Download Real Gases

CHEM 331
Physical Chemistry
Fall 2014
Real Gases
During our development of the Historical Gas Laws we had to admit that real gas behavior was
distinctly different than that of an Ideal Gas. For instance, even below 1 atm, P data for
gaseous CO2 at 0oC shows this product is not constant.
P [atm]
0.2500
0.3333
0.5000
0.6667
1.0000
[L/mol]
89.5100
67.0962
44.6794
33.4722
22.2643
P [L atm/mol]
22.3897
22.3654
22.3397
22.3148
22.2642
Low Pressure PV Data for CO2
22.4
PV [L atm / mol]
22.38
22.36
22.34
22.32
22.3
22.28
22.26
22.24
0
0.2
0.4
0.6
0.8
1
1.2
Pressure [atm]
Although the properties of an Ideal Gas are derived from the properties of real gases, an Ideal
Gas is simply a model of a real gas that happens to work well at moderate temperatures and
pressures. Its behavior is extrapolated from that of real gases. We now want to consider what
happens at much higher pressures; 500-1000 atm. It is much more difficult to develop an
Equation of State for a real gas because the behavior of the gas depends not only on the state
variables (T, P, ), but on the chemical composition of the gas.
We know, for an Ideal Gas:
So, if we define a Compressibility Factor as:
then, for an Ideal Gas, Z = 1. For a real gas, we have:
What if we are not in the limit of zero pressure, where the gas behaves as an Ideal Gas? What if
we are at a pressure of 1000 atm? The dependence of the compressibility factor on pressure is a
convenient method of presenting experimental data so as to note the deviations from Ideal
behavior.
We begin by examining P- -T data for a particular real gas. In this case, I randomly selected
data for Ammonia gas from Thermodynamic Functions of Gases, Ed. by F. Din (1956).
Calculations of Z are easy enough. For instance, at P = 1 atm and T = 320 K, we have
26.069 L/mol. So,
=
Z =
= 0.9927
This data is typically plotted in the form of isotherms of Z vs. P. This is done for Ammonia for
isotherms at 320K, 340K, 360K, 380K, 400K, 420K, 440K, 460K and 480K. The general trend
in the data is typical. Note that for low pressures, the slope of the Z vs. P curve tends to skate
upward with increasing temperature.
Z = Comprssiblity Factor
Ammonia Gas (320 - 480K)
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
-10
0
10
30
50
70
90
110
130
150
Pressure [atm]
For another example, consider P- -T for gaseous Ethane taken from the International Critical
Tables. Z vs. P isotherms are plotted for 300K and 350K.
Compressibility Factor
Gaseous Ethane
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
T = 300K
T = 350K
0
100
200
300
400
500
600
Pressure [atm]
When Z < 1, Attractive Intermolecular
Forces Dominate
When Z>1, Repulsive Intermolecular
Forces Dominate
When the attractive forces between the gaseous molecules dominate the intermolecular
interactions, the gas will hug in on itself, giving rise to a molar volume that is smaller than it
would be if the gas were Ideal. So, Z will have values below unity. If repulsive forces dominate,
the gas will expand slightly; meaning will be larger than it would be if the gas were Ideal.
This will result in Z values greater than unity. This behavior is observed at very high pressures.
In fact, at these high pressures the gas may be more of a super-critical fluid or even a liquid; not
a gas in the traditional sense.
Additional Notes:


Z, at any given temperature or pressure, depends on the composition of the gas.
The initial slope of the Z vs. P plot depends on the temperature of the gas; increasing with
increasing temperature.
A Textbook of Physical Chemistry
Arthur W. Adamson
If the initial slope is zero, then Z ~ 1 over an extensive pressure range and the gas obeys Boyle's
Law. The temperature at which this occurs is defined as the Boyle Temperature, TB. Formally,
we define the Boyle Temperature as the temperature when:
= 0
at TB
In the above graphs, we note that gaseous N2 at 100oC is very close to its Boyle Temperature.
The Boyle temperature for several gases is presented below:
Gas
He
H2
N2
Ar
CH4
CO2
C2H4
NH3
M [g/mol]
4
2
28
40
16
44
28
17
TB [K]
23.8
116.4
332
410
506
600
624
995
Let us return to our gaseous Ammonia data and consider the 380 K isotherm. Below ~ 20 atm,
we see the isotherm is fairly linear.
Z
Ammonia Gas at 380K
1.02
1
0.98
0.96
0.94
0.92
0.9
0
5
10
15
20
25
Pressure [atm]
Thus, we can write:
Z = 1 + B'(T) P
where the "constant" B'(T) is the slope of the above line and is dependent on temperature. This
constant is referred to as the Second Virial Coefficient. If we push beyond 20 atm in pressure,
the data begins to show significant curvature.
Ammonia Gas at 380K
1.1
1
Z
0.9
0.8
0.7
0.6
0
20
40
60
80
Pressure [atm]
To properly represent this data, we should add a quadratic term to our equation above:
Z = 1 + B'(T) P + C'(T) P2
In fact, if we wish to represent the data over an even broader range of pressures, we should
expand Z in terms of higher powers of P:
Z = 1 + B'(T) P + C'(T) P2 + D'(T) P3 + …..
This gives us a practical Equation of State for a real gas. All that is needed is to determine the
values of B'(T), C'(T), D'(T), etc. for the gas in question.
