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SNU NAOE
Youngsub Lim
EOS
SNU NAOE
Youngsub Lim
Whatโs the Equation of State?
โข We can find a value for a state by using diagram or table often inconvenient
โข Scientists seek an equation that can explain measured variables from experiment
2
Ideal Gas equation of state ๐๐ = ๐๐ ๐ (๐๐ ๐๐ฃ = ๐ ๐)No size(volume)No intermolecular forces
Boyleโs law:P of an ideal gas is inversely proportional to the volume if the temperature is constant.
๐ โ1
๐๐๐
๐1๐1 = ๐2๐2
Charles's law:When the P is constant,V of the gas is directly proportional to T
๐ โ ๐๐๐
๐2๐1
=๐2๐1
Avogadro's law:the V and n(moles) of the gas are directly proportional if T, P is constant
๐ โ ๐๐๐
๐1๐1
=๐2๐2
Benoรฎt Paul-ร mile Clapeyron
(1799-1864)
SNU NAOE
Youngsub Lim
Real molecule
โข An atom contains positively charged nucleus(+) surrounded by negatively charged electron cloud.
3http://www.physchem.co.za/OB11-mat/bonding.htm
๐ฟ + ๐ฟ โ
Dipole
SNU NAOE
Youngsub Lim
Dipole moment
Molecule ๐
HCl 1.08
H2O 1.85
H2, N2, O2 0
Ne, Ar 0
CH4, C2H6, C3H8โฆ(methane, ethane, propaneโฆ)
0
CO2 0
4
H
Cl
โ
+ +++
โโโ
๐
http://www.school-for-champions.com/chemistry/bonding_types.htm#.VRHvsPmUdFo
http://www.gcsescience.com/o10.htm
SNU NAOE
Youngsub Lim
Dipole-dipole force (Keesom force)
5http://chemwiki.ucdavis.edu/Wikitexts/UC_Davis/UCD_Chem_2B/UCD_Chem_2B%3A_Larsen/Unit_II%3A_States_of_Matter/Intermolecular_Interactions/11.2%3A_Intermolecular_Forces
SNU NAOE
Youngsub Lim
Dipole-induced dipole force (Debye force)
6
โข Dipole-induced dipole force (Debye force)
H
Cl
โ
+ +++
โโโ
Ar
+
โ
Ar
SNU NAOE
Youngsub Lim
Dispersion Force
โข Instantaneously induced dipole-induced dipole force (London Force)
7http://chemwiki.ucdavis.edu/Physical_Chemistry/Quantum_Mechanics/Atomic_Theory/Intermolecular_Forces/London_Dispersion_Interactions
Ar Ar+ +โโ
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Youngsub Lim
Van der Waals forces
โข Collection of
โ dipole-dipole force
โ dipole-induced dipole force
โ induced dipole-induced dipole force (Dispersion force)
8
Intermolecular potential Energy
SNU NAOE
Youngsub Lim
Equation of State
โข Ideal gas Pv=RT
9
โข Real gas
P
V
Isotherm T1
P
V
Isotherm T1
liq
vap
Vap/Liqdome
SNU NAOE
Youngsub Lim
PT diagram and Pv diagram
10
T=500K PropaneT=400K
T=370K=Tc
T=340K
T=320K
Isotherm (๋ฑ์จ์ )
320K340K
370K=Tc
P
T
SaturationPressureCurve
T=400K
SNU NAOE
Youngsub Lim
Pv diagram and isotherm
11
T=500K PropaneT=400K
T=370K=Tc
T=340K
T=320K
Isotherm
Vapor-liquid domeVapor domeSaturation dome
Similar with ideal gas law(P=RT/v)
Cannot be described with ideal gas law
SNU NAOE
Youngsub Lim
van der Waals Equation
12
๐ =๐ ๐
๐ฃ โ ๐โ
๐
๐ฃ2๐ =
๐ ๐
๐ฃ
Reflect the reducing volumedue to the size of molecule.
