2 equation of state1
TRANSCRIPT
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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.
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Assumption:
1) the gas consists of a large number of molecules, whichare in random motion and obey Newton's laws of
motion;
2) the volume of the molecules is negligibly smallcompared to the volume occupied by the gas; and
3) no forces act on the molecules except during elastic
collisions of negligible duration.
PV = RT
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1. Van der Waals equation of state
2. RedlichKwong equation of state3. Soave modification of RedlichKwong
4. PengRobinson equation of state
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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 (Vb) instead of V.
We consider molecules attracting others within adistance of several molecules' radii affects pressure
we get dengan (P+ a/V2) instead of P.
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RTbVV
aP
2
where V is molar volume
The substance-specific constants a and b can be
calculated from the critical properties Pc, Tc, and Vcas
c
2
c
2
c
2
c
2
P
TR
64
27
P
TR
64
27a
c
c
c
c
P
TR
8
1
P
TR
8
1b
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Cubic form of vdW eos
2V
a
bV
RT
P
0
P
abV
P
aV
P
RTbV
23
0ABZAZ1BZ 23
2
r
r22
TP
6427
TRaPA
r
r
T
R
8
1
RT
bPB
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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
c
rT
TT
c
rP
PP
c
rV
VV
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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 stateprinciple demands that they behave alike.
In this case, the two conditions correspond to one
another, and we are to expect those gases to havethe same properties.
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Reduced form of vdW EOS:
rr2
r
r T81V3V
3
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 Vrregardless of the fluid.
As long as two gases are at corresponding states (samereduced 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.
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The compressibility factor at the critical point, which is
defined as
c
cc
cRT
VPZ
Zcis predicted to be a constant independent ofsubstance by many equations of state; the Van der Waals
equatione.g. predicts a value of 0.375
http://en.wikipedia.org/wiki/Van_der_Waals_equationhttp://en.wikipedia.org/wiki/Van_der_Waals_equationhttp://en.wikipedia.org/wiki/Van_der_Waals_equationhttp://en.wikipedia.org/wiki/Van_der_Waals_equationhttp://en.wikipedia.org/wiki/Van_der_Waals_equationhttp://en.wikipedia.org/wiki/Van_der_Waals_equationhttp://en.wikipedia.org/wiki/Van_der_Waals_equation -
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Substance Value
H2O 0.23
He 0.30
H2 0.30
Ne 0.29
N2 0.29Ar 0.29
Zcof various substances
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Standing-Katz
Compressibility
Factor Chart
Application of PCS
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The Redlich
Kwong equation is adequate for calculationof gas phase properties when:
bVVa
bVRTP
c
2
c
2
P
TR42748.0a
c
c
P
TR08662.0b
cc T2
T
P
P
21
rT
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Cubic form of RK eos
bVV
a
bV
RTP
0ABZBBAZZ 223
2
r
r
22
T
P42748.0
TR
PaA
r
r
T
P08662.0
RT
bPB
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bVV
a
bV
RTP
c
2c
2
P
TR42748.0ac
c
P
TR08662.0b
2
5.0r2 T115613.055171.148508.01
r2 T30288.0exp202.1:HorF
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Cubic form of SRK eos
bVV
a
bV
RTP
0ABZBBAZZ 223
2
r
r
22
T
P42748.0
TR
PaA
r
r
T
P08662.0
RT
bPB
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22bbV2V
abV
RTP
c
2
c
2
PTR45724.0a
c
c
P
TR07780.0b
25.0
r
2T12699.054226.137464.01
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Cubic form of PR eos
0BBABZB3B2AZB1Z 32223
2
r
r
22
T
P45724.0
TR
PaA
r
r
T
P07780.0
RT
bPB
22bbV2V
a
bV
RTP
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0cZcZcZ 01223
eos c2 c1 c0
vdW B1 A AB
RK 1 ABB2 AB
SRK 1 ABB2 AB
PR B1 A2B3B2 ABB2B3
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0cZcZcZ01
2
2
3
27
ccK
2
1
2
01232
c27cc9c2221L
4
L
27
KD
23
(determinant)
Calculate:
31
D2
LM
31
D2
LN
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Case 1: D > 01 real root and 2 imaginary roots
3
cNMZ 2
1
Case 2: D = 0three real roots and at least two are equal
3
cNMZ 21
3
cNM
2
1ZZ 2
32
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Case 3: D < 0three, distinct, real roots
3
ck120cos
3
K2Z 2
i
Where k = 0 for i = 1k = 1 for i = 2
k = 2 for i = 3
27K
4Lcos 3
2
1
The minus sign applies when B > 0,
The plus sign applies when B < 0.
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...
V
D
V
C
V
B1Z
32
2P'CP'B1
RT
PVZ
RT
B'B
2
2
RT
BC'C
3
3
RT
B2BC3D'D
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RTVaRTebVP
Rb4aT
c 22ceb4
aP b2Vc
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For mixtures, we apply the same equation,but weimpose 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 jijjim
ayya
i
iim byb
jiijij aak1a