
r22 T
P42748.0TRPaA
r
r
TP08662.0
RTbPB
PENG-ROBINSON EOS
22 bbV2Va
bVRTP
c
2c
2
PTR45724.0a
c
c
PTR07780.0b
25.0r
2 T12699.054226.137464.01
Cubic form of PR eos
0BBABZB3B2AZB1Z 32223
2r
r22 T
P45724.0TRPaA
r
r
TP07780.0
RTbPB
22 bbV2Va
bVRTP
SOLVING CUBIC EQUATION
0cZcZcZ 012
23
eos c2 c1 c0vdW – 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
27ccK
21
2
01232 c27cc9c2
221L
4L
27KD
23
(determinant)
Calculate:
31
D2LM
31
D2LN
Case 1: D > 01 real root and 2 imaginary roots
3cNMZ 2
1
Case 2: D = 0 three real roots and at least two are equal
3cNMZ 2
1
3cNM
21ZZ 2
32
Case 3: D < 0three, distinct, real roots
3ck120cos
3K2Z 2
i
Where k = 0 for i = 1k = 1 for i = 2k = 2 for i = 3
27K4Lcos 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
VB1Z 32
2P'CP'B1RTPVZ
RTB'B 2
2
RTBC'C
3
3
RTB2BC3D'D
DIETERICI EOS
RTVaRTebVP
Rb4aTc 22c eb4
aP b2Vc
MIXTURE• For mixtures, we apply the same equation, but we
impose certain mixing rules to obtain “a” and “b”, which are functions of the properties of the pure components.
• We create a new “pseudo” pure substance that has the average properties of the mixture.
i j
ijjim ayya
i
iim byb
jiijij aak1a
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