Reflect the reducing pressuredue to the interaction of molecules.
Number of molecules per unit volume: ๐ โ๐
๐
Number of combination:
๐๐ถ2 =๐ ๐ โ 1
2โ ๐2 โ
๐
๐
2
=1
๐ฃ2Then, if you adjust a and b, You can make a shape that is similar to the isotherm!
v goes to very large
๐ ๐
๐ฃ โ ๐โ๐ ๐
๐ฃ
โ๐
๐ฃ2โ 0
๐ =๐ ๐
๐ฃ
v near b
๐ ๐
๐ฃ โ ๐โ โ
SNU NAOE
Youngsub Lim
Cubic EOS
โข 2 parameters (a: attraction, b: size)โ Van der Waals
โข ๐ =๐ ๐
๐ฃโ๐โ
๐
๐ฃ2or (๐๐ฃ3 โ ๐ ๐ + ๐๐ ๐ฃ2 + ๐๐ฃ โ ๐๐ = 0)
โข ๐ =27
64
๐ ๐๐2
๐๐=
0.42188๐ 2๐๐2
๐๐, ๐ =
๐ ๐๐
8๐๐= 0.125๐ ๐๐/๐๐
โ Redlich-Kwong
โข ๐ =๐ ๐
๐ฃโ๐โ
๐
๐๐ฃ(๐ฃ+๐)
โข ๐ = 0.42748๐ 2๐๐2.5/๐๐ , ๐ = 0.08664๐ ๐๐/๐๐
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๐ =๐ ๐
๐ฃ โ ๐โ
๐
๐ฃ2๐ฃ3 โ
๐ ๐
๐+ ๐ ๐ฃ2 +
๐
๐๐ฃ โ
๐๐
๐= 0
SNU NAOE
Youngsub Lim 14
Tc
If T>=Tc, only one v for all P
T=500K>Tc
If T<Tc, v must have at least two different roots at the saturation pressure.
T=340K<Tc
In this graph, the critical point means that the number of roots change.Then how about decide a and
b to make the equation has triple roots at the critical point?
๐ฃ3 โ๐ ๐
๐+ ๐ ๐ฃ2 +
๐
๐๐ฃ โ
๐๐
๐= 0
๐ฃ โ ๐ฃ๐3 = 0
Letโs set a and b to make theequation becomes a form of
๐ =27
64
๐ ๐๐2
๐๐=0.42188๐ 2๐๐
2
๐๐
๐ =๐ ๐๐8๐๐
= 0.125๐ ๐๐/๐๐
T=320K<Tc
SNU NAOE
Youngsub Lim
Analytic solution of cubic equation
15
๐ด๐ฅ3 + ๐ต๐ฅ2 + ๐ถ๐ฅ + ๐ท = 0
๐ฅ = ๐ก โ๐ต
3๐ด
๐ก3 + ๐๐ก + ๐ = 0 ๐ =โ๐ต2 + 3๐ด๐ถ
3๐ด2๐ =
2๐ต3 โ 9๐ด๐ต๐ถ + 27๐ด2๐ท
27๐ด3๐ข + ๐ฃ = ๐ก
๐ง2 + ๐๐ง โ๐3
27= 0 ๐ง1 = ๐ข3 = โ
๐
2+
๐2
4+๐3
27๐ง2 = ๐ฃ3 = โ
๐
2โ
๐2
4+๐3
27
Case 1, ๐ท:๐2
4+
๐3
27โฅ 0One real root for ๐ฅ
๐ฅ = ๐ก โ๐ต
3๐ด= ๐ข + ๐ฃ โ
๐ต
3๐ด=
3
โ๐
2+
๐2
4+๐3
27+
3
โ๐
2โ
๐2
4+๐3
27โ
๐ต
3๐ด
Case 2, ๐ท:๐2
4+
๐3
27< 0Three real roots for ๐ฅ
๐ฅ๐ = ๐ก๐ โ๐ต
3๐ด= 2 โ
๐
3cos
1
3arccos
3๐
2๐โ3
๐+2
3๐๐ โ
๐ต
3๐ด, ๐ = 0,1,2
SNU NAOE
Youngsub Lim
Example with real Pv diagram
โข When T>=Tc
16
SaturationPressure
P
T
We are finding this area
T=400K
T=400K
P=45bar
SNU NAOE
Youngsub Lim 17
P
T
Example with EOST>Tc
(1) T>Tc, EOS has only one root when you solve it.
