equation of states

7/29/2019 Equation of States

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“EQUATIONS OF STATE”



The thermodynamic state of a single homogeneous fluid may be specified by

properties such as pressure, temperature and volume. An equation of state is a

functional relationship between these three variable and it may be written as-

F(P,V,T)=0

The most common equation of state is the one applicable for ideal gases.

On a molecular level, an ideal gas may be treated as the one for which-

The size of the molecules is very small compared to the distance between them

so that volume of the molecules is negligible in comparison with the total volume

of the gas.

The intermolecular forces are very small .

Introduction-



Equation of state for IDEAL GASES

given by

Perfect gas law is inadequate to explain the behavior of real gases. For real gases to

behave ideally the molecular interaction should be negligible, and apparent volume and

pressure effect should not present in gas, which is impossible in practical situation so

there are different types of equation are proposed to defining the behavior of gases„„

VAN-DER WAALS equation.

REDLICH-KWONG equation.

REDLICH-KWONG-SOAVE equation.

PENG-ROBINSON equation.

BENEDICT-WEBB-RUBIN equation.

VIRIAL equation.



VAN-DER WAALS

equation:-

Where, Vm

is molar volume, and a & b are substance-specific constants. They can be

calculated from the critical properties P C

, T C

& V C (V

C is the molar volume at the critical

point).

In order to compensate for the attracting forces molecules term (a/V² ) is introduced in

ideal gas equation, and for overcoming the problem of the actual volume occupied by gas

term (b) is subtracted in overall volume of gas.

The constants are calculated as below:-

; or,

; or,



REDLICH-KWONG

equation:-

Introduced in 1949, the Redlich-kwong equation was a considerable improvement over

other equations of the time. It is still of interest primarily due to its relatively simple form.

While superior to the Vander-Waals equation of state, it performs poorly with respect to the

liquid phase and thus cannot be used for accurately calculating vapour liquid equilibria .

However, it can be used in conjunction with separate liquid-phase correlations for this

purpose.

The Redlich-Kwong equation is adequate for calculation of gas phase properties when the

ratio of the pressure to the critical pressure (reduced pressure) is less than about one-half of

the ratio of the temperature to the critical temperature (reduced temperature):



REDLICH-KWONG-SOAVE

equation:-

Where is the acentric factor for the species.

This formulation for is due to Graboski and Daubert. The original formulation from Soave is:

for hydrogen:

In 1972 Soave replaced the 1/√(T ) term of the Redlich-Kwong equation with a function

involving the temperature and the acentric factor. The function was devised to fit the vapour

pressure data of hydrocarbons and the equation does fairly well for these materials.

•For most fluids, the acentric factor =0 ,

•For more complex fluid acentric factor ˃0.



PENG-ROBINSON equation:-

The Peng –

Robinson equation was developed in 1976 in order to satisfy the following goals:- The parameters should be expressible in terms of the critical properties and the acentric

factor.

The model should provide reasonable accuracy near the critical point, particularly for

calculations of the compressibility factor and liquid density.

The equation should be applicable to all calculations of all fluid properties in natural gas

processes.

For the most part the Peng – Robinson equation exhibits performance similar to the Soave

equation, although it is generally superior in predicting the liquid densities of many materials,

especially non-polar ones.



BENEDICT-WEBB-RUBIN(BWR)

equation:-This equation was proposed in 1940.being a multi parameter model it is a complex, but more accurate

then the cubic equations of state discussed above. Despite its complexity it is widely used in petroleum

and natural gas industries for determining the thermodynamic properties of light hydrocarbons and their

mixtures.

Where,

p = pressure

ρ = the molar density and A˳ B˳ C ˳ D˳ are constant.



VIRIAL equation:-It is observed experimentally that the product of PV for gases along an isotherm is almost constant

and equals to RT , as pressure tends to zero or volume tends to infinity. This suggest the possibility of

expressing PV/ RT as power series of p or 1/v.

The ratio of PV/RT is termed as “compressibility factor ”, so virial equation denotes the

compressibility factor as power series of P and V.

Z, compressibility factor

Coefficients are known as second coefficient (B), third coefficient(C ) so on.

The coefficient also can be given physical interpretation .The virial coefficient account for molecular

interaction, the second virial coefficient take into account deviation from ideal behaviour which result

from molecular interaction between two molecules.

Although usually not the most convenient equation of state, the virial equation is important because it

can be derived directly from statistical mechanics. This equation is also called the Kamerlingh-Onnes

equation.

Here, a and b are Vander-Waals constant.

equation of states

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