real gases part 2 - the edelstein center for the analysis ... · 04/07/2010 · a gas may be...
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The Critical State (again): “Last Exit Before the Toll”
P
T
Phase Diagram of H2O
S
L
G Fluid
Tc
Explain
the 3
Physical Processes
critical point
triplepoint
Pc
Tc = the highest temperature at which
a gas may be liquefied
Real Gases – Part 2
1 © Prof. Zvi C. Koren 21.07.10
T<<Tc T=TcT<Tc
Tem
per
atu
re (
0C
)davg
davg = at+b
Gas Problems: 35. 2 © Prof. Zvi C. Koren 21.07.10
Experimental Isotherms
of CO2
At T = Tc,
substance is called
a “fluid” (זורם)
Note the behavior of
the gas at T > Tc !!!
3 © Prof. Zvi C. Koren 21.07.10
L+V
L G
Consider 2 paths:B–i–f–A: 2-phase L-V path
B–D–C–E–A: 1-phase Fluid path
Principle ofContinuity of States
The liquid state can be
considered as a continuation
of the gaseous state:
That is, a liquid can be
considered as a highly
compressed (very dense) gas.
Thus, an equation of state for
a gas at high pressure may
also be applicable to the
liquid state.
5 © Prof. Zvi C. Koren 21.07.10
Theoretical Isotherms of
CO2 According to the
van der Waals Equation
Experimental(from previous slide)
RT b-VV
aP 2
6 © Prof. Zvi C. Koren 21.07.10
Equivalency of the
vdW Eqn. and
Experimental Results
Experimental
vdW
Equal Areas
7 © Prof. Zvi C. Koren 21.07.10
Determination of the van der Waals Constants, a & b
Two Methods:
1. Algebraic –
Polynomial Properties
2. Differential –
Point of Inflection Properties
CO2
0 P
VP
VP
PbRTV
23
aba
3 roots. At:
T > Tc 1 real root& 2 imaginary ones
T < Tc 3 real roots (b,c,d)T = Tc 3 identical real roots (a)
1. Algebraic – Polynomial Properties:
vdW is a Cubic Eqn. in V:
8 © Prof. Zvi C. Koren 21.07.10
vdW at (Tc,Vc):
STRATEGY:
• We already know that vdW is a cubic eqn. in V.
• So, create a general cubic eqn in V, and then compare coefficients:
Expand “V” about “Vc” for any order desired:
c
c
c
c
c
ccc
ccc
cc
ccc
P
abV
P
aV
P
bPRTV
P
ab
P
a
P
bPRT
VVV
, 3 , 3
0 V VV
0 V 3 V 3 V
32
23
3223
Comparing coefficients:
0 V 0 V V , TAt 3 cccc VVVT
9 © Prof. Zvi C. Koren 21.07.10
cc PVa2
3
From before:
c
c
c
c
c
ccc
P
abV
P
aV
P
bPRTV , 3 , 3
32
a & b in terms of (Tc,Pc):
a & b in terms of (Vc,Pc):3cV
b
c
c
c
c
P
RTb
P
TRa
8 ,
64
2722
375.08
3
c
ccc
RT
VPZ
c
cc
T
VPR
3
8
according to vdW, for all gases:
van der Waals states that:
1. Zc is a constant, AND
2. Zc = 0.375
Experimentally:
Zc 0.3 (for many relatively non-polar gases)10 © Prof. Zvi C. Koren 21.07.10
2. Differential – Point of Inflection Properties:
CO2
P
V
0
(P/V)Tc=0
[(P/V)/V]Tc = (2P/V2)Tc = 0
Tc
0Vc
dP/dV 0
horizontal
point of inflection
11 © Prof. Zvi C. Koren 21.07.10
(P/V)Tc=0
(2P/V2)Tc = 0
At the critical point (point of inflection):
van der Waals eqn.:
