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
Sir
John Edward
Lennard-Jones
1894 – 1954
England
U-
A
r
6

B
r
n
, 9  n  12
Attraction Repulsion
U,
Lennard-Jones Potential Energy:
req
r
dU
Fdr
2
A
B
F  7  n 1 , 9  n  12
r
r
© Prof. Zvi C. Koren
21.07.10
Z  Compressib ility (or Compressio n) Factor  P V / RT
(00C)
3
© Prof. Zvi C. Koren
21.07.10
Compressibility Factors
Find TB, the Boyle Temperature
N2
4
CH4
© Prof. Zvi C. Koren
21.07.10
Variation of Z with T: General Considerations
For any real gas:
Temp.
> TB
= TB
(TB)
< TB
Z
>1
1
<1
>1
5
dZ/dP
>0
0
Low P: < 0
Medium P: > 0
High P: > 0
© Prof. Zvi C. Koren
21.07.10
van der Waals Equation of State & Intermolecular Forces
Johannes Diderik van der Waals
1837–1923
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)
“Corrections” for P & V in Ideal Gas Equation of State
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
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:
 (n/V)2
= a(n/V)2 = n2a/V2,
a  effective intermolecular force parameter
Rewrite Ideal Gas Eqn. of State as: Pideal
·Voccupied = nRT
Preduction
7
2 

 P  n a   V - nb  nRT

© Prof. Zvi C. Koren
21.07.10
V2 
van der Waals Equation of State (continued)

n2a 
 P  2 V - nb   nRT
V 

a  effective intermolecular force parameter
b  effective molar volume parameter
a, b  f(T)
Units of “a”: ______________
Units of “b”: ______________
a 

 P  2 V - b   RT
V 

8
Molar Volume  V or Vm  V/n
© Prof. Zvi C. Koren
21.07.10
van der Waals
Constants for
Various Gases
(a in atmּL2/mol2;
b in L/mol)
He
Ne
Ar
Kr
Xe
9
Gas
Formula
Ammonia
Argon
Carbon dioxide
Carbon disulfide
Carbon monoxide
Carbon tetrachloride
Chlorine
Chloroform
Ethane
Ethylene
Helium
Hydrogen
Hydrogen bromide
Methane
Neon
Nitric oxide
Nitrogen
Oxygen
Sulfur dioxide
Water
NH3
Ar
CO2
CS2
CO
CCl4
C12
CHCl3
C2H6
C2H4
He
H2
HBr
CH4
Ne
NO
N2
O2
SO2
H2O
a
b
4.17
1.35
3.59
11.62
1.49
20.39
6.49
15.17
5.49
4.47
0.034
0.244
4.45
2.25
0.211
1.34
1.39
1.36
6.71
5.46
0.0371
0.0322
0.0427
0.0769
0.0399
0.1383
0.0562
0.1022
0.0638
0.0571
0.0237
0.0266
0.0443
0.0428
0.0171
0.0279
0.0391
0.0318
0.0564
0.0305
© Prof. Zvi C. Koren
21.07.10
Comparison of Ideal Gas Law and van der Waals Equation
(at 1000C)
Hydrogen
Observed
P
P
%
P Calc.
Carbon Dioxide
%
P
Calc. Devia- van der Devia- Calc.
(atm)
Ideal
tion
50
48.7
-2.6
75
72.3
100
95.0
Waals
%
P Calc.
%
Devia-
van der
Devia-
tion
Waals
tion
tion
Ideal
50.2
+0.4
57.0
+14.0
49.5
-1.0
-3.6
75.7
+0.9
92.3
+17.3
73.3
-2.3
-5.0
100.8
+0.8
133.5
+33.5
95.8
-4.2
Note:
For each observed P there is an observed V and that is plugged into the
equation to calculate a theoretical P.
10
© Prof. Zvi C. Koren
21.07.10
“Cubic” Equations of State
vdW: The van der Waals eqn. is an example. It is cubic in “V”:
RT
a
a 

 2
Original form:  P  2 V - b   RT or P 
V-b V
V 

Multiply out and by V2 and rearrange:
Cubic form:
 RT  Pb  2
a
 ab 
V 
V   V     0
P


P
 P 
3
Redlich-Kwong (1949):


 P  1/2
T V V






 V    RT

Show the cubic form:
11
© Prof. Zvi C. Koren
21.07.10
Berthelot Equation of State
Low-Pressure form (P  1 atm):
2


9PTc  6Tc 
PV  RT 1 
1 2 
T 
 128Pc T 

Tc = critical temperature
Pc = critical pressure
No additional constants
12
© Prof. Zvi C. Koren
21.07.10
Beattie-Bridgeman Equation of State
(5 constants)
13
Explicit in P:
RT 


