real gases part 1 - the edelstein center for the … · real gases –part 1 treatments: 1....
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Real Gases – Part 1
Treatments:1. Lennard-Jones Potential Energy Function
2. Z Compressibility Factor (מקדם דחיסות) 3. van der Waals Equation of State
4. Redlich-Kwong Eqn.
5. Berthelot Eqn.
6. Virial (Kamerlingh Onnes) Eqn.
7. Beattie-Bridgeman Eqn.
8. Reduced Equations of State
9. Activity Coefficient - based on G (next semester)
10. Many Many Others
1 © Prof. Zvi C. Koren 21.07.10
Intermolecular Forces: Attractions & Repulsions
req
dr
dU- F
U,
r
12n9 ,r
B
r
A F
1n7
12 n9 r
B
r
A- U
n6,
Attraction Repulsion
Lennard-Jones Potential Energy:
SirJohn Edward
Lennard-Jones1894 – 1954
England
2 © Prof. Zvi C. Koren 21.07.10
(TB)
For any real gas:
dZ/dPZTemp.
> 0> 1> TB
0 1= TB
Low P: < 0< 1
< TB Medium P: > 0
High P: > 0> 1
Variation of Z with T: General Considerations
5 © Prof. Zvi C. Koren 21.07.10
van der Waals Equation of State & Intermolecular Forces
Johannes Diderik van der Waals1837–1923
Netherlands
Nobel Laureate in Physics, 1910
for his work on
The equation of state for gases and liquids
Definition of “van der Waals forces”:
Any force between neutral molecules!
6 © Prof. Zvi C. Koren 21.07.10
van der Waals Equation of State (continued)
Rewrite Ideal Gas Eqn. of State as: Pideal ·Voccupied = nRT
Pideal > Preal:Attractive forces reduce the FREQUENCY and FORCE of collisions with the walls
of the container.
The reduction in EACH factor is proportional to the concentration of the gas, n/V:
Preduction (n/V)2
= a(n/V)2 = n2a/V2,
a effective intermolecular force parameter
Voccupied by ideal > Voccupied by real:V occupied by ideal gas = V of entire container
V occupied by real gas = V of entire container – V of molecule themselves
= V – nb,
b effective molar volume parameter
“Corrections” for P & V in Ideal Gas Equation of State
nRT nb-VV
anP
2
2
7 © Prof. Zvi C. Koren 21.07.10
nRT nb-VV
anP
2
2
van der Waals Equation of State (continued)
a effective intermolecular force parameter
b effective molar volume parameter
a, b f(T)
RT b-VV
aP 2
Units of “a”: ______________
Units of “b”: ______________
V/n V V VolumeMolar m or
8 © Prof. Zvi C. Koren 21.07.10
baFormulaGas
0.03714.17NH3Ammonia
0.03221.35ArArgon
0.04273.59CO2Carbon dioxide
0.076911.62CS2Carbon disulfide
0.03991.49COCarbon monoxide
0.138320.39CCl4Carbon tetrachloride
0.05626.49C12Chlorine
0.102215.17CHCl3Chloroform
0.06385.49C2H6Ethane
0.05714.47C2H4Ethylene
0.02370.034HeHelium
0.02660.244H2Hydrogen
0.04434.45HBrHydrogen bromide
0.04282.25CH4Methane
0.01710.211NeNeon
0.02791.34NONitric oxide
0.03911.39N2Nitrogen
0.03181.36O2Oxygen
0.05646.71SO2Sulfur dioxide
0.03055.46H2OWater
van der Waals
Constants for
Various Gases(a in atmּL2/mol2;
b in L/mol)
He
Ne
Ar
Kr
Xe
9 © Prof. Zvi C. Koren 21.07.10
Comparison of Ideal Gas Law and van der Waals Equation
(at 1000C)
Carbon DioxideHydrogen
%P Calc.%P%P Calc.%PObserved
Devia-van derDevia-Calc.Devia-van derDevia-Calc.P
tionWaalstionIdealtionWaalstionIdeal(atm)
-1.049.5+14.057.0+0.450.2-2.648.750
-2.373.3+17.392.3+0.975.7-3.672.375
-4.295.8+33.5133.5+0.8100.8-5.095.0100
10 © Prof. Zvi C. Koren 21.07.10
Note:
For each observed P there is an observed V and that is plugged into the
equation to calculate a theoretical P.
