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