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Real Gases
All real gases are imperfect
Forces of attraction and repulsion come into play
within range of influence.
If there were no forces of attraction, all matter
would be gaseous, since there would be nothing to
bring the molecules together in the solid and liquid.
The behavior of matter in condensed phases is
determined by the balance of the forces of
attraction and repulsion.
Evans Adei CHEM 155
Intermolecular Forces
4ε 𝝈 ∕ 𝒓 𝟏𝟐 = repulsive component of Ep (short -
Range interaction repulsive dorminant)
Ep
0 𝒓𝒐
r
δ
ε
- 4ε 𝝈 ∕ 𝒓 𝟔 = attractive
component of Ep (long range
interaction; attractive
dorminant)
Potential energy (Ep) is a relative quantity and it is usual to take it to be zero
when the molecules are separated by an infinite distance. Evans Adei CHEM 155
The force F between two molecules is related to the potential energy Ep or U
From experimental and theoretical studies ……………. (4)
…………. (5)
At infinity separation between molecules, the intermediate Ep is zero
…………….. (6)
This function is known as the Lennard-Jones 6-12 function
The intermolecular potential energy (Eq.6) is usually written in a somewhat different
form.
………… (7)
is one of the values of r for which Ep (r) = 0 ( the other being r = ) and ε is the
energy at the minimum in the potential- energy curve.
Because of the theoretical difficulty of calculating the constant kr and ka, it is usual to
make use of experimental value of and ε. Evans Adei CHEM 155
Real Gases & The Virial Equation
The compression factor (z) - A measure of deviation from ideality.
- The gas laws treated so far hold fairly well for most gases over a limited range of
pressures and temperatures.
- However real gases deviate from ideal behavior when the range and the accuracy
of experimental measurements were extended and improved.
- A more convenient technique often used to show the deviation from ideal behavior
involves the use of graphs or tables of the compression (compressibility) factor.
- Gas imperfection or deviations from the ideal gas law, may be stated in terms of
the compressibility factor z, which may be defined by:
- For ideal gases , z = 1
Evans Adei CHEM 155
The compression factor (z)
2.0
H2
z He
1.0 perfect
200 400 P/atm
CH4
C2H4
NH3
z = 1: ideal, no interaction
z > 1: high pressure, repulsion
z < 1: intermediate attraction
Evans Adei CHEM 155
Real Gases & The Virial Equation
𝒛 = 𝑷𝑽 𝒏𝑹𝑻 = 𝑷𝑽𝒎 𝑹𝑻
At low pressure, all the gases in the figure have z 1 and are behaving nearly
perfectly.
As pressure approaches zero independent of the nature of the gas (z 1)
At intermediate pressures, most of the gases have z <1, indicating that the
attractive forces are dominant and favor compression.
At high pressure, z 1 repulsive forces dominant.
Evans Adei CHEM 155
CONDENSATION OF GASES- Critical points
The distinction among the three types of phases: gas, liquid and solid is valid for most substances under ordinary conditions.
Gas classification arose that distinguished liquefiable gases-called vapour from permanent gases.
Permanent gases are substances for which no liquid state was known, this include O2 ,N2, CO, Ar, Ne, He, CH4.
Eventual success at liquefying them depended on the nature of the liquid-gas transition at high temperature.
Thomas Andrews (1869) studied the behavior of a gas in the neighborhood of its critical point.
‘Permanent gases’ liquefaction led to the understanding of the concept of critical temperature.
Evans Adei CHEM 155
G
P/atm
𝝏𝑷 ∕ 𝝏𝑽 𝑻𝒄 = 𝝏𝟐𝒑 ∕ 𝝏𝒗𝟐 𝑻𝒄 = 𝟎
D
Pc supercritical fluid
20oC
H
31.04 50oC
40
C B
F 40oC
20
A
E critical isotherm
0.2 0.4 0.6 Vm/(Lmol-1)
Evans Adei CHEM 155
EQUATIONS OF STATE: REAL GASES
A number of equations have been developed to represent P-V-T
data for real gases.
Such an equation is called equation of state because it relates state
properties for substances at equilibrium.
The most general way to fit data to an equation or represent PVT
data for real gases.
Virial Equation - Representation as power series of density (n/v)
The use of power series is the most general way to fit data to an
equation.
Since deviations from ideality depend on the density of the gas, it
would be reasonable to represent the equation of state as a power
series in n/V and to include as many terms as may be necessary to
represent the experimental PVT data with the desired accuracy.
Evans Adei CHEM 155
Viral Equation
- A number of equations have been developed to represent P-V-T data for real gases.
Such an equation is called equation of state because it relates state properties for
substances at equilibrium
- Since deviations from ideality depend on the density of the gas, it would be
reasonable to represent the equation of state as a power series in n/V
-
Some of these equations of states are entirely empirical, obtained by simply fitting
an equation with adjustable parameters to the observed data. With no implication
of any physical significance to the various terms.
In 1901, Kamerlingh Onnes proposed an equation of state for real gases, which
expresses the compressibility factor z as a power series in 1/v for a pure gas.
- The equation so obtained is called a viral equation from the Latin vir, power
-
- The viral equation can be written as:
The coefficient B (T), C (T), etc are called the second, third, etc viral coefficients.
They are functions of temperature.
Evans Adei CHEM 155
The Van der Waals’ Equation
- The perfect gas law PV = nRT/V is replaced by P1 (V- nb) = nRT
- The attractive force exerted on a single molecule about to strike the wall is proportional to
the square of the density of the gaseous molecules.
-
- The total equation then becomes
- Van der Waals parameters a and b are much better regarded as empirical parameters than as
precisely defined molecular properties.
Evans Adei CHEM 155
TEST OF EQUATION OF STATE; OXYGEN AT 0oC
Molar volume Ideal P VdWP virial B Real P
0.2241 100atm 89atm 94 atm 94 atm
0.112 200 171 190 184
0.0560 400 492 446 440
0.0448 500 1040 650 675
It is evident that the: viral equation is useful over a much greater range of
pressure than the Vander Waals equation. This improvement follows for the
increased number of adjustable parameters. Evans Adei CHEM 155
Other equations of state
It is too optimistic to expect a single simple expression to be the true equation of state
of all substances.
When one equation fails, use is made of other equations of state that have been
proposed, invent a new one or go back to the viral equation.
The Berthelot equation of state
The Dieterici equation of state
Redlich-Kwong equation of state
Evans Adei CHEM 155
THE PRINCIPLE OR LAW OF CORRESPONDING STATES
- The choice of a related fundamental property of the same kind and the setting up
of a relative scale on that basis to compare the properties of objects is an important
general technique in science.
- The ratios of PV and T to the critical values Pc,Tc and Vc are called reduced
variables (reduced pressure, volume and temperature respectively)
Normalization:
Gaseous behavior (especially at moderate pressures) is very much the same when
normalized.
The observation that all gaseous substances would obey the same equation of state
in terms of the reduced variables, PR, TR, VR,- i.e, VR = f(PR, TR) was pointed out
by van der Waals in 1881, and he proposed to call this empirical rule :
Law of Corresponding States. Evans Adei CHEM 155
When the compressibility factors are plotted as a function of the reduced
pressure at a given reduced temperature, the points for various substances
fall on the same curve.
This regularity permits the prediction of the compressibility factor over a
wide range of temperature and pressure from the knowledge of the critical of
the substances concerned.
TR = 2.00
1.0
H2
z
1.0
TR = 1.50
C2H4 z = PV/RT
CH4 TR = 1.20
NH3
1.0 3.0 5.0 Pr
Evans Adei CHEM 155