“Cubic” Equations of State
vdW: The van der Waals eqn. is an example. It is cubic in “V”:
RT b-VV
aP 2
Original form:
Multiply out and by V2 and rearrange:
0 P
VP
VP
PbRTV
23
abaCubic form:
Redlich-Kwong (1949):
RT V
VVTP
1/2
or2
V
a
b-V
RTP
Show the cubic form:
11 © Prof. Zvi C. Koren 21.07.10
Berthelot Equation of State
Low-Pressure form (P 1 atm):
2
2
T
61
T128
9P1RT VP
cT
P
T
c
c
Tc = critical temperature
Pc = critical pressure
No additional constants
12 © Prof. Zvi C. Koren 21.07.10
Beattie-Bridgeman Equation of State
432
VVVV
RT P
(5 constants)
Explicit in P:
Explicit in V:
2
32P
RTP
RTRTP
RT V
δγβ
200T
RRT
cABβ
2
000
T
RRT
cBaAbB
2
0
T
R
bcB
Very accurate even for P 100 atm and
T -1500C
(Table on next slide)13 © Prof. Zvi C. Koren 21.07.10
c x 10-4bBoaAoGas
0.00400.014000.059840.0216He
0.10100.020600.21960.2125Ne
5.9900.039310.023281.2907Ar
0.050-0.043590.02096-0.005060.1975H2
4.20-0.006910.050460.026171.3445N2
4.800.0042080.046240.025621.4911O2
4.34-0.011010.046110.019311.3012Air
66.000.072350.104760.071325.0065CO 2
12.83-0.015870.055870.018552.2769CH4
33.330.119540.454460.1242631.278(C2H5)2O
BEATTIE-BRIDGEMAN CONSTANTS FOR SOME GASES*(P in atm, Vm in L/mol)
*J. Am. Chem. Soc., 50, 3136 (1928). See also Maron and Turnbull, Ind. Eng. Chem., 33, 408 (1941).
14 © Prof. Zvi C. Koren 21.07.10Gas Problems: Real Gases: 14-17, 22.
The Kamerlingh Onnes Virial Equation (vires = forces)
Pin a as VP Express :Equation Virial
RTVP :Equation Gas Ideal
esPower Seri
DP CP BP A VP 32
2nd virial coefficient. Units =
3rd virial coefficient. Units =
= RT
Notes:A, B, C, … = f(T)
A > 0
B < 0 (low T), B = 0 (certain T), B > 0 (higher T)
A >> B >> C >>
BP A VP :) 15( P low relativelyAt atm
:0B A VP ,TTat Also B
(טור חזקות)
(האיבר הראשון)
2CP BP A VP :)atm 05( P mediumAt
HeikeKamerlingh
Onnes1853 – 1926,
Holland
15 © Prof. Zvi C. Koren 21.07.10
E x 1011D x 108C x 105B x 102At (°C)
Nitrogen
4.657-14.47014.980-2.879018.312-50
1.704-6.9108.626-1.051222.4140
0.9687-3.5344.4110.666230.619100
0.7600-2.3792.7751.476338.824200
Carbon Monoxide
6.225-17.91117.900-3.687818.312-50
1.947-7.7219.823-1.482522.4140
0.9235-3.6184.8740.403630.619100
0.7266-2.4493.0521.316338.824200
Hydrogen
1.022-1.7411.1641.202718.312-50
0.7354-1.2060.78511.3638 22.4140
0.1050-0.16190.10031.797463.447500
VIRIAL COEFFICIENTS OF SOME GASES
(P in atm, Vm in L/mol)
0B
A VP ,TTAt
T)(higher 0 B
T),(certain 0 B
T), (low 0 B
0 A
f(T) C, B, A,
C B A
:Notes
B
16 © Prof. Zvi C. Koren 21.07.10
-4
-2
0
2
-50 0 50 100 150 200
t (0C)B
x 1
02
E x 1011D x 108C x 105B x 102At (°C)
6.225-17.91117.900-3.687818.312-50
1.947-7.7219.823-1.482522.4140
0.9235-3.6184.8740.403630.619100
0.7266-2.4493.0521.316338.824200
VIRIAL COEFFICIENTS OF CO (P in atm, Vm in L/mol)
Determination of TB: Graphical & Regression
:0B A VP ,TTAt B
17 © Prof. Zvi C. Koren 21.07.10
Z Meets Virial
PRT
D P
RT
C P
RT
B 1 Z 32
RT
VP Z DP CP BP A VP 32
PD' PC' PB' 1 Z 32
RT
D D' ,
RT
C C' ,
RT
B B'
A = RT
We can also express “Z” as an inverse power series in the molar volume
Vδ V γ Vβ 1 Z-3-2-1
These two forms obey the conditions for “Good” Mathematical-Chemical Functions:
V1 )V Z(lim
0P1 Z(P)lim
and
:1הצגה
:2הצגה
18 © Prof. Zvi C. Koren 21.07.10
Relationships Between the Two Sets of Coefficients
PD' PC' PB' 1 Z /RTVP 32
Vδ V γ Vβ 1 Z /RTVP -3-2-1
) Vδ V γ Vβ VRT( P -4-3-2-1 Solve for P = f(V) from 2nd eqn.:
V γ Vβ 1 Z -2-1
P D' P C' P B' 1 Z 32
) Vδ V γ Vβ V((RT)D'
) Vδ V γ Vβ V((RT) C'
) Vδ V γ Vβ V( RT B' 1 Z
34-3-2-1-3
24-3-2-1-2
-4-3-2-1
V](RT)C' RTβ[B' VRTB' 1 Z -22-1
= B’RT = (B/RT)RT B =
= B’RT + C’(RT)2 = B + CRT = B2 + CRT C = ( –2)/(RT)
Substitute P = f(V) into 1st eqn.:
Compare with 2nd eqn.:
:השוואת מקדמים של חזקות שוות
19 © Prof. Zvi C. Koren 21.07.10Gas Problems: Real Gases: 18, 19, 25.