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 wit ID: 617779
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Slide1
EQUATION OF STATE
CHAPTER 2Slide2
An equation of state is a relation between state variablesIt 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 stateSlide3
Assumption:the gas consists of a large number of molecules, which are in random motion and obey Newton's laws of motion;
the volume of the molecules is negligibly small compared to the volume occupied by the gas; and
no forces act on the molecules except during elastic collisions of negligible duration.CLASSICAL IDEAL GAS LAW
PV = RTSlide4
CUBIC EQUATIONS OF STATE
Van der Waals equation of state
Redlich–Kwong equation of stateSoave modification of Redlich–KwongPeng–Robinson equation of stateSlide5
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
/
V
2
)
instead of
P.Slide6
where V is molar volumeThe substance-specific constants a and b can be calculated from the critical properties Pc, T
c
, and Vc asSlide7
Cubic form of vdW eosSlide8
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 followsSlide9
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.Slide10
Reduced form of vdW EOS:
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.Slide11
The compressibility factor at the critical point, which is defined asZc
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 0.375Slide12
SubstanceValueH2O
0.23
He0.30H20.30
Ne
0.29
N
2
0.29
Ar
0.29
Z
c
of various substancesSlide13
Standing-Katz Compressibility Factor ChartApplication of PCSSlide14
REDLICH-KWONG EOS
The
Redlich–Kwong equation is adequate for calculation of gas phase properties when:Slide15
Cubic form of RK eosSlide16
SOAVE-REDLICH-KWONG EOSSlide17
Cubic form of SRK eosSlide18
PENG-ROBINSON EOSSlide19
Cubic form of PR eosSlide20
SOLVING CUBIC EQUATION
eos
c2c1
c
0
vdW
– B – 1
A
– AB
RK
– 1
A – B – B
2
– AB
SRK
– 1
A – B – B
2
– AB
PR
B – 1
A – 2B – 3B
2
AB – B
2
– B
3Slide21
(determinant)
Calculate:Slide22
Case 1: D > 0 1 real root and 2 imaginary roots
Case 2: D = 0
three real roots and at least two are equalSlide23
Case 3: D < 0 three, distinct, real roots
Where k = 0 for
i = 1 k = 1 for i = 2 k = 2 for i = 3
The minus sign applies when B > 0,
The plus sign applies when B < 0. Slide24
NON CUBIC EQUATIONS OF STATE
VIRIAL EOSSlide25
DIETERICI EOSSlide26
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.