2V-V
RT P
a
b
0
V
2
-V
RT
V
P :)T,V,(PAt 3
c
2
c
c
T
ccc
c
a
b
0
V
6
-V
2RT
V
P :)T,V,(PAt 4
c
3
c
c
T
2
2
ccc
c
a
b
2ccc27
P , 27
8 T ,3 V
b
a
Rb
ab
c
c
c
c
P
RTb
P
TRa
8 ,
64
2722
8
3
c
ccc
RT
VPZ (as before)
12 © Prof. Zvi C. Koren 21.07.10
Critical Constants and van der Waals Parameters
van der WaalsExperimentalb (L/mol)a (L2atm/mol2)ZcVc (L/mol)dc (g/cc)Pc (atm)tc (oC)Gas
0.235111.5132.4Ammonia
0.53148-122Argon
0.46073.030.98Carbon dioxide
0.31135-139Carbon monoxide
0.57376.1144.0Chlorine
0.2148.832.1Ethane
0.275563.1243.1Ethyl alcohol
0.2250.99.7Ethylene
0.06932.26-267.9Helium
0.031012.8-239.9Hydrogen
0.48425.9-228.7Neon
0.5265-94Nitric oxide
0.311033.5-147.1Nitrogen
0.43049.7-118.8Oxygen
0.22642.0196.81Propane
0.29241.6320.6Toluene
0.307219.5374.4Water
13 © Prof. Zvi C. Koren 21.07.10
Principle of Corresponding States
Define dimensionless variables.
(An important “trick” in science and engineering.)
Define reduced variables: Pr=P/Pc, Vr=V/Vc, Tr=T/Tc,
Using, for example, van der Waals’ eqn., and recalling that:
c
cc
T
VPR
3
8cc PVa
23
3cV
b
RT -VV
P 2
b
a
Substitute these into vdW eqn.:
T3
8
3-V
V
3P 2
2
c
ccccc
T
VPVPV
(continued)14 © Prof. Zvi C. Koren 21.07.10
T3
8
3-V
V
3P 2
2
c
ccccc
T
VPVPV
PcVc:
cc
c
c T
T
V
V
V
V
P
P
3
8
3
132
2
rr
r
r TVV
P 8 133
2
van der WaalsReduced Equation
of State
Benefits of this eqn.:
1. No direct parameters specific for the gas molecules
2. Universally true
3. Similar to the Ideal Gas Equation of State.
Principle of Corresponding States:
Gases with the same reduced volume and the same reduced temperature
have the same reduced pressure; hence, they are in corresponding states.15 © Prof. Zvi C. Koren 21.07.10
Universal van der Waals
(Reduced Eqn. of State)
0.4
0.9
1.4
1.9
2.4
2.9
3.4
0.5 1 1.5 2 2.5Vr
Pr
0.9
0.95
1.0
1.05
1.1
1.15
1.4
ideal
log-log
0.1
1
10
100
0.1 1 10
log Vr
l o g
P r
PV=k, [T]
logP + logV=logk
logP = –logV+logk
Tr
Note the behavior of the gas at T > Tc !!!
1
slope = – 1
P vs. V
16 © Prof. Zvi C. Koren 21.07.10
Universal van der Waals Equation
(Reduced vdW Equation of State)
3DAll Reduced Variables
17 © Prof. Zvi C. Koren 21.07.10
How graph is obtained:
For N2, for example:
If Pr=3.0, P=3.0Pc=known
If Tr=1.0, T=1.0Tc=known
At this P & T, V is measured
Zexp = PV/RT=known
Experimental Z’sas a function of Pr
Tr =
Pr18 © Prof. Zvi C. Koren 21.07.10
Expansion and Compression Abilities
Coefficient of thermal expansionThermal expansivityVolume expansivity
Cubic expansion coefficient PT
V
V
1 βor α
• For an ideal gas, “expansivity” = 1/T
(Noteeach
factor)
Isothermal compressibility
TP
V
V
1 or
(“kappa”)
• For an ideal gas, = 1/P
(chem) (eng)
(eng) (chem)
Just Do it!
Just Do it!
Gas Problems: 36-42.
19 © Prof. Zvi C. Koren 21.07.10