Real Gases

1
Chapter 8

• Real Gases

2
Homework
Real Gases
Physical Chemistry

• Section 8.2 8.2, 8.3
• Section 8.4 8.7, 8.11
• Section 8.5 8.17, 8.18
• Section 8.7 8.20, 8.21
• Section 8.8 8.26
• General 8.35, 8.38, 8.40, 8.45

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Real Gases
Physical Chemistry
Estimate a molar volume
Estimate the molar volume of CO2 at 500 K and 100
atm by treating it as a van der Waals gas. (the
van der Waals coefficients of CO2 are a3.592
atmL2mol-2, b4.267?10-2 L mol-1)
Answer.
Rearrange equation (8.2) for a molar volume
4
Real Gases
Physical Chemistry
Estimate a molar volume
Estimate the molar volume of CO2 at 500 K and 100
atm by treating it as a van der Waals gas. (the
van der Waals coefficients of CO2 are a3.592
atmL2mol-2, b4.267?10-2 L mol-1)
Answer.
5
Real Gases
Physical Chemistry
Estimate a molar volume
Answer.
Must solve
6
Real Gases
Physical Chemistry
Estimate a molar volume
Estimate the molar volume of CO2 at 500 K and 100
atm by treating it as a van der Waals gas. (the
van der Waals coefficients of CO2 are a3.592
atmL2mol-2, b4.267?10-2 L mol-1)
Answer.
Solve for the molar volume
For a perfect gas
7
Real Gases
Physical Chemistry
Equation of State (eos)
The van der Waals and R-K equations are cubic
equations of state.
A cubic algebraic equations always has three
roots.
Above the critical temperature Tc, two of the
roots are complex numbers, one will be a real
number.
At the critical temperature Tc, three equal real
roots.
Below the critical temperature Tc, three unequal
real roots.
A cubic equation of states isotherm in the
two-phase region below Tc will resemble the
dotted line in Fig. 8.3.
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Real Gases
Physical Chemistry
Condensation
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Real Gases
Physical Chemistry
virial equation of state
(8.5)
A more convenient expansion is (in many
applications)
first second third coefficients
The virial equation is an example of a common
procedure in physical chemistry, in which a
simple law is treated as the first term in a
series in powers of a variable (P or Vm).
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Real Gases
Physical Chemistry
virial equation of state
For a perfect gas
For a real gas
There may be a temperature at which Z?1 with zero
slope at low pressure. (Boyle T)
The properties of the real gas do coincide with
those of a perfect gas as P?0.
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Real Gases
Physical Chemistry
virial equation of state
The compression factor approaches 1 at low P, but
does not so with different slopes. For a perfect
gas, the slope is zero, but real gases may have
either positive or negative slopes, and the slope
may vary with T. At the Boyle T, the slope is
zero and the gas behaves perfectly over a wider
range of conditions than that at other T.
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van der Waals Equation
Assuming
When dT 0
For real gases,
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Real Gases
Physical Chemistry
Molecular interactions
Real gases do not obey the perfect gas equation
exactly.
Real gases show deviations from the perfect gas
equation because molecules interaction with each
other.
Repulsive forces between molecules assist
expansion and attractive forces assist
compression.
Repulsions dominant
The variation of the potential energy of two
molecules with their separation.
Attractions dominant
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Real Gases
Physical Chemistry
Molecular interactions
Repulsive forces between neutral molecules are
significant only when the molecules are almost in
contact
They are short-range interactions!
Repulsions can be expected to be important only
when the molecules are close together on average.
At high P when a large number of molecules occupy
a small V.
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Real Gases
Physical Chemistry
Molecular interactions
Attractive forces have a relatively long range
and are effective over several molecular
diameters.
They are important when the molecules are fairly
close together but not necessarily touching.
At low P when the molecules occupy a large V, the
molecules are far apart.
At low P the molecular forces play no significant
role and the gas behaves perfectly.
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Real Gases
Physical Chemistry
Molecular interactions
At moderate P when the molecules are on average
only a few molecule diameters apart, the
attractive forces dominate the repulsive forces.
The gas can be expected to be more compressive
than a perfect gas (the forces are helping to
draw the molecules together).
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Real Gases
Physical Chemistry
Condensation
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Real Gases
Physical Chemistry
H2O phase diagram P T
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Real Gases
Physical Chemistry
Condensation
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Real Gases
Physical Chemistry
Calculation of liquid-vapor equilibria
For 200oC isotherm in Fig. 8.3, J and N
correspond to liquid and vapor in equilibrium.
The phase-equilibrium condition
21
Real Gases
Physical Chemistry
Calculation of liquid-vapor equilibria
The phase-equilibrium condition
(Helmholtz function A)
(8.23)
The Gibbs equation
At const. T
22
Real Gases
Physical Chemistry
Condensation
P
H
H2O
Y
G
U
T
S
W
M
400 oC
R
J
N
374 oC
L
300 oC
V
L
200 oC
L V
K
Vm
Isotherms of H2O
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Real Gases
Physical Chemistry
Calculation of liquid-vapor equilibria
(Integration from J to N along the path JKLMN)
Where eos indicates that the integral is
evaluated along the equation-of-state isotherm
JKLMN.
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Real Gases
Physical Chemistry
Calculation of liquid-vapor equilibria
(8.24)
The area under the dotted line JKLMN.
equal areas
This area will equal the rectangle area only if
the areas of the regions labeled I and II in Fig.
8.5 are equal.
II
I
Maxwells equal-area rule
Fig. 8.5
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Real Gases
Physical Chemistry
Calculation of liquid-vapor equilibria
For the R-K equation (8.3), equation (8.24)
becomes
(8.25)
In addition to satisfy equation (8.25), the R-K
equation (8.3) must be satisfied at point J for
the liquid and at point N for the vapor, giving
the equations
(8.26)
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Real Gases
Physical Chemistry
Calculation of liquid-vapor equilibria
Excel spreadsheet