For theoretical reasons, the real gas Equation of State is typically given as an expansion of Z in
terms of . This equation of state is referred to as the Viral Equation of State, and the
coefficients are referred to as Virial Coefficients; with B(T) being the second virial coefficient,
C(T) the third, etc.
Z = 1 + B(T)
+ C(T)
+ D(T)
+ …..
This form is more useful than [Z vs. P] because molar volume [ ] is a measure of the average
distance between molecules and an expansion in terms of is thus an expansion in terms of
intermolecular distance. The virial coefficients can then in turn be estimated by means of various
theories for intermolecular forces of attraction and repulsion.
A Textbook of Physical Chemistry
Arthur W. Adamson
So, an expansion of Z in terms of P is more experimentally useful. But, an expansion of in terms
of
is more theoretically valid. Not to worry, one expansion can be expressed in terms of the
other. So, measured values of B'(T), C'(T), D'(T) etc. can be used to determine B(T), C(T), D(T),
etc. Of course, the reverse is also true.
Let's show this. First, use the Virial expansion in
to determine P:
P =
Now equate the two expansions:
1 + B(T)
+ C(T)
+ ….. = 1 + B'(T) P + C'(T) P2 + D'(T) P3 + …..
+ D(T)
Insert our expression for P into the expansion on the right:
1 + B(T)
+ C(T)
+ …..
+ D(T)
= 1 + B'(T) [
]
2
+ C'(T) [
+ …..
On the right, collect terms in powers of
1 + B(T)
+ C(T)
+ D(T)
+ …..
+ [B'(T) RT B(T) + C'(T) (RT)2]
= 1 + B'(T) RT
Coefficients of
:
+ …..
on either side of this equation must be equal. So,
B(T)
= RT B'(T)
C(T)
= B'(T) RT B(T) + C'(T) (RT)2 = (RT)2 [(B'(T))2 + C'(T)]
etc.
As mentioned, this process can be inverted and we can solve for B'(T), C'(T), etc. in terms of
B(T), C(T), etc. This gives:
B'(T) =
C'(T) =
etc.
Finally, real gases also differ from an Ideal gas in that if cooled or compressed, they will
typically condense into a liquid. This behavior can be illustrated by examining P- isotherms for
the gas.
According to Boyle's Law, for a given temperature, pressure varies inversely with volume.
Ethane at 350K
100
Pressure [atm]
80
60
40
P
20
P (Ideal)
0
0
5000
10000
Molar Volume [cc/mole]
If the gas is sufficiently cool, at some point, as it is compressed, it will condense. During the
compression, as the gas is condensed into a liquid, the pressure will remain constant. Once the
condensation to a liquid has occurred fully, further compression of the liquid will cause the
pressure to rise rapidly because of the incompressibility of the liquid.
At higher pressures and temperatures, the gas is, again, behaving more as a super-critical fluid
and so will not "condense" into a liquid. It is already extremely dense and behaving somewhat
as a liquid already.
The density of a supercritical fluid is more like that of a liquid than a gas, but is significantly less
than for the liquid at ordinary conditions. For example, … the density of H 2O at the critical point
[is] 0.32 g/cm3, compared with 1.00 g/cm3 at room T and P. In ordinary room-temperature liquids,
there is little space between the molecules, so diffusion of solute molecules through the liquid is
slow. In supercritical fluids, which have a lot of space between molecules, diffusion of solutes is
much faster that in ordinary liquids and the viscosity is much lower than in ordinary liquids.
Moreover, the properties of supercritical fluids in the region near the critical point vary very
rapidly with P and T, so these properties can be "tuned" to desired values by varying P and T.
Supercritical CO2 is used commercially as a
solvent to decaffeinate coffee and to extract
fragrances from raw materials for use in
perfumes. Supercritical and near-critical water
are good solvents for organic compounds and
are being studied as environmentally friendly
solvents for organic reactions.
Physical Chemistry
Ira N. Levine
Above the Critical Isotherm (Tc), only the super-critical fluid can exist. The Critical Pressure
(Pc) and Critical Volume ( ) occur at the inflection point of the critical isotherm.
Values for the critical point of many gases are reported below:
A Textbook of Physical Chemistry
Arthur W. Adamson
It should be noted that experimental determinations of
not typically accurate.
are difficult to make and the results are
Near the Critical Point, the fluid will exhibit Critical Opalescence.
A capillary tube is evacuated and then partly filled with liquid, the remaining space containing no
foreign gases but only vapor of the substance in question; the tube is then sealed. On heating,
opposing changes take place. The liquid phase increases its vapor pressure, and the vapor density
increases as vaporization occurs; this acts to diminish the volume of the liquid phase. On the other
hand, the liquid itself expands on heating. If just the proper degree of filling of the capillary was
achieved, these two effects will approximately balance, and the liquid-vapor meniscus will remain
virtually fixed in position as the capillary is heated. A temperature will then be reached at which
the meniscus begins to become diffuse and then no longer visible as a dividing surface. At this
temperature the system often shows opalescence; the vapor and liquid densities are so nearly the
same and their energy difference is so small that fluctuations can produce transient large liquidlike
aggregates in the vapor and vice versa in the liquid.
A Textbook of Physical Chemistry
Arthur W. Adamson
Carbon Dioxide at the Critical Point
http://en.wikipedia.org/wiki/File:Critical_carbon_dioxide.jpg