use it as v
Solve EOS
๐ =๐ ๐
๐ฃ โ ๐โ
๐
๐ฃ2
Get one root: v
T=400K v
SNU NAOE
Youngsub Lim
Example with real Pv
18
T<Tc with real isothermal line
P
T
We are finding this area
T=320K
(1) P=Psat
T=320K
(1) T<Tc, P=Psat
smallest one for vlsat, largest one for vv
sat
(2) P>Psat
(2) T<Tc, P>Psat, one v for liquid
(3) P<Psat
(3) T<Tc, P<Psat, one v for vapor
SNU NAOE
Youngsub Lim 19
P
T
Example with EOS
(2-2)
(3-1) T<Tc, P<Psat, three roots for v when you solve EOS
use the largest v as vv
(3-2) T<Tc, P<Psat, one roots for v when you solve EOS
use the v for vv
T<Tc
(1) T<Tc, P=Psat, three roots for v when you solve EOS
use the smallest v as vlsat, and the largest v as vv
sat
T=320K
(1) P=Psat
(2-1)P>Psat
(2-1) T<Tc, P>Psat, three roots for v when you solve EOS
use the smallest v as vl
(2-2) T<Tc, P>Psat, one root for v when you solve EOS
use the v as vl
(3-1)P<Psat
(3-2)
SNU NAOE
Youngsub Lim
Flowchart for solving vdW EOS for pure substance
Calculate EOS coefficients
๐ =27
64
๐ 2๐๐2.5
๐๐, ๐ =
๐ ๐๐8๐๐
Read database for given substance ๐๐ , ๐๐ , ๐ด, ๐ต, ๐ถ ๐๐๐ ๐ด๐๐ก๐๐๐๐ ๐๐.
Calculate cubic eq. coefficients of EOS
๐ =๐ ๐
๐ฃ โ ๐โ
๐
๐ฃ2โ
๐๐ฃ3 โ ๐๐ + ๐ ๐ ๐ฃ2 + ๐๐ฃ โ ๐๐ = 0
แ๐ด = ๐
๐ต = โ(๐๐ + ๐ ๐)
แ๐ถ = ๐
๐ท = โ๐๐
๐ =3 แ๐ด แ๐ถ โ ๐ต2
3 แ๐ด2๐ =
2 ๐ต3 โ 9 แ๐ด ๐ต แ๐ถ + 27 แ๐ด2๐ท
27 แ๐ด3
๐ท =๐2
4+๐3
27
Solve cubic eq. and calculate vโ if D>=0,
๐ฃ =3โ๐
2+ ๐ท +
3โ๐
2โ ๐ท โ
๐ตฬ
3 แ๐ดโก if D<0,
๐ฃ๐ = 2 โ๐
3cos
1
3arccos
3๐
2๐โ3
๐+2
3๐๐ โ
๐ตฬ
3 แ๐ด
๐ = 0,1,2
Calculate Psat at given T from Antoine eq.
๐๐ ๐๐ก = 105 exp ๐ด โ๐ต
๐ + ๐ถ
if ๐ < ๐๐ ๐๐ก, return Max ๐ฃ๐ as vapor volumeif ๐ > ๐๐ ๐๐ก, return Min ๐ฃ๐ as liquid volumeif ๐ = ๐๐ ๐๐ก, return Max ๐ฃ๐ as sat. vapor volume
and Min ๐ฃ๐ as sat. liquid volume
Start with T, Pwith given T, P