P 
 2 3 4
V V
V
V
Explicit in V:
RT
β
γ
δ
2
V 


P

P
3
P RT RT 2
RT 
Rc
β  RTB0  A0  2
T
RcB0
   RTB0b  A0 a  2
T
RB0bc
 
T2
Very accurate even for
P  100 atm and
T  -1500C
(Table on next slide)
© Prof. Zvi C. Koren
21.07.10
BEATTIE-BRIDGEMAN CONSTANTS FOR SOME GASES*
(P in atm, Vm in L/mol)
Gas
Ao
a
Bo
b
c x 10-4
He
0.0216
0.05984
0.01400
0
0.004
Ne
0.2125
0.2196
0.02060
0
0.101
Ar
1.2907
0.02328
0.03931
0
5.99
H2
0.1975
-0.00506
0.02096
-0.04359
0.050
N2
1.3445
0.02617
0.05046
-0.00691
4.20
O2
1.4911
0.02562
0.04624
0.004208
4.80
Air
1.3012
0.01931
0.04611
-0.01101
4.34
CO 2
5.0065
0.07132
0.10476
0.07235
66.00
CH4
2.2769
0.01855
0.05587
-0.01587
12.83
(C2H5)2O
31.278
0.12426
0.45446
0.11954
33.33
*J. Am. Chem. Soc., 50, 3136 (1928). See also Maron and Turnbull, Ind. Eng. Chem., 33, 408 (1941).
14
Gas Problems: Real Gases: 14-17, 22.
© Prof. Zvi C. Koren
21.07.10
The Kamerlingh Onnes Virial Equation
Ideal Gas Equation :
Virial Equation :
(vires = forces)
PV  RT
Express P V as a Power Series in P
)‫(טור חזקות‬
Heike
PV  A  BP  CP 2  DP 3     Kamerlingh
Onnes
1853 – 1926,
Holland
)‫ = (האיבר הראשון‬RT
2nd virial coefficient. Units =
Notes:
3rd virial coefficient. Units =
A, B, C, … = f(T)
A>0
B < 0 (low T), B = 0 (certain T), B > 0 (higher T)
A >> B >> C >>   
At relatively low P ( 15 atm) : P V  A  BP
Also at T  TB , P V  A   B  0 :
15
At medium P (  50 atm ) : P V  A  BP  CP 2 © Prof. Zvi C. Koren
21.07.10
VIRIAL COEFFICIENTS OF SOME GASES
(P in atm, Vm in L/mol)
t (°C)
A
-50
0
100
200
18.312
22.414
30.619
38.824
-50
0
100
200
18.312
22.414
30.619
38.824
-50
0
500
18.312
22.414
63.447
16
B x 102 C x 105 D x 108 E x 1011
Nitrogen
-2.8790 14.980 -14.470 4.657
-1.0512 8.626 -6.910
1.704
0.6662
4.411 -3.534 0.9687
1.4763
2.775 -2.379 0.7600
Carbon Monoxide
-3.6878 17.900 -17.911 6.225
-1.4825 9.823 -7.721
1.947
0.4036
4.874 -3.618 0.9235
1.3163
3.052 -2.449 0.7266
Hydrogen
1.2027
1.164 -1.741
1.022
1.3638 0.7851 -1.206 0.7354
1.7974 0.1003 -0.1619 0.1050
Notes :
A  B  C
A, B, C,   f(T)
A0
B  0 (low T),
B  0 (certain T),
B  0 (higher T)
At T  TB , P V  A
  B0
© Prof. Zvi C. Koren
21.07.10
Determination of TB: Graphical & Regression
VIRIAL COEFFICIENTS OF CO (P in atm, Vm in L/mol)
A
B x 102
C x 105 D x 108
E x 1011
-50
18.312
-3.6878
17.900
-17.911
6.225
0
22.414
-1.4825
9.823
-7.721
1.947
100
30.619
0.4036
4.874
-3.618
0.9235
200
38.824
1.3163
3.052
-2.449
0.7266
2
At T  TB , P V  A   B  0 :
0
B x 10
2
t (°C)
-50 0
-2
17
-4
0
t ( C)
50 100 150 200
© Prof. Zvi C. Koren
21.07.10
Z Meets Virial
PV  A  BP  CP 2  DP 3    
A = RT
B
C 2
D 3
Z  1
P 
P 
P  
RT
RT
RT
PV
Z 
RT
Z  1  B' P  C' P 2  D' P3    
:1 ‫הצגה‬
B
C
D
B' 
, C' 
, D' 
RT
RT
RT
We can also express “Z” as an inverse power series in the molar volume
-1
Z  1  βV  γ V
-2
-3
 δV    
:2 ‫הצגה‬
These two forms obey the conditions for “Good” Mathematical-Chemical Functions:
18
lim Z(P)  1
P0
and
lim Z(V)  1
V 
© Prof. Zvi C. Koren
21.07.10
Relationships Between the Two Sets of Coefficients
PV/RT  Z  1  B' P  C' P 2  D' P3  
PV/RT  Z  1  βV -1  γV -2  δV -3  
Solve for P = f(V) from 2nd eqn.: P  RT( V -1  βV -2  γV -3  δV -4  )
Substitute P = f(V) into 1st eqn.:
Z  1  B' P  C' P 2  D' P3  
Z  1  B' RT  (V -1  βV -2  γV -3  δV -4  )
 C' (RT) 2 (V -1  βV -2  γV -3  δV -4  ) 2
 D' (RT) 3 (V -1  βV -2  γV -3  δV -4  ) 3  
Z  1  B' RT V -1  [B' RTβ  C' (RT) 2 ]V -2  
Compare with 2nd eqn.: Z  1 
βV -1 
γV -2

:‫השוואת מקדמים של חזקות שוות‬
 = B’RT = (B/RT)RT  B = 
 = B’RT + C’(RT)2 = B + CRT = B2 + CRT  C = ( –2)/(RT)
19
Gas Problems: Real Gases: 18, 19, 25.
© Prof. Zvi C. Koren